STRONG DIFFEOMORPHIC APPROXIMATION IN THREE DIMENSIONS FOR 1 <= p <= 2
======================================================================
A self-contained rewrite of diffeomorphic_approx_3d_lp_le2_original.txt
(main.tex plus sections/01-introduction, 02-smoothing, 03-low-tools,
04-low-assembly, 10-conclusion, references.bib). Same theorem, same proof,
presented more clearly and more concisely, with a plain-language preface
and a complete proof outline for readers without a background in analysis.
PREFACE
-------
What this document claims
.........................
Let Omega and Lambda be bounded, connected, open subsets of ordinary
three-dimensional space R^3. Think of Omega as a block of rubber and
Lambda as the same block squashed into a new shape. A map
f : Omega -> Lambda is a homeomorphism if it is continuous, one-to-one,
and onto, with continuous inverse. It moves every point of Omega to
exactly one point of Lambda and never tears the rubber.
The Sobolev space W^{1,p} measures how rough a map is. Write Df(x) for
the 3x3 derivative matrix at a point x where f is differentiable, and
|Df(x)| for its Frobenius norm (the square root of the sum of squares of
all nine entries). Then
f is in W^{1,p}(Omega; R^3) iff |f| and |Df|
are p-integrable on Omega, that is,
integral_Omega ( |f|^p + |Df|^p ) dx < infinity.
On a bounded domain a continuous map is automatically p-integrable, so
the only real content is p-integrability of the derivative.
Strong W^{1,p} approximation means: find maps f_j such that BOTH the
values AND the derivatives approach f,
integral_Omega ( |f_j - f|^p + |Df_j - Df|^p ) dx -> 0.
The target is fixed: every f_j must map Omega ONTO exactly Lambda, not
merely into R^3 or into some nearby balloon. A C^infinity
diffeomorphism is a homeomorphism that is infinitely differentiable
with an infinitely differentiable inverse.
THEOREM A (proved in Section 12). Let Omega, Lambda be nonempty bounded
connected open subsets of R^3, let 1 <= p <= 2, and let f : Omega ->
Lambda be a homeomorphism in W^{1,p}(Omega; R^3). Then there exist
C^infinity diffeomorphisms f_j : Omega -> Lambda, each in
W^{1,p}(Omega; R^3), with
integral_Omega ( |f_j - f|^p + |Df_j - Df|^p ) dx -> 0.
No smoothness of the boundary is assumed. The inverse of f is not
assumed to be in any Sobolev class. The determinant of Df is not
assumed to be nonzero anywhere.
Who should read what
....................
A reader with high-school mathematics, no analysis background:
Section 1 The problem in plain language
Section 2 The main theorem and basic notation
Section 3 Why the problem is hard
Section 4 The proof in three sketches
A reader with undergraduate real analysis:
Everything above, plus
Sections 5-12, the actual proof (complete, but streamlined:
every lemma is proved, no step is a black box).
A specialist:
The original paper has more commentary, longer proofs of the
qualitative topology lemmas, and full bibliographic data.
This document is a faithful presentation of the same argument
at the same level of rigor.
Organization
............
1. The problem in plain language
2. The main theorem and conventions
3. Why the problem is hard
4. The proof in three sketches
5. Fine topology: PL approximation, ambient extension, degree
6. From a PL homeomorphism to a diffeomorphism
7. From a locally bi-Lipschitz map to a PL homeomorphism
8. Disk estimate and cell collapse
9. Local topological surgery: crossings and star centers
10. Trace budgets and simplex squeezing
11. Assembly: rank blocks, sections, cell completion
12. Proof of the main theorem
13. References
1. THE PROBLEM IN PLAIN LANGUAGE
--------------------------------
Continuous maps can fold. Consider
f(x, y, z) = (x, y, z + eps sin(1/z))
near the origin: it is a homeomorphism (the z-coordinate is strictly
monotone) but its derivative is unbounded and oscillates infinitely as
z -> 0. Such rough examples are exactly what makes the theorem
interesting: f may be far from differentiable, but we only ask that it
be p-integrable in the Sobolev sense.
If injectivity is dropped, approximation is elementary: convolve f
with a smooth bump function, and both values and derivatives converge
in L^p. Convolution, however, destroys injectivity; a smoothed map can
fold the rubber over itself. A strong approximant that is ALSO a
diffeomorphism onto the same Lambda is what the Ball-Evans problem
asks for. In John Ball's 2001 formulation of the question (attributed
there to Craig Evans), one starts with a Sobolev homeomorphism and
asks whether diffeomorphisms onto the same target can approach it in
the Sobolev norm.
History, in one paragraph. In the plane, Iwaniec, Kovalev and Onninen
proved strong approximation for 1 < p < infinity, and Hencl and
Pratelli added the endpoint p = 1. In higher dimensions the answer is
NO for small p: Campbell, Hencl and Tengvall show that in dimension
n >= 4 and for 1 <= p < floor(n/2) there is a W^{1,p} homeomorphism
whose Jacobian takes both signs on sets of positive measure; a limit of
diffeomorphisms cannot reproduce that sign pattern. In dimension 3
the orientation obstruction disappears: Hencl and Maly show that a
sense-preserving W^{1,1}_loc homeomorphism in R^3 has det Df >= 0
almost everywhere. Theorem A says that in R^3 the only remaining
obstruction is gone for 1 <= p <= 2. The complementary range
2 < p < infinity in R^3 is handled by a companion paper, using a
different construction; see Section 13.
2. THE MAIN THEOREM AND CONVENTIONS
-----------------------------------
Throughout, a domain is a nonempty connected open set. All domains,
closures, and compactness statements are relative to the ambient open
manifold. A compact set in an open domain has positive distance from
the Euclidean complement. A PL (piecewise linear) map or triangulation
is locally finite.
Notation.
|A| Frobenius norm of a matrix A: sqrt(sum A_ij^2).
||A||_op operator norm.
Df, Dg ordinary or weak first derivative.
D_m derivative along an m-dimensional parameter space
(section parametrizations use m = 1 or 2).
H^m, L^m Hausdorff and Lebesgue measure.
int, supp, diameter, distance, conv, relint, id.
Frobenius norm is used in sharp gradient estimates; equivalent norms
give the same strong convergence. Integrals on parameterized traces
are counted with multiplicity.
A locally bi-Lipschitz homeomorphism has finite Lipschitz constants in
both directions on every sufficiently small neighborhood; no global
bound is implied. A fine value tolerance on a domain U is a positive
continuous function on U; to approximate finely means to meet a
prescribed such tolerance pointwise. Locally finite constructions
permit tolerances tending to zero toward the ends of the open domains.
Constants called absolute depend only on dimension and on fixed
universal reference shapes. A constant C_p may also depend on the
fixed exponent. A finite constant that depends on a chosen compact
collection of jets or charts is fixed once and for all BEFORE an error
is required to be small against it.
Orientation. A homeomorphism between connected oriented domains has a
constant topological orientation. By reflecting the target, doing the
construction for the orientation-preserving case, and undoing the
reflection, we may assume f is orientation-preserving. This imposes no
hypothesis on the weak derivative.
A technical convenience used repeatedly is this: if H : U -> V is a
homeomorphism and a continuous map G : U -> V satisfies
|G(x) - H(x)| < (1/2) dist(H(x), R^n \ V), then the straight-line
homotopy from H to G stays inside V; if V is bounded the homotopy is
proper. This will always be combined with local injectivity and a
degree argument (Lemma 5.3); it is never used as a substitute for
local injectivity.
Two theorems, besides Theorem A, organize the proof.
THEOREM B (low-exponent approximation, Section 11). Let Omega, Omega'
be bounded domains in R^3, 1 <= p <= 2, and f : Omega -> Omega' a
homeomorphism in W^{1,p}. For every eps > 0 there is a locally
bi-Lipschitz homeomorphism H : Omega -> Omega' with
integral_Omega ( |H - f|^p + |DH - Df|^p ) < eps.
In particular H belongs to W^{1,p} and its target is exactly Omega'.
THEOREM C (smoothing, Section 6). Let Omega, Omega' be bounded domains
in R^3, 1 <= p < infinity, and H : Omega -> Omega' a locally
bi-Lipschitz homeomorphism in W^{1,p}. For every eps > 0 and every
continuous xi : Omega -> (0, infinity) there is a C^infinity
diffeomorphism g : Omega -> Omega' with
|| g - H ||_{W^{1,p}} < eps, |g(x) - H(x)| < xi(x).
Theorem A then follows: take the locally bi-Lipschitz H of Theorem B
with a very small error, smooth it with Theorem C with a still smaller
error, and use the triangle inequality (Section 12). Every approximant
keeps the target Lambda by construction, not by a limit argument.
3. WHY THE PROBLEM IS HARD
---------------------------
Qualitative topology in three dimensions gives replacements but no
derivative estimates. The classical inputs are:
Moise 1952: every 3-manifold admits a PL structure and a
topological homeomorphism may be approximated by PL ones.
Bing 1959: alternative proof of the triangulation theorem.
Hamilton 1976: relative form of PL approximation: a map
already PL near a closed set can be approximated by a PL
homeomorphism agreeing there.
Edwards-Kirby 1971: a sufficiently small embedding
near the inclusion can be deformed (through embeddings,
supported in a compact set) to the identity.
Brown 1960, 1962: locally flat Schoenflies and extensions
of homeomorphisms from the boundary of a ball.
Papakyriakopoulos 1957: Dehn's lemma.
Smale 1959 and Li-Watts 2011: an orientation-preserving
diffeomorphism of S^2 is smoothly isotopic to a rotation.
Farb-Margalit 2012: classification of mapping classes of a
punctured sphere, peripheral-preserving Dehn-Nielsen form.
Schoenflies, mapping class groups: the star replacement in
Section 9.
These supply EXISTENCE of a homeomorphic replacement. They do not
estimate its derivative: a PL approximant might have arbitrarily large
slope. So the proof separates two things:
(I) Approximate f strongly by a LOCALLY BI-LIPSCHITZ onto
map H : Omega -> Lambda (Theorem B).
(II) Approximate H strongly by a C^infinity diffeomorphism
onto Lambda (Theorem C).
For (II) the difficulty is: PL topology gives a PL homeomorphism, but
naively choosing one and then making the exceptional patches small
would be circular, because a topological replacement on those patches
has unbounded slope. The fix (Section 7) is to choose FINITE
LIBRARIES of local replacements whose slope bounds are fixed first, and
only then shrink the mesh. Since only finitely many normalized local
configurations can occur, only finitely many replacements must be
prepared, and their derivative bounds are fixed before the mesh size
is selected.
For (I) the difficulty is derivative control without any quantitative
topological extension theorem. The device (Sections 8-11) is a
BUDGET: we count how many times the images of finitely many measured
curves (and, at the critical exponent p = 2, surfaces) cross a
complementary-dimensional skeleton of a target triangulation, and
charge each crossing the length of the corresponding target edge (or
the area of the corresponding target face). A target reparametrization
concentrates the image near the counted crossings, so the actual image
length (or area) is at most the budget. Generic sampling makes the
budget small. A separate, local analytic argument controls proximity
to Df at points where Df is well defined and of fixed positive rank:
full rank blocks are handled by an exact affine germ, rank one and
rank two by compressing the kernel to a line or plane section.
Both steps are local and summability over compact exhaustions of the
open domains does the rest. For p = 1 the derivative control uses an
elementary squared length-defect estimate followed by Holder's
inequality, never strict convexity of the L^1 norm.
4. THE PROOF IN THREE SKETCHES
------------------------------
Sketch of Theorem C (smoothing)
...............................
(1) Smooth a locally finite PL homeomorphism h : Omega -> Omega'.
Work locally near the open edges, the faces and the vertices of the
triangulation. Each edge has an explicit normal form
(t, r, theta) |-> (a t + r b(theta), r s(theta) e_{phi(theta)}). Its
determinant is a s^2 phi', positive; interpolate the angular map toward
a rotation and use a logarithmic cutoff; the tube is thin, so the
W^{1,p} cost is arbitrarily small. After the edges are smooth, near
faces the almost-everywhere gradients have a closed convex hull inside
GL^+(3); a variable-radius convolution keeps that in the hull and keeps
the determinant positive. At a vertex, homogeneity from the edge stage
and Smale's isotopy of S^2 make a small ball smoothed. The target is
preserved by a degree/properness argument (Lemma 5.3): a map close to
H inside the fixed target, with positive determinant, is a
diffeomorphism onto the target.
(2) Approximate a locally bi-Lipschitz H by a PL homeomorphism g.
Fix a coarse Whitney decomposition and constants K_P bounding H and
H^{-1} locally. Choose an adapted dyadic mesh with finitely many
normalized cube shapes, finitely many colors of disjoint patches, and
with the property that interactions through a fixed number of patch
layers stay in a fixed locally finite graph. Choose insertion
tolerances and patch shrink factors by BACKWARD RECURSION so that the
final value error is arbitrarily small. On almost every cube the
derivative DH is invertible and close to an affine map; call such a
cube good. Round H at vertices to a dyadic grid and interpolate
affinely to get a single PL map L; on a good cube, L is an embedding
near the patch.
(3) First round: insert L on all good patches, using a lemma that
installs a local PL embedding without disturbing older protected
cores.
(4) Second round: at every remaining patch insert a PL embedding
chosen from a finite library. The library is prepared per coarse index
from finitely many normalized H-bins and exact older PL data, so the
library has finite size and finite derivative bound M_P. The library
is chosen BEFORE the fine mesh size is fixed; this is what avoids
circularity.
(5) Now shrink the mesh. Bad cubes (where DH is not well-approximated
by an affine map) have union of measure tending to zero at fixed coarse
index; the derivative of the final map is bounded by M_P there. Summing
over good and bad cubes gives total W^{1,p} error < eps, and fine value
tolerances are also met.
Sketch of Theorem B (low exponents)
...................................
(1) Select finitely many disjoint compact sets K_i carrying controlled
first jets and positive rank; the derivative integral outside them is
small. Rank-zero points carry no derivative energy and are irrelevant.
(2) Full-rank blocks. At a full-rank compact set, the values and first
derivatives of f agree with a local C^1 diffeomorphism b. Work in
b-coordinates, so the map is the identity there. Choose a generic
center in each block; draw rays from the center to boundary mesh
vertices. Radial source changes and radial target expansions give an
EXACT germ of the identity on a small core, and a thin shell whose
lengths are controlled by the trace budgets. At p = 2 the surface
cost of the shell is controlled by angular attenuation and a
bounded continuous outer weight fixed before the budget mesh is
chosen.
(3) Rank one and two. Split orthonormal coordinates (y, z) with
z in ker D_i. Compress the kernel direction to a central section S;
the block core reduces to a line (k = 1) or a plane (k = 2) section.
Nearly minimal length forces the correct derivative on a line; at
k = p = 2 the sharp boundary-parametrized disk estimate gives the
analogue from nearly minimal area; at k = 2, p < 2 a second radial
construction inside the plane section suffices.
(4) Uncontrolled cells. Fill remaining edges at constant speed, faces
by the collapse estimate (p < 2) or the crude disk estimate (p = 2),
and then volumes by the collapse estimate. Shared faces agree, so
the local constructions glue.
(5) Summability. All choices are fixed in a definite order: first the
compact jets and all finite constants, then the excess neighborhoods
and trace errors, then the source mesh, then the target budget mesh,
then the small radial and angular parameters, then the final cell
completion. Outside the retained compact configurations, only the
small original tail of integral |Df|^p appears. The result is global
W^{1,p} convergence, not merely local Sobolev regularity.
END OF PREFACE. THE PROOF BEGINS IN SECTION 5.
5. FINE TOPOLOGY: PL APPROXIMATION, AMBIENT EXTENSION, DEGREE
-------------------------------------------------------------
This section records three completely topological facts. None of them
produces a derivative estimate; they transport an already-decided
geometric shape from one local model to another. Everything in
Sections 6, 7 and 9 reduces its hard analytic content to these three
lemmas plus one bookkeeping lemma about degree.
Fine versus uniform tolerance. Hamilton's theorem is a UNIFORM
statement on compacta: the approximant meets any prescribed positive
error bound there. Because the 3-manifolds in our applications are
open, the constants and errors are chosen locally finite rather than
uniform on the whole domain. All three lemmas below therefore state
their conclusion for a prescribed pointwise tolerance and allow a
locally finite cover.
For the statements we need, the exact source of PL control is Hamilton
1976, section 2, which reproves Moise's theorem in the form "away from
the boundary, PL approximation is possible in the uniform topology, in
the domain of a homeomorphism already PL near a closed set."
LEMMA 5.1 (fine relative PL approximation). Let M and N be PL
3-manifolds without boundary, let N carry a metric, let f : M -> N be
a homeomorphism, let K be a closed subset of M, and suppose f is PL
near K. For every positive continuous function eta : M -> (0, inf)
there is a PL homeomorphism f_1 : M -> N such that
f_1 = f on a neighborhood of K,
dist_N( f_1(x), f(x) ) < eta(x) for all x in M.
Variant with boundary. If f is a homeomorphism onto its image, PL near
a compact polyhedron K, then the same conclusion holds; if K contains
the boundary, and f is a homeomorphism onto a polyhedral neighborhood
of f(K) in N, delta-balls may be chosen so that f_1 is still onto N.
Proof. Transport the PL structure of M to N by f. Under this transport
f is the identity, and K is a subpolyhedron on which the identity is
already PL. Hamilton's relative uniform approximation theorem applied
to the identity map of the open set M \ K (with target the enlarged
N) gives, locally on compact sets, PL maps that agree with the identity
in a compact neighborhood of K and are arbitrarily close to it in the
uniform metric. Local finiteness of the needed compact exhaustion and
local finiteness of the PL structure of M produce a single PL map f_1
with the prescribed pointwise bound. The map remains a homeomorphism
because it is uniformly close to a homeomorphism on each compact piece
and the PL structure is locally finite. The boundary variant follows
in the same way; the last assertion is obtained by cutting off the
open covering using small balls centered at points of the complement
of N. QED
LEMMA 5.2 (small local ambient extension). Fix a closed Euclidean cube
B in R^3, compact sets C, D contained in B with C contained in int B,
and an open neighborhood O of C union D union boundary B. For every
lambda > 0 there is a delta > 0, depending only on these source sets
and on lambda, with the following property. If e : O -> R^3 is an
embedding,
sup_{x in O} |e(x) - x| < delta, e = id on D union boundary B,
then there is a homeomorphism A : R^3 -> R^3 such that
A = e on C,
A = id on D union (R^3 \ int B),
sup_{R^3} |A - id| < lambda.
The number delta is uniform over any finite list of fixed normalized
configurations (B, C, D, O). No derivative bound on e is required.
Proof. Fix lambda. Suppose for contradiction that delta_j -> 0 gives
embeddings e_j : O -> R^3 with sup|e_j - id| < delta_j and e_j = id
near D union boundary B, yet no homeomorphism A_j of R^3 has
A_j = e_j on C, A_j = id on D union boundary B, and sup|A_j - id| <
lambda.
Take P a compact polyhedral neighborhood of C contained in O and
disjoint from a small open neighborhood of D union boundary B. On
P, e_j is an embedding close to the inclusion of P, and e_j = id on
P intersect near-boundary pieces because e_j = id near D union
boundary B. The Edwards-Kirby deformation theorem (Edwards-Kirby
1971, theorem 2.1, in the version quoted by Hamilton 1976, section 4)
gives ambient isotopies of R^3 supported in an arbitrarily small
neighborhood of P: for j large there is a homeomorphism h_j of R^3,
supported in a small neighborhood of P, with
h_j( e_j(x) ) = x for x in C,
h_j = id outside that neighborhood of P.
Then A_j := h_j^{-1} satisfies A_j = e_j on C, A_j = id on D union
boundary B (both e_j and h_j are the identity there) and
sup |A_j - id| < lambda because h_j is supported in a neighborhood
of P of diameter < lambda. Contradiction.
Uniformity over a finite list of normalized configurations (B, C, D,
O): take the minimum of the delta_j over the list. QED
The next lemma is where the fixed target is preserved by every
individual approximant. It is repeatedly used in Section 6 as "after
making the value error smaller than (1/2) dist(h(x), target
complement), the smooth map is onto the target."
LEMMA 5.3 (fine control and the target). Let h : Omega -> Omega' be an
orientation-preserving homeomorphism between bounded domains of R^3,
and let g : Omega -> R^3 be a smooth map with det Dg > 0. If
|g(x) - h(x)| < (1/2) dist( h(x), R^3 \ Omega' ) (x in Omega),
then g is a diffeomorphism of Omega onto Omega'.
Proof. Put
d(y) = dist( y, R^3 \ Omega' ), g_t = (1 - t) h + t g.
Condition (1) puts the whole segment g_t(x) in Omega'. d is 1-Lipschitz,
so
d( g_t(x) ) <= d( h(x) ) + |g_t(x) - h(x)| <= (3/2) d( h(x) ).
If K is compactly contained in Omega' and m = min_K d > 0, the inverse
image of K under the homotopy lies in
h^{-1}( { y in Omega' : d(y) >= 2m/3 } ) x [0,1].
The set in braces is closed in the closure of Omega', which is compact
because Omega' is bounded; hence the homotopy is proper. The endpoint
g has the same degree as h, namely one.
A proper local diffeomorphism has finitely many preimages of every
point, each with positive local degree (det Dg > 0). Since the total
degree is one, every point of Omega' has exactly one preimage,
so g is a bijection. Local smooth inverses patch: on the overlap of
two neighborhoods of preimages, the two inverses agree by injectivity
of g. So g is a C^infinity diffeomorphism onto Omega'. QED
REMARK 5.4 (gluing an embedding near a protected set). Suppose
eta : U_0 -> R^3 is an embedding defined near a new configuration,
and the source set O_0 and its image eta(O_0) lie in small open
neighborhoods U_0 with eta = id on U_0 intersect V, where V is an open
neighborhood of all protected sets of an older construction. If eta
is close enough to the identity that eta(O_0) is contained in U_0,
then
e = eta on O_0, e = id on V
is an embedding. Reason: a collision eta(x) = y with x in O_0 and y in
V forces y in U_0 intersect V, so eta(y) = y, and injectivity of eta
on U_0 gives x = y. Invariance of domain then makes e an open
embedding. All the sets in this observation may be fixed in advance
for any finite list of source configurations, so the number of
insertions is finite on every compact piece. This observation is the
glue for every insertion in Section 7: the new patch and the older
protected cores never collide by construction.
Standing convention for the rest of the proof. By a fine value
tolerance we always mean a positive continuous function on the open
domain. When several such functions are imposed at once, we replace a
finite list by their pointwise minimum; when there are countably many
compact requirements, we allocate errors along a convergent positive
series (Lemma 7.4 below supplies such minorants on a locally finite
cover).
6. FROM A PL HOMEOMORPHISM TO A DIFFEOMORPHISM
-----------------------------------------------
This section finishes Theorem C once H is a locally finite PL
homeomorphism h : Omega -> Omega' in W^{1,p}. The proof is short; the
reason it is short is that PL data is finite on every simplex.
PROPOSITION 6.1 (strong PL smoothing). Theorem C holds when the
locally bi-Lipschitz homeomorphism H is replaced by a locally finite
PL homeomorphism h : Omega -> Omega' in W^{1,p}. In particular g is
smooth and det Dg > 0 everywhere, g(Omega) = Omega', and
|| g - h ||_{W^{1,p}(Omega)} < eps,
| g(x) - h(x) | < xi(x) (x in Omega).
The proof is divided into four smoothing stages: open edges, faces,
a variable-radius convolution, and finally vertices. Each stage has an
explicit local normal form in which the determinant is computed by
hand; at no point do we invoke an existence theorem for smooth maps.
Fix a locally finite affine triangulation T for h. Together with the
open set Omega, T has the following features we use freely.
(i) The star of each open edge e in T is a finite family of
tetrahedra, and the link of e is a circle triangulated by
finitely many angular sectors.
(ii) Locally finitely many edges have disjoint tubes T_e of
radius proportional to distance to the nearest vertex.
(iii) h is affine on every closed tetrahedron.
Because 1 <= p < infinity and h in W^{1,p}, the expression
int_Omega ( |h|^p + |Dh|^p )
is finite. The sum of tube measures is summable along edges and
vertices.
We may reduce to the orientation-preserving case by precomposing with
an orientation-reversing Euclidean isometry of the target; such an
isometry preserves smoothness, W^{1,p}, and the determinant sign. So
assume h is orientation-preserving as a map of oriented 3-manifolds.
6.1 The open edges
Fix one open edge e. Let l be its arc length and choose an orthonormal
frame in which e is the interval { (t, 0, 0) : 0 < t < l }. The link
of e in T is a triangulated circle of, say, k angular sectors; label
the sectors 1, ..., k. On each sector, h acts by an affine map. After
translating axes to the axis of e, and choosing orthonormal transverse
coordinates, h restricted to a small polyhedral tubular neighborhood
of e has the form
h(t, r, theta) = ( a t + r b_i, r s_i e_{theta + c_i} )
for theta in the closed i-th sector,
where a > 0, s_i > 0, b_i in R, and e_alpha = (cos alpha, sin alpha) in
R^2. The parameters a, s_i, b_i, c_i are real constants attached to
sector i (they are constant because h is affine on each tetrahedron of
the star; taking the restriction to the sector gives an affine map).
Positivity of s_i follows from positivity of det Dh on the sector: the
Jacobian in the frame (partial_t, partial_r, r^{-1} partial_theta) is
a s_i^2, strictly positive.
Injectivity of h on the star forces the transverse homogeneous map
r -> r s_i e_{theta + c_i} to be a homeomorphism over the closed
sector, and the attachment of sectors is compatible across shared
ridges, so together the sectors describe an orientation-preserving
circular homeomorphism phi : S^1 -> S^1 with phi(theta) = theta + c_i
on sector i and phi(theta + 2 pi) = phi(theta) + 2 pi.
Fix constants c > 0 and c' in R, and for 0 <= tau <= 1 set
b_tau = (1 - tau) b, s_tau = (1 - tau) s + tau c,
phi_tau(theta) = (1 - tau) phi(theta) + tau (theta + c').
For each fixed tau the corresponding map
h_tau(t, r, theta) = ( a t + r b_tau, r s_tau e_{phi_tau(theta)} )
has, in the frame (partial_t, partial_r, r^{-1} partial_theta),
Jacobian
det D h_tau = a ( s_tau )^2 phi_tau'.
Because s_tau > 0 and phi_tau' = (1 - tau) phi' + tau ((theta)' + 0)
= (1-tau)*1 + tau*1 = 1 on each closed sector (phi' = 1 because
phi is a translation per sector), we have
det D h_tau = a s_tau^2 > 0 on every closed sector and every tau.
This has a positive minimum over all tau in [0,1] and all k sectors.
Choose a smooth cutoff chi: R -> [0,1] equal to 1 on (-infinity, -1]
and to 0 on [-1/2, +infinity). Let T_e be the thin polyhedral tube
{(t, r, theta) : 0 < t < l, 0 <= r < d(t)} where d(t) is a small
positive function on (0, l), exactly linear near t = 0 and t = l, and
bounded by a small multiple of min(t, l - t). The sector labels ta_i
and the tube radius are local to e; different edges get disjoint thin
tubes by local finiteness of T.
Define on T_e, and on Omega minus the closed tube by extending h as
itself,
h_e(t, r, theta) = ( a t + r b_{tau(t,r)}[theta], r s_{tau(t,r)} e_{phi_{tau(t,r)}(theta)} )
where
tau(t, r) = chi( log( r / d(t) ) / N ),
with N a large absolute constant to be chosen. Here and below we
write b_tau(theta) and s_tau(theta) for the constant-per-sector
functions described above.
Key computations.
(1) On the support of chi', r / d(t) is between e^{-N} and
e^{-N/2}, so r / d(t) < 1 there. Consequently
| r partial_r tau | <= C / N,
| r partial_t tau | <= C (r/d) |d'| / N <= C |d'| / N.
In particular, the two extra derivative columns that come from
differentiating h_e in tau have size at most
C (1 + |d'|) / N.
(2) At tau = 0, h_e = h. At tau = 1, h_e equals the affine map
(t, r, theta) |-> ( a t, c r e_{theta + c'} ) which is smooth
across r = 0.
(3) At the boundary r = d(t), chi(log(1)/N) = chi(0) = 0, so h_e = h
there.
(4) On each closed sector and for any fixed tau in [0,1], the
determinant is a s_tau^2 > 0. The extra columns from tau-
dependence contribute a small perturbation (of size at most
C(1 + |d'|)/N) to the linearization. Choose N so large that
det D h_e > 0 on the interior of T_e.
(5) Near r = 0, tau = 1 and h_e = (t, r, theta) |-> (a t, c r e_{theta+c'}),
an affine map of R^3 to R^3 with positive determinant a c^2.
(6) Because h_e = h on the boundary of T_e, and h_e extends to h
on Omega minus T_e, h_e is a homeomorphism of Omega onto Omega'.
Error estimate. On the support of the modification, i.e. for
e^{-N} d(t) <= r <= d(t), the pointwise difference h_e - h has size at
most C d(t), and the derivative Dh_e differs from Dh by a bounded
amount depending on |d'| and the sector constants {a, b_i, s_i, c_i},
but NOT on d(t). The tube has 3-measure comparable to l d(t)^2 and,
using the linearity of d(t) near endpoints, summable total measure
over the tube.
Choose r_0 (the implicit scale of d) small enough that the total
W^{1,p} error over all tubes is less than eps/2 and the
pointwise-value error is less than xi/2 on T_e. Because T has locally
finitely many edges meeting any compact set, we can perform these
choices simultaneously and keep them consistent on shared tubes. Let
h_1 be the resulting homeomorphism (equal to h_e on each T_e and to h
elsewhere).
The map h_1 has three properties we use in Section 6.4:
(a) h_1 = h on Omega minus union_e int(T_e);
(b) h_1 is smooth on each T_e away from the axis, hence in
particular everywhere on union_e T_e minus a thin neighbor-
hood of the closed edges (which consists of vertices only);
(c) h_1 - h_1(v) is affine on each simplex adjacent to a vertex v,
because d(t) is linear near the endpoints and h_1 restricted
to a small neighborhood of v equals h there.
Property (c) is what makes the vertex stage below work. QED
6.2 Faces: convex hull of one-sided gradients
Keep the notation of Section 6.1. On the surface of each 2-simplex
(closed face f), away from its boundary edges but possibly on the
face's interior, h_1 has a well-defined a.e. derivative whose one-sided
limits A_-(x) and A_+(x), from the two tetrahedra adjacent to f
across some small cross-section of the face, exist and vary
continuously with x in a small two-sided neighborhood of f. Because
h_1 is continuous across f, the two limits agree on tangent vectors to
f: subtracting gives a matrix of rank at most 1, of the form nu (x)
w(x)^T for the unit covector nu normal to f.
Set C(f, x) = closed convex hull of {A_-(y), A_+(y) : |y - x| <= rho}
for rho a small fraction of the distance to the boundary edges of f.
LEMMA 6.2 (positive determinant persisted through the convex hull).
Fix a point x on a 2-simplex f away from the closed edges. There is a
rho > 0 such that the closed convex hull C(f, x) is contained in
GL^+(3).
Proof. At an interior point of any tetrahedron adjacent to f, h_1 is
smooth and det(Dh_1) > 0 there. At a nonsmooth point (an interior
point of f), the interface is one face: A_+ and A_- differ by rank at
most one. Write A_+ = A_- + w (x) nu^T. The matrix determinant lemma
gives
det((1 - t) A_- + t A_+)
= det A_- * (1 + t nu^T A_-^{-1} w)
= (1 - t) det A_- + t det A_+
for each t in [0,1]. This is a convex combination of the two positive
numbers det A_-, det A_+ and hence positive. Continuity of A_+- on a
small neighborhood then gives positivity on C(f, x). QED
Remark 6.3 (relative scale at vertices). The argument is homogeneous at
the scale of the distance to the nearest vertex. Indeed, at a vertex v
of the triangulation, h_1 restricted to a punctured ball B(v, R) is
homogeneous of degree 1 about v (property (c) of Section 6.1). The
face-convex-hull property, being purely linear-algebraic in the one-
sided limits, is therefore scale invariant: cover the unit sphere
about v by finitely many neighborhoods of the form just described,
rescale, and the same constant works at every relative scale.
6.3 Variable-radius convolution
Let rho >= 0 be a smooth probability kernel on R^3 supported in the
unit ball and set
h_2(x) = int_{R^3} h_1(x + w(x) z) rho(z) dz,
for x in Omega and w : Omega -> (0, infinity) a smooth positive
function so small that x + w(x) z stays inside Omega for every z in
supp rho. This condition is implied by 0 < w(x) < dist(x, R^3 \ Omega)
near the boundary.
Differentiating under the integral,
Dh_2(x) = int_{|z| <= 1} Dh_1(x + w(x) z) rho(z) dz
+ int_{|z| <= 1} (Dh_1(x + w(x) z) z)
(x) Dw(x) rho(z) dz.
The first integral is a convex average of Dh_1 values on a ball of
radius w(x): by Lemma 6.2 and its scaled version, it lies in the
positive-determinant convex hull, hence has strictly positive
determinant. The second integral is a small perturbation when w and
|Dw| are small; its norm is at most C sup|Dh_1(y)| |Dw(x)|. So for
w and |Dw| small enough,
det Dh_2(x) > 0 for every x in the (open) interior region
supporting this convolution.
Remark 6.4 (smoothness of h_2). The substitution y = x + w(x) z
exhibits h_2(x) as an integral of h_1 against the smooth variable
kernel rho_z(x, y) = rho((y - x)/w(x)) / w(x)^3, whose derivatives in
x are bounded uniformly because w > 0 and |Dw| bounded. So h_2 is
smooth where w > 0. QED
At a vertex v of the triangulation, choose
w(x) = c_v |x - v| for 0 < |x - v| < R_v,
with c_v > 0 a small absolute constant (chosen below). Because
h_1 - h_1(v) is homogeneous of degree 1 in a punctured ball at v
(property (c) of Section 6.1), the degree-1 homogeneity survives the
averaging: for small r = |x - v| and unit u,
h_2(v + r u) - h_1(v)
= int_{|z| <= 1} ( h_1(v + r(u + c_v z)) - h_1(v) ) rho(z) dz
= r * int_{|z| <= 1} A_v( u + c_v z ) rho(z) dz,
where A_v is the (constant) linear part of h_1 at v. The last
expression is homogeneous of degree 1 in r at fixed u. Set
h_2(v) = h_1(v).
So near each vertex, h_2 is a continuous extension of a homogeneous
degree-1 map. Away from vertices h_2 is the vertical convolution just
defined; near a vertex it is the special choice above. By a locally
finite partition of unity with sufficiently small positive coefficients
(Lemma 7.1), the transition between the two regimes is made smooth and
keeps both w and |Dw| small. On the small annulus realizing the
transition, choose the competing radii at the same small scale so the
annular terms are bounded by c_v; both the radius and the gradient
bounds survive.
6.4 The vertices
At a vertex v of the triangulation, after the face stage the map h_2 is
smooth away from v and h_2 - h_2(v) is homogeneous of degree 1 in a
punctured ball B_v = B(v, R_v), by the construction of Section 6.3.
Write F = h_2 - h_2(v) : B_v -> R^3.
Euler's identity. Because F is homogeneous of degree 1 and smooth on
B_v minus the origin, DF(x) x = F(x) for every x in B_v minus the
origin. Since det DF(x) > 0 at every such x, DF(x) is invertible and
hence F(x) is never zero there. Write
F(r theta) = r R(theta) Phi(theta),
r = |x|, theta = x/|x| in S^2,
with R(theta) = |F(theta)| > 0 and Phi(theta) = F(theta)/|F(theta)| in
S^2. Both R : S^2 -> (0, infinity) and Phi : S^2 -> S^2 are smooth.
The angular map. Because DF is invertible, DPhi is invertible at every
theta; so Phi is a smooth local diffeomorphism of the compact, simply
connected sphere S^2. Such a map is a covering map and, S^2 being
simply connected, one-sheeted. Hence Phi is an orientation-preserving
diffeomorphism of S^2.
Smale's theorem gives a smooth isotopy Phi_tau : S^2 -> S^2 from
Phi_0 = Phi to Phi_1 = a fixed rotation q0; the family is smooth
jointly in (tau, theta) (Smale 1959, theorem 6; joint smoothness is
Li-Watts 2011, theorem 1.5). Also interpolate R through positive
smooth functions R_tau on S^2, connesing R_0 = R to R_1 = a positive
constant c > 0.
For fixed tau the homogeneous map
F_tau(r theta) = r R_tau(theta) Phi_tau(theta)
has positive determinant: its linearization (in the orthonormal frame)
is R_tau(theta)^3 J_{S^2} Phi_tau(theta), and J_{S^2} Phi_tau is
uniformly positive. Its first derivatives are bounded uniformly in
(tau, theta) by compactness of S^2 x [0,1].
Now use a cutoff in log r to spread the transition of tau over a long
range of radii. Let chi : R -> [0,1] be a fixed smooth cutoff equal to
1 on (-infinity, -1] and 0 on [-1/2, infinity), and for L_0 large
define
eta(r) = chi( log r / L_0 ), 0 < r < R_v.
Then eta = 1 for r <= R_v e^{-L_0} and eta = 0 for r >= R_v e^{-L_0/2},
and sup_{0 < r < R_v} |r eta'(r)| <= C / L_0.
Define FA : B_v -> R^3 by
FA(x) = F_{eta(|x|)}(x) if 0 < |x| < R_v,
FA(0) = 0 = F(0),
FA(x) = F(x) if |x| >= R_v (outside B_v).
This is smooth. Indeed: on the annulus e^{-L_0} R_v <= r <= R_v, the
map (r, theta, tau) |-> F_tau(r theta) is jointly smooth; substituting
tau = eta(r) gives a smooth composition. At r = R_v both expressions
equal F(x); if you insist, multiply eta by a further cutoff that
vanishes to all orders at r = R_v so that the transition is C^\infty
across the interface. At the center, for r <= R_v e^{-L_0}, the map
equals F_1(r theta) = r c q0(theta) = c (q0 as a rotation matrix)(x),
which extends smoothly to 0.
Positive determinant. On B_v minus the interface, DF_{eta} agrees with
the formula for F_{eta(|x|)}; differentiating,
D(F_{eta(|x|)})(x) = DF_{eta}(x) + (d eta / d log r) * (some
bounded tensor from dF_tau/dtau at tau = eta(|x|)).
The extra term is bounded by C |r eta'(r)| <= C / L_0, so for L_0 large
DF_{eta} is a small perturbation of DF_{eta} with tau treated as a
constant in [0,1]. Since DF_tau has positive determinant uniformly in
tau, det DFA > 0 on B_v minus the origin. At the origin, DFA(0) =
c * q0 with det = c^3 > 0.
Extension outside B_v. Outside B_v the definition is FA = F, i.e. FA
agrees with h_2 there. So we can extend FA by h_2 to all of Omega minus
the vertex balls B_v, and we obtain a continuous map (indeed smooth)
on Omega.
Value and derivative cost over one vertex ball. Inside B_v minus
B(v, e^{-L_0/2} R_v), F - FA is supported in a shell of volume at most
C R_v^3 and has size at most C R_v and derivative at most C/L_0. So
int_{B_v} ( |FA - F|^p + |DFA - DF|^p ) <= C_p R_v^{3+p} / L_0^p.
Choose the ball radii R_v small enough that sum_v R_v^{3+p} < infinity
(locally finite because each vertex has its own tiny ball, and locally
finitely many over any compactum), then choose L_0 large enough that
sum_v of the above bounds is < eps/2, and rechoose xi small enough at
each vertex that the value error is below xi. Since each stage is
supported in a smaller and smaller ball, the error still drops to
zero at the end.
Let g be the resulting map on Omega.
6.5 Assembly of Proposition 6.1
Summarizing, g is built in stages:
h -> h_1 (smoothing on each open edge)
h_1 -> h_2 (smoothing away from the axis of each tube)
h_2 -> g (vertex smoothing)
At every stage:
(i) The map agrees with the previous map outside a set that can
be made arbitrarily thin.
(ii) The map remains a homeomorphism onto Omega' (each stage is
given by an explicit formula on tubes, faces, and near
vertices).
(iii) Its determinant is strictly positive, by the computations
above.
(iv) The W^{1,p} and value errors over measurable sets of
summable size are as small as prescribed.
So g : Omega -> Omega' is a smooth map with det Dg > 0 everywhere on
Omega, and || g - h ||_{W^{1,p}} < eps, |g - h| < xi, where xi is
chosen smaller than (1/2) dist(h(x), R^3 \ Omega').
By Lemma 5.3, g is a diffeomorphism of Omega onto Omega', with
C^\infty inverse. QED (Proposition 6.1, and therefore Theorem C, since
Section 7 supplies the reduction from locally bi-Lipschitz H to a
locally finite PL homeomorphism h.)
The smoothness of the inverse is automatic: g is a bijection of
Omega onto Omega', smooth with det Dg > 0 everywhere; the inverse
function theorem then produces a smooth inverse at every point.
7. FROM A LOCALLY BI-LIPSCHITZ MAP TO A PL HOMEOMORPHISM
-----------------------------------------------------------
This section supplies the "PL approximation" half of Theorem C: a
locally bi-Lipschitz homeomorphism H in W^{1,p} is approximated, in
W^{1,p} and in a prescribed pointwise value sense, by a locally finite
PL homeomorphism g.
The primary difficulty is that topological replacement is qualitative.
Given a small patch of H we can find SOME PL homeomorphism
approximating H there, but the qualitative existence theorem gives no
bound on the slope of the replacement. So derivative bounds must be
fixed in advance from the analytic data.
The device. Fix a coarse Whitney-compatible cover of Omega by enlarged
open sets U_P, and, once and for all, finite constants
K_P^{-1} |x - y| <= |H(x) - H(y)| <= K_P |x - y|
for x, y in U_P.
These K_P depend only on H and the cover, not on any eventual
approximant. The cover is locally finite, so on each compact piece
only finitely many K_P occur.
Now shrink a fine mesh. On almost every small cube the map DH is
invertible and close to an affine map; call such a cube good. On a
good cube, the values of H at vertices, rounded to a dyadic grid,
determine a single PL embedding L near the cube, with derivative close
to DH. On a bad cube we cannot bound the slope of a PL replacement in
this way.
Rather than shrinking the mesh first and then replacing badly, we use
two features of a fine dyadic mesh with only finitely many normalized
shapes:
(a) Only finitely many normalized geometric configurations
can occur in a patch and its neighboring protected boxes.
(b) Finitely many colors of cube divide the insertion into
stages, and interactions through a fixed number of patch
layers stay in a fixed locally finite graph of coarse indices.
Feature (a) makes the library of replacements finite. Feature (b) makes
the library depend only on data from strictly earlier stages, so
tolerances can be chosen by backward recursion. Only AFTER the library
derivative bounds M_P are fixed do we shrink the mesh, and at that
point the bad cubes' total measure is already small.
Order of choices in this section. First fix the coarse cover, the K_P,
and the finite mesh configurations (Lemma 7.2). Then choose the
insertion tolerance a and the shrink factors by backward recursion
(Section 7.4). Then, INDEPENDENT of any final mesh, prepare the finite
PL library and its derivative bound M_P (Sections 7.5 and 7.6). Then
choose a fine mesh and execute the two rounds of insertions. Only then
do the bad-cube measures enter (Section 7.7). This order is what
prevents circularity.
Notation. B_Q(s) = c_Q + [-s l_Q, s l_Q]^3 for a cube Q with center
c_Q and side length l_Q. P(Q) is a coarse index of Q (see Lemma 7.2
(iv)).
7.1 Smooth local tolerances
LEMMA 7.1 (smooth local tolerances). Let U subset R^n be open and let
a : U -> (0, infinity) be continuous. For every L > 0 there is a
positive L-Lipschitz function d on U with
d(x) <= min{ a(x), 1, L dist(x, R^n \ U) }.
There are also positive smooth minorants satisfying prescribed
positive continuous upper bounds on values and first derivatives. One
may additionally impose positive constant bounds on a locally finite
family of compact regional tests. Finite collections of such
requirements can be combined, and error budgets on a countable compact
exhaustion can be made summable.
Proof. Extend e(x) = min{a(x), 1, L dist(x, R^n \ U)} by zero outside
U, and put
d(x) = inf_{y in R^n} { e(y) + L|x - y| }.
Then d is L-Lipschitz and d <= e. It is positive at an interior point
x: on a small closed ball about x, e has a positive minimum; outside
that ball the distance term has a positive lower bound. So d is
positive on U.
For the smooth minorant. Fix a smooth locally finite partition of
unity chi_i with compact supports in U and put b = sum_i c_i chi_i,
with positive constants c_i. Given positive continuous bounds v, w,
choose
c_i <= 2^{-i-2} min{ inf_{supp chi_i} v,
inf_{supp chi_i} w / (1 + |D chi_i|_inf) }.
Then b > 0, b <= v, and |Db| <= w. Replace a finite list of such
requirements by their minimum. For countably many regional tests, use
a locally finite compact cover with compact enlargements, refine the
bounds on each intersecting support, and allocate total errors by a
convergent positive series. Only finitely many regional requirements
meet any given compact support. QED
7.2 Adapted dyadic meshes
LEMMA 7.2 (adapted dyadic meshes). There is an absolute integer n_0
with the following property. Given arbitrary positive local upper
bounds on the fine scale, there is a locally finite dyadic cube
decomposition Q of Omega with centers c_Q and side lengths l_Q for
which, writing
B_Q(s) = c_Q + [-s l_Q, s l_Q]^3,
the following hold.
(i) B_Q(100) is compactly contained in Omega; cubes meeting
B_Q(50) have side lengths comparable to l_Q by absolute
constants.
(ii) The normalized cube configurations in B_Q(6) belong to a
finite list. There is a conforming affine triangulation of Q
with finitely many normalized nondegenerate shapes on these
configurations.
(iii) The cubes have n_0 colors so that closed boxes B_Q(6) of a
single color are pairwise disjoint.
(iv) Attach to Q a coarse index P(Q) containing c_Q using a fixed
convention on coarse boundaries. The initial scale bounds can
be reduced so that all interactions through any fixed number
of neighboring patch layers are confined to a fixed locally
finite graph of coarse indices, independent of further mesh
refinement. The corresponding patches lie in neighborhoods on
which the required bounds in the local bi-Lipschitz hypothesis
hold.
The upper bounds on the fine scale remain arbitrarily prescribable.
Proof. Choose a positive size function s on Omega with sufficiently
small absolute Lipschitz constant, below all the prescribed local
bounds and below 10^{-3} dist(x, R^3 \ Omega). Lemma 7.1 supplies such
a minorant. Take the maximal dyadic cubes for which
l_Q <= inf_Q s.
Failure of the condition for the dyadic parent gives
s(x) <= (2 + C Lip(s)) l_Q for x in Q.
The small Lipschitz constant then gives local comparability on B_Q(50).
For example, choosing it sufficiently small makes every dyadic ratio
in this neighborhood belong to {1/2, 1, 2}. The distance bound puts
B_Q(100) inside Omega. Maximality gives a decomposition, and the
positive lower bound for s near every compact subset gives local
finiteness.
The cube vertices have dyadic coordinates. Bounded local scale ratios
and bounded normalized distance therefore give only finitely many
normalized configurations. To triangulate, decompose each cube face
into its square contact facets with adjacent cubes. Put every contact
vertex and hanging subdivision point on the perimeter of each such
square. Cone these perimeter segments from the square center, and then
cone the resulting triangulation of each cube boundary from the cube
center. The constructions on shared facets agree. They give
nondegenerate simplices, and their normalized shapes range over a
finite list.
Local comparability bounds the degree of the graph joining cubes whose
B_Q(6) boxes meet. A coloring with one more than that degree proves
(iii). Finally choose the initial local upper bounds so small relative
to the coarse Whitney sizes that the finitely many required patch
layers stay in fixed coarse neighborhoods. Since the coarse enlarged
cover is locally finite, its interaction graph is locally finite.
Taking a still smaller minorant s does not change any of these
conclusions. QED
Remark 7.3 (finite list of normalized configurations). The finite list
in Lemma 7.2 records actual normalized geometric configurations,
including the positions of their faces and vertices. When protected
boxes are subsequently shrunk by one of finitely many fixed amounts,
their intersections and all margins used below still range over
finitely many geometric configurations. This is what makes the finite
library of Section 7.6 finite.
7.3 Insertion with protected cores
The next lemma is the technical heart of the section. It inserts a new
PL patch on a cube while leaving the values on older protected cores
untouched, and it does so using only a uniform closeness hypothesis
between the new PL patch, the current homeomorphism G, and the target
homeomorphism H. No derivative bound on G or h_Q enters.
Notation. Fix a patch Q and a finite collection of older protected
boxes with the relative sizes and positions provided by Lemma 7.2. For
every older box prescribe a larger half-shrunk box and a smaller
fully-shrunk box, separated by a fixed positive normalized margin.
LEMMA 7.4 (insertion with protected cores). Consider the data above.
Let G : Omega -> Omega' be a homeomorphism and let h_Q be a PL
embedding on a neighborhood of B_Q(3). Assume that h_Q = G on the
intersection of B_Q(3) with each half-shrunk box.
For every lambda > 0 there is an a > 0, depending only on lambda, the
normalized source configuration, and the local constant K_P of Q,
such that
sup_{B_Q(5)} |G - H| < a l_Q,
sup_{B_Q(3)} |h_Q - H| < a l_Q
imply the following conclusion. There is a domain homeomorphism
A supported in B_Q(5), with
sup |A - id| < lambda l_Q,
such that G o A = h_Q on a neighborhood of B_Q(2) and G o A = G on
the fully-shrunk older boxes. The tolerances are uniform over any
finite list of source configurations. They do not depend on
derivatives of G or of h_Q.
Proof. Scale the patch to l_Q = 1. First reduce a using K_P so that
h_Q(B_Q(3)) is contained in G( int B_Q(4) ).
Indeed, for x in B_Q(3) and z in boundary B_Q(4), the distance
|H(z) - H(x)| is bounded below by K_P^{-1}. The small value errors in
the hypothesis of the lemma preserve this separation. Degree on
int B_Q(4), comparing G with H on the boundary, shows that h_Q(x) is
in its image. The same statement holds with a small margin around
B_Q(3).
Thus, on a neighborhood of B_Q(3), the map
A_0 := G^{-1} o h_Q
is well defined. It is a homeomorphism onto its image and, because
h_Q is uniformly close to H and G^{-1} is uniformly close to H^{-1} on
a bounded neighborhood, A_0 is a small perturbation of the identity:
sup |A_0 - id| < C a.
Set A to equal A_0 on B_Q(2), and interpolate A towards the identity
on the annulus between B_Q(2) and B_Q(3). Take A to equal the identity
outside B_Q(3), though we extend its support formally to B_Q(5) so that
it is supported away from the neighboring patches. When |A_0 - id| <
lambda/2 and the interpolation is made small, all changes are
supported in B_Q(5) and satisfy sup|A - id| < lambda.
Compatibility with older cores. On the intersection of B_Q(3) with
each half-shrunk older box, we have h_Q = G by hypothesis, so that
A_0 = G^{-1} h_Q = id there; the interpolation is identical to the
identity there because it is already the identity on a neighborhood of
that intersection. Hence G o A = G on the fully-shrunk part of every
older box. In particular, A acts as identity on a smaller
neighborhood of every old protected core (Remark 5.4 patched
embedding observation).
Conclusion. Equality G o A = h_Q holds on B_Q(2) after the change,
on a smaller neighborhood B_Q(5/4), and the displacement is bounded
by lambda. Because the source configuration and the local bi-Lipschitz
constant K_P are the only inputs, the tolerance a is uniform over the
finite list of normalized configurations. QED
7.4 Shrink factors and tolerances by backward recursion
At every insertion we shrink the older protected cores a small fixed
amount, so that protection is lost slowly enough to allow recursion
through many stages.
Boundary data of the recursion. Two rounds of insertions will be made.
Write S = 2 n_0 for the total number of stages. A newly installed
patch is protected at radius 2 (in its own B_Q coordinates). At every
subsequent stage, decrease every older protected patch's radius by
d_* = (4 n_0 + 4)^{-1}.
Its final radius after S stages is therefore greater than 3/2. In
particular, its closed original cube Q remains inside its protected
core throughout, and at every insertion we may use half of the current
prescribed decrease for agreement of the new model with the old map,
while the full decrease is used for the sets fixed by the ambient
change. Both are positive absolute constants; they are exactly the
margins of Lemma 7.4.
Recursion for the value tolerances. For a patch insertion supported in
B_Q(5) we apply the update
|G(A(x)) - H(x)| <= |G(A(x)) - H(A(x))| + K_P |A(x) - x|.
Divide by the local cube size l_Q. Interacting patches bring only the
bounded ratios from Lemma 7.2 into this inequality. So for a fixed
number of stages the tolerances are assigned by backward recursion:
Start with the desired final value bound;
choose a small allowed displacement at the last stage;
use Lemma 7.4 to obtain its input tolerance;
require the preceding value error to be a small fraction
of that tolerance;
continue backward.
At each coarse index P take the minimum over P's finitely many
interacting indices Q at each step. Only finitely many steps occur,
every tolerance remains positive, and none depends on a PL derivative
bound selected later. This is the whole reason for choosing the
libraries before the mesh resolution.
7.5 Good cubes and rounding to the dyadic grid
At each vertex v of the fine triangulation round H(v) to a target
dyadic grid whose spacing is a small dyadic multiple of the smallest
incident cube size. The dyadic multiple is fixed from the coarse
indices of the incident cubes, small enough that all corresponding
normalized rounding errors are at most delta_P for the coarse index
P of the cube. Interpolate the rounded values affinely on the
conforming triangulation of Lemma 7.2 (ii) to obtain a continuous
piecewise affine map L : Omega -> R^3. We do not assert that L is
globally injective.
A cube Q with coarse index P = P(Q) is called good if there is an
affine map A_Q of bi-Lipschitz constant at most 2 K_P such that
sup_{B_Q(20)} |H - A_Q| <= delta_P l_Q,
( (1/|Q|) int_Q |DH - DA_Q|^p )^{1/p} <= delta_P.
The factor 20 could be replaced by any larger absolute constant.
LEMMA 7.5 (rounding interpolation near a good cube). Let Q be good
with P = P(Q). Then
|DL - DA_Q| <= C delta_P near B_Q(3),
sup_{B_Q(3)} |L - H| <= C delta_P l_Q,
where C depends only on dimension and the absolute shape constants.
Proof. Subtract A_Q from L on the tetrahedra meeting B_Q(3). The
vertex errors are at most C delta_P l_Q by the definition of good cube
and the rounding rule. The inverse matrix of the three edge vectors at
each tetrahedron has norm at most C / l_Q, because the finitely many
normalized shapes are nondegenerate. Multiplying edge-vector
coordinates by vertex errors gives
|DL - DA_Q| <= C delta_P on B_Q(3).
The value estimate then follows by affine interpolation of the value
errors at the vertices, combined with the good-cube value bound. QED
LEMMA 7.6 (L is a local embedding near a good cube). On a slightly
larger convex box containing B_Q(3), the map L - A_Q is Lipschitz with
constant at most C delta_P. Integrating this bound along segments,
|L(x) - L(y)| >= ( (2 K_P)^{-1} - C delta_P ) |x - y| > 0.
So L is injective on a neighborhood of B_Q(3); by invariance of domain
it is an open embedding there.
Proof. The derivative bound was proved in Lemma 7.5; Lipschitz
constants integrate along segments in a convex set. The positivity of
the lower bound is stable because delta_P is a small absolute multiple
of K_P^{-1}. Invariance of domain promotes injective-openness to an
embedding onto an open image. QED
Remark 7.7 (almost every cube is good, bad cubes have small union).
By differentiability and Lebesgue-point arguments, DH is invertible
and well approximated by an affine map on almost every x. Formally,
fix a Whitney-compatible partition of the derivative field into
finitely many compact pieces where DH is close (on average in L^p and
uniformly on balls) to an affine representative; the union of bad
cubes at each fixed coarse index P shrinks to a measurable set whose
measure tends to zero as the local mesh upper bound tends to zero. We
postpone the summability of bad-cube errors to Section 7.7.
7.6 Finite PL libraries before resolution selection
This is the step that removes circularity. We prepare a finite library
of local PL replacements and fix their derivative bounds M_P BEFORE
choosing the fine resolution of the mesh.
What is binned. For a given stage s (1 <= s <= S) and coarse index P,
collect all possible normalized H data on a patch of coarse index P
that obey the fixed coarse bil-Lipschitz bound K_P. Include in the bin
exactly the following:
(a) normalized values of L at vertices of B_Q(4) (finitely
many, up to dyadic grid rounding),
(b) translated coordinates (bounded dyadic vectors by Lemma 7.2),
(c) exact PL data of the designated older protected cores
inside B_Q(6), and
(d) the small discrete data of the current stage (which color,
which subset of older cores is being protected).
Because the vertex values are rounded to a dyadic grid with a bound
determined by K_P and the coarse index, and each localized H family is
locally Lipschitz bounded, the set of configurations in (a)-(d) is
finite for each (s,P). Let B_{s,P} be that finite set.
What is stored. For each configuration in B_{s,P}, prepare one local
PL embedding
h^* : neighborhood of B_Q(3) -> R^3
agreeing exactly with (c) on the corresponding subset of older
protected cores, agreeing exactly with L at the vertices in (a), and
with bounded first derivatives on the compact patch. The number of
stored maps is finite, and each stored map is PL on a neighborhood of a
fixed compact patch, so it has finitely many affine pieces there and a
finite derivative bound. Let
M_P < infinity
be the maximum of all stored derivative bounds over all stages and
over the finitely many stored maps of coarse index P. M_P depends only
on the data of H, K_P, and the finite library; it is INDEPENDENT of the
eventual fine resolution of the mesh.
The existence of h^* from an H-bin. Given a configuration in B_{s,P},
there is an actual H and an actual normalized mesh exhibiting it; by
Hamilton 1976 in the relative form of Lemma 5.1 applied to a small
polyhedral body around B_Q(3) and identity on the older protected
cores, there is a PL embedding of that body into R^3 agreeing with H
on the protected cores and with L at the dyadic-rounded vertices. This
PL embedding is our h^*.
Consistency across stages. At an insertion at stage s and coarse index
P, the actual tuple (stage, coarse index, normalized H data, older PL
data) is one of the finitely many configurations in B_{s,P}, so
exactly one h^* has been stored. The chosen h^* is the "model" of the
insertion. Because all possible normalized meshes obeying the fixed
coarse bounds were included when deciding which combinations occur, a
single stored model is prepared for every combination that can arise.
Finite derivative bound M_P. By induction on s:
(a) the set of possible exact PL data is finite after each
stage, because the insertion of h^* can be expressed in
finitely many ways on finitely many normalized shapes;
(b) there are finitely many stored maps per local coarse
combination, and finitely many stages;
hence every intermediate and final map has derivative bounded
by M_P almost everywhere on any cube of coarse index P.
The constants M_P can be arbitrarily large; they are chosen BEFORE the
mesh resolution.
7.7 Assembly and final proof of Proposition 7.16
Recall the two rounds of insertions (first round on good cubes, second
round on the rest).
PROPOSITION 7.16 (strong PL approximation of a locally bi-Lipschitz
map). Under the hypotheses of Theorem C with H in place of the
abstract input, there is a locally finite PL homeomorphism
G : Omega -> Omega' satisfying
|| G - H ||_{W^{1,p}(Omega)} < eps,
| G(x) - H(x)| < xi(x) (x in Omega).
Proof. Fix the coarse neighborhoods, the local bounds K_P, the finite
mesh configurations of Lemma 7.2, and the finite library and constants
M_P of Section 7.6. Until the final resolution choice, we describe the
two rounds as rules uniform over all admissible meshes.
First round. Start with G_0 = H. For stages 1..n_0 process the colors
in order, inserting a patch only if it is good, and taking its stored
model to be h_Q = L. On every overlap with an older protected core the
insertion agrees with the already installed map, because both equal
the single global map L there. Lemma 7.5 and the chosen thresholds
supply the input accuracy of Lemma 7.4. Supports B_Q(5) of one color
are disjoint, so these insertions are simultaneous. Their local
finiteness makes the union a domain homeomorphism. The value
invariants follow from the update inequality of Section 7.4. At the
end of this round the map agrees with L on every good cube and on a
protected neighborhood of it.
Second round. For stages n_0+1..S process the colors in order,
inserting a stored model of the library on every remaining patch.
Here the finiteness prepared in Section 7.6 does the work. At each
coarse index P a finite family of candidate models is available, and
the choice at stage s depends only on strictly earlier stages at the
finitely many interacting indices. Its derivative bound M_P has been
fixed in advance. Value accuracy and exact agreement with older
protected cores follow from storage constraints and from Lemma 7.4
after undoing normalization.
Simplification of the mesh. When both rounds finish, every cube has a
protected PL neighborhood, and the map is locally PL; taking common
refinements gives a single locally finite affine triangulation. The
map is onto Omega' because each stage was a precomposition by a domain
homeomorphism (Lemma 7.4). On every good cube the map equals L, and
the protected neighborhood preserves that agreement.
Error budget. On a good cube Q, Lemma 7.5 and Equation (7.x) give
int_Q |DG - DH|^p <= (C delta_P)^p |Q|.
On a bad cube Q, use the fixed bound |DG| <= M_P and the local
bi-Lipschitz bound |DH| <= K_P:
int_Q |DG - DH|^p <= (M_P + K_P)^p |Q_{bad}|.
Choose positive numbers b_P with sum_P b_P < eps^p/8, and decrease the
local mesh upper bounds until
(M_P + K_P)^p | union of bad cubes with coarse index P | < b_P.
The bounds can be imposed simultaneously using Lemma 7.2. The M_P
fixed in Section 7.6 are unaffected by this refinement. On good cubes,
summing the principal derivative errors and choosing the mesh fine
enough makes their total contribution below eps^p/2; then the bad
cube contributions are below eps^p/2. So the total p-th power
derivative error is below eps^p. Fine value tolerances are met by
choosing the recursion tolerances below xi. QED
This finishes the reduction from a locally bi-Lipschitz H to a locally
finite PL homeomorphism g, and (together with Section 6) Theorem C.
8. FINE ESTIMATES ON DISKS AND CELLS
-------------------------------------
Two analytic lemmas are used in Section 11. Neither is topological:
the disk estimate uses Ahlfors-Bers' measurable Riemann mapping theorem
to conformally straighten a metric on the source square; the cell
collapse uses the integrability of r^{m-1-p} when p < m.
Definitions. Let Q = [-1/2, 1/2]^2. For a bi-Lipschitz embedding h : Q
-> R^d, put
A(h) = int_Q J_2 Dh, L(h) = int_{bdry Q} |D_t h|^2,
where J_2 Dh is the signed area of the 2-vector Dh(e_1) ^ Dh(e_2). For
a bi-Lipschitz injective embedding the sign is constant, so A(h) is the
area of h(Q).
We say a bi-Lipschitz map H on a polyhedral cell has ESSENTIAL
TANGENTIAL COMPATIBILITY if the trace of the distributional derivative
DH on each closed facet agrees, on a full (m-1)-measure subset of the
facet, with the weak relative derivative of the restriction. Every
bi-Lipschitz map has this property by Rademacher on a manifold-with-
boundary. This is used to make sense of |D_t(H o sigma)| in (8.2).
8.1 Boundary-parametrized disk estimate
LEMMA 8.1 (disk estimate). Let Q = [-1/2, 1/2]^2 and let h : Q -> R^d
be a bi-Lipschitz embedding with essential tangential compatibility. For
every eps > 0 there is a bi-Lipschitz self-homeomorphism rho : Q -> Q,
equal to the identity on bdry Q, such that
int_Q |D(h o rho)|^2 <= C ( A(h) + L(h) ) + eps, (8.1)
with C absolute.
Sharp form. Let h_j be bi-Lipschitz embeddings with sup_j L(h_j) <
infinity, A(h_j) <= 1 + o(1), and h_j|_{bdry Q} -> iota|_{bdry Q}
uniformly, where iota(x) = (x, 0) in R^d. The rho_j may be chosen so
that
int_Q |D(h_j o rho_j)|^2 <= 2 + o(1),
D(h_j o rho_j) -> D iota in L^2(Q). (8.2)
The bi-Lipschitz constants of the rho_j may depend on j. The constant 2
in (8.2) is optimal: D iota has Frobenius norm squared identically 2.
Proof of Lemma 8.1. Three stages.
Stage A (a metric majorant). Pull the Euclidean metric of R^d back to
Q by h: g_0 = h^* = Dh^T Dh. This is a bounded measurable
positive-definite 2-tensor; int_Q sqrt(det g_0) = A(h). Fix delta > 0
and let Q_delta = (1-delta) Q be the concentric square. Mollify g_0 in
L^1 against a smooth kernel, project onto the positive-definite cone,
and place the result on Q_delta. Because sqrt(det) is convex in the
entries of a positive-definite matrix (AM-GM in eigenvalues), the
mollification is done so that
int_{Q_delta} sqrt(det g_0) <= A(h) + eps/3. (8.3)
Extend g_0 smoothly to a positive-definite g on all of Q by blending,
via a smooth radial cutoff supported in the shell Q \ Q_delta, to the
constant scalar M^2 I. The number M > 0 is fixed by the size of g_0 on
the shell (finite because h is locally bi-Lipschitz). On a two-sided
neighborhood of bdry Q the metric g equals M^2 I exactly.
Stage B (conformal straightening). Because g is bounded, uniformly
positive-definite, and smooth, the Ahlfors-Bers theorem (1960) gives a
homeomorphism Phi : Q -> R, conformal from (Q, g) to (R, Euc), for a
Euclidean rectangle R = (0, W) x (0, S). In the interior of Q, Phi is
a diffeomorphism (Beltrami equation, elliptic regularity). Since the
solution is unique up to a conformal map of the target, we normalize to
the rectangle R.
Boundary regularity. Because g = M^2 I in a two-sided neighborhood of
bdry Q, the conformal map satisfies the Cauchy-Riemann equations near
bdry Q. Schwarz reflection extends Phi smoothly across each side, and
reflection across the two perpendicular sides at each corner further
extends Phi past each corner. Consequently Phi and Phi^{-1} are
bi-Lipschitz up to bdry Q.
Stage C (Cauchy-Schwarz on R). Let \hat h = h o Phi^{-1} : R -> R^d.
The two coordinate functions of \hat h are \hat X, \hat Y : R -> R.
Because Phi is conformal from (Q, g) to (R, Euc), and by the conformal
invariance of the Dirichlet energy in 2D,
int_R | D \hat h |^2_{Euc} dy = 2 A(h). (*)
The boundary values of \hat X and \hat Y are the coordinate pullbacks:
\hat X(0, .) = 0, \hat X(W, .) = S; \hat Y(., 0) = 0, \hat Y(., S) = S.
Hence
int_R partial_1 \hat X dy = S^2,
int_R partial_2 \hat Y dy = S W.
Expanding,
int_R (|partial_1 \hat X - S/W|^2 + |partial_2 \hat X|^2)
= int_R |D \hat X|^2 - 2 (S/W) int_R partial_1 \hat X
+ (S/W)^2 (W S)
= int_R |D \hat X|^2 - S^3/W =: Defect_X >= 0, (8.4)
int_R (|partial_1 \hat Y|^2 + |partial_2 \hat Y - 1|^2)
= int_R |D \hat Y|^2 - S W =: Defect_Y >= 0. (8.5)
Adding and using (*):
2 A(h) = S^3/W + S W + Defect_X + Defect_Y. (8.6)
Also S^3/W + S W = S (S^2/W + W) >= 2 S^2 by AM-GM, so
A(h) >= S^2. (8.7)
Sharp case. Suppose A(h_j) <= 1 + o(1) and h_j|_{bdry Q} -> iota|_{bdry
Q} uniformly. The image \hat h_j(R_j) then converges to the unit square
in R^2 x {0}, so W_j S_j -> 1 and W_j, S_j -> 1. By (8.7) this forces S_j
-> 1, W_j -> 1. Then (8.6) gives Defect_X + Defect_Y -> 0, i.e.
partial_1 \hat X_j -> 1, partial_2 \hat X_j -> 0,
partial_1 \hat Y_j -> 0, partial_2 \hat Y_j -> 1 in L^2(R_j).
After the rescaling (y, t) -> (y/W_j, t/S_j) of R_j to Q, which preserves
L^2 energy up to 1 + o(1), we obtain (8.2) with rho_j = Phi_j^{-1}
followed by the rescaling.
Weak case (8.1). If boundary closeness is not assumed, one no longer
has S_j, W_j -> 1. Blend rho on the outer shell Q \ Q_delta to the
identity, with a cutoff going from rho at the inner boundary of the
shell to the identity at bdry Q. The tangential derivative of the
radial schedule is arranged not to interfere with D_t h. After the
blend,
int_Q |D(h o rho)|^2 <= 2 A(h) + C delta L(h) + eps',
and choosing delta small enough absorbs the boundary energy term, with
C absolute. Combined with the interior bound 2 A(h) + eps/3 this gives
(8.1). QED
Two uses in Section 11:
(i) Sharp form, applied at rank 2 and p = 2 to critical tiles,
where the budgets force A(h_j) <= 1 + o(1) and boundary
closeness to the identity square. The value 2 + o(1) is
what we compare the actual derivative energy to.
(ii) Crude form, applied at rank 2 and p = 2 to noncritical
tiles and at p = 2 and m = 2 generally, where the cell
collapse of 8.2 does not apply because p = m.
8.2 Cell collapse
LEMMA 8.2 (cell collapse). Let m in {1, 2, 3}; allow 1 <= p < infinity
when m = 1, and 1 <= p < m when m >= 2. Let D subset R^m be a convex
polyhedral cell of scale ell, with inradius >= c ell and diameter <= C
ell. Let H : D -> R^d be bi-Lipschitz with essential tangential
compatibility. Let sigma : bdry D -> bdry D be face-preserving
cellwise bi-Lipschitz. For every eps > 0 there is a bi-Lipschitz
parametrization F : D -> H(D) with F|_{bdry D} = H o sigma and
int_D |DF|^p <= C_{m,p,c,C} ell *
int_{bdry D} |D_t(H o sigma)|^p + eps. (8.8)
The boundary data on shared cells are retained exactly: if two
neighboring cells D, D' share a facet and prescribe the same boundary
values H o sigma there, the F's agree on that facet.
Proof. Scale ell = 1 and multiply back at the end. Use the shape chart
placing an incenter of D at the origin, with radial coordinates
x = r z, z in bdry D, 0 <= r <= 1.
The chart x -> (r, z) is bi-Lipschitz with joint constant depending only
on c and C.
Case m = 1. D = [-1,1], bdry D = {-1, +1}. If sigma fixes both
endpoints, take F = H. If sigma swaps them, take F(x) = H(-x). In either
case int_D |DF|^p dx <= 2 int_{bdry D} |D_t(H o sigma)|^p. For an
unprescribed edge of source length H and image length L, constant-speed
parametrization gives exactly
int_e |DF|^p = L^p H^{1-p},
the m = 1 bound in the form used in Section 11.6.
Case m >= 2. Fix 0 < alpha < 1 and 0 < delta < 1 and put
q_{alpha,delta}(r)
= (1-delta) r / alpha (0 <= r <= alpha),
= 1 - delta + delta (r-alpha)/(1-alpha) (alpha <= r <= 1).
Then q(0) = 0, q(alpha) = 1 - delta, q(1) = 1, and q is a bi-Lipschitz
homeomorphism [0,1] -> [0,1]. Define
F_{alpha,delta}(r z) = H( q(r) sigma(z) ) (0 < r <= 1),
F_{alpha,delta}(0) = H(0).
Boundary agreement. F(1 z) = H(sigma(z)), so F|_{bdry D} = H o sigma.
Bi-Lipschitz. The map (r, z) -> q(r) sigma(z) is bi-Lipschitz on each
half and agrees at r = alpha; composing with H gives a bi-Lipschitz F.
Inner energy (0 < r <= alpha). Every derivative of F is at most
((1-delta)/alpha) times the corresponding derivative of H: the radial
coordinate of H is differentiated with the extra factor q'(r) =
(1-delta)/alpha, and the angular coordinate with the extra factor q(r) =
(1-delta) r/alpha <= 1 - delta. So
int_{B_alpha} |DF|^p dx <= ((1-delta)/alpha)^p alpha^m
sup_D |DH|^p
<= C alpha^{m-p} sup_D |DH|^p,
which is O(alpha^{m-p}) and hence -> 0 as alpha -> 0.
Outer energy (alpha <= r <= 1). Here q'(r) = delta/(1-alpha) -> 0 as
delta -> 0, and q(r) -> 1 uniformly on alpha <= r <= 1. So
DF = DH(q(r) sigma(z)) ( q'(r) sigma(z + rD sigma ... ) )
-> DH(sigma(z)) ( 0, D sigma ... ) + o(1) |DH|
as delta -> 0 at fixed r in (alpha, 1). By essential tangential
compatibility, the limit integrand is |D_t(H o sigma)(z)|^p, and the
volume element r^{m-1} dr dz separates as r^{m-1} dr dz. Hence
int_{B_1 - B_alpha} |DF|^p dx
-> ( int_alpha^1 r^{m-1-p} dr )
int_{bdry D} |D_t(H o sigma)(z)|^p dz,
and int_alpha^1 r^{m-1-p} dr = (1 - alpha^{m-p})/(m-p) <= 1/(m-p).
Conclusion. Choose alpha so small that the inner energy is below eps/2.
With alpha fixed, choose delta so small that the outer energy is below
(int_alpha^1 r^{m-1-p} dr) int_{bdry D} |D_t(H o sigma)|^p + eps/2.
Combining, we obtain (8.8).
Boundary gluing. On a facet F |_shared = H o sigma, so two neighboring
cells prescribing the same boundary data give F's that agree on the
shared facet.
Uniformity. The construction is uniform over a finite list of fixed
shape-controlled cells and finitely many normalized sigma; the shape
constants enter C_{m,p,c,C}. QED
The m = 2, p = 2 case (excluded because p = m) is provided by the crude
form (8.1) of the disk estimate. After rescaling Q to the parameter cell,
int_Q |D(h o rho)|^2 <= C (A(h) + L(h)) + eps,
which is exactly the bound the cell-completion argument needs at p = 2
in dimension 2.
9. LOCAL TOPOLOGICAL SURGERY
----------------------------
Two local operations are needed to attach the constructions of
Section 11 to the qualitative topology of Section 5:
(a) cleaning an isolated surface/line crossing to a regular
transverse crossing with a rigid affine germ nearby;
(b) replacing a homeomorphism at a "star center" (several
labelled rays through a point) by one that agrees with the
identity on a small ball, certifiable by a finite test.
Neither operation is quantitative: neither supplies a derivative bound.
The derivative estimates all come from Sections 8, 10, 11.
The classical inputs we use are listed once and for all:
locally flat Schoenflies and extensions of homeomorphisms
from a sphere (Brown 1960, 1962);
Dehn's lemma (Papakyriakopoulos 1957);
classification of mapping classes of punctured spheres,
peripheral-preserving Dehn-Nielsen form (Farb-Margalit 2012,
Theorem 8.8 and Proposition 3.19);
Smale 1959 and Li-Watts 2011 for isotopies of S^2.
Marked points on a sphere carry no tangent framing; an isotopy fixing
such a point need not fix a tangent direction there. This affects only
the last normalization step of Proposition 9.5.
9.1 Cleaning an isolated crossing
LEMMA 9.1 (cleaning an isolated crossing). Let P be a locally flat
embedded surface disk and let I be an interval straight in specified
local topological coordinates. Suppose z in P intersect I is interior
to both, is isolated in P intersect I, and has a neighborhood
containing no other surface sheet under consideration. There is an
ambient modification of P, supported in an arbitrarily small
neighborhood of z, after which the intersection in that neighborhood
is either empty or a single standard transverse crossing. At a
retained crossing the disk can be taken flat across the straight
interval. No initial local flatness of the pair (P, I) is assumed.
If I passes from one side of P to the other at z, then a sufficiently
small modification retains one crossing.
Proof. Choose a small topological ball B about z in which P is a
proper unknotted disk. All changes will be in the inner third of B.
Step 1 (chart and slabs). Straighten I in its supplied coordinates
near z and choose two cap levels on opposite sides of z along a
segment of the axis whose only intersection with P is z. In a parameter
disk for P choose round disks D_1 compactly inside D_0 about the
parameter of z, so that P(D_0) is compactly contained in the open slab
between the cap levels and in the straight-axis chart.
Step 2 (thin cylinder). Choose a sufficiently thin polygonal cylinder C
about the intervening axis interval. Its caps miss P; the side annulus
A intersects P only in P(D_1) once the radius is small enough. Every
side loop and its bounded parameter subdisk lie in D_1. All disks used
below for linking lie in the cap slab and cannot meet the extended
axis.
Step 3 (general position). Put P in PL general position near A without
changing it near the axis or near boundary B; this is possible because
P is a locally flat disk in a chart, and PL approximation of that chart
near the prospective side intersections (Lemma 5.1) produces a
configuration supported off the axis. Then P intersect A is a finite
family of polygonal simple closed curves.
Step 4 (surgery). Disk replacements (on P-side innermost disks of A,
and on A-side innermost disks of P) reduce the intersection to curves
that do not bound disks in A. The replacement disks stay in already-
arranged regions by choosing cylinders and parameter disks small enough.
Step 5 (linking). Each remaining essential side curve has linking
number plus or minus one with the extended straight axis. So its
parameter disk in P must meet that axis. Since surgeries have added no
axis intersections, this disk contains the original isolated
intersection. The disks bounded by the remaining curves are therefore
nested. Replace the outermost one by a tame proper disk in C with the
same rim and with exactly one flat transverse crossing of the axis. One
way to see this: isotop the essential rim in the side annulus to a
circular section, and extend that isotopy across an outer collar of a
meridian disk.
Step 6 (ambient extension). The result is a locally flat proper disk in
B agreeing with the original near boundary B. Both disks split B into
balls; the locally flat Schoenflies theorem followed by extension on
the two sides gives an ambient homeomorphism of B, fixed on boundary
B, carrying the original disk to the constructed disk. Extend by
identity outside B. Supports can be made successively smaller.
Opposite-side case. If I passes from one side of P to the other at z,
retain two nearby axis points on opposite sides of z outside the
modification support. The separation forces the modified disk to meet
the intervening axis interval; since at most one intersection is
produced, exactly one. QED
COROLLARY 9.2 (affine germs at cleaned crossings). Let h : M -> N be a
homeomorphism of open 3-manifolds. A crossing of a regular source
surface with the preimage of a regular target line can be cleaned by
applying Lemma 9.1 to the image of that surface. A crossing of a
regular source curve with the preimage of a target surface is handled
by applying Lemma 9.1 to h^{-1}. If both closed subsets have C^1
straightening charts near the crossing, then at the retained crossing
h may be made affine on a smaller ball of the local chart, carrying
the regular source stratum to the straight-line (or plane) target
stratum. The support of the modification is arbitrarily small in both
source and target.
Proof. After cleaning, work in the two C^1 charts. Choose a small outer
ball B_o and a much smaller inner ball B_i; prescribe an affine map A
on B_i, sending the source stratum to the target stratum and its
ambient normal filling to the ambient normal filling. The map A
extends as a homeomorphism from boundary B_i to boundary B_o by
coning across the planar annulus between the two equators of the two
spheres. The two half-extensions (across the equatorial plane) agree
on the planar annulus, so we obtain an ambient homeomorphism of
B_o \ B_i that fixes the boundary of B_o and equals A on B_i. Choose
a smooth cutoff supported near B_o; extend by the identity. In the
inverse homeomorphism, the inverse affine germ is affine again. Both
supports can be made small because h and h^{-1} are continuous.
QED
9.2 Two-point cut of an unknot
LEMMA 9.3 (two-point cut of an unknot). Let K be a tame unknot in the
3-sphere and let B be a locally flat ball whose boundary meets K in
exactly two standard transverse crossings. Then the proper arc K
intersect B is boundary-parallel in B. In particular, the ball--arc
pair is the standard unknotted interval pair.
Proof. Choose a tube about K meeting boundary B in two meridian disks.
Such tubes are constructed on the two cut intervals, starting with
product charts at the crossings and continuing regular neighborhoods
along interiors. Glue the endpoint disks to obtain a tubular
neighborhood T_K of K.
The exterior X of T_K in S^3 has pi_1 = Z, generated by a meridian
(the unknot is standard). Its two pieces obtained by cutting along
the annular band boundary B minus the two meridian disks are glued
along that annulus, and the annulus inclusion induces an isomorphism
on pi_1. Its inclusions into the two pieces are therefore injective.
The injective-amalgam form of Seifert-van Kampen then embeds each
piece's pi_1 in pi_1(X) = Z. The common pi_1 already maps onto Z, so
each piece inclusion is an isomorphism onto Z.
Combining with the standard product structure of S^3 \ K, both pieces
of X are one-holed thickened spheres with a straight arc (the proper
interval K intersect B) running through. Any such structure is
topologically standard by Brown-Schurle-type regular neighborhood
recognition, so the proper arc K intersect B is unknotted. QED
Consequence. In any locally flat ball crossing a tame unknot
transversely in exactly two points, the proper interval inside is
ambiently isotopic (relative endpoints) to a straight proper interval,
so it is boundary-parallel.
9.3 Punctured-product sphere
We need one general product recognition, used in Stage 6 of the star
replacement.
LEMMA 9.4 (punctured product sphere). Let M be homeomorphic to S^2 x I,
with boundary spheres boundary^- = S^2 x {0} and boundary^+ = S^2 x
{1}. Let S be a disjoint union of properly embedded unknotted
intervals (arcs) L_1, ..., L_n, each joining boundary^- to boundary^+.
Let F_1 be a properly embedded surface in M separating boundary^-
from boundary^+, with boundary circles on boundary^- union boundary^+
each meeting one L_i exactly once, transversely. Suppose the induced
isomorphism
pi_1(F_1) -> pi_1(M \ S)
preserves the individual peripheral conjugacy classes of boundary
circles. Then (M, S) is the standard product M \ S = P x [a, b] with
P = S^2 minus n marked points and S = {marked points} x [a, b], and
the stated boundary parametrization identifies product ends.
Proof. Cut M along F_1 and along the tubular neighborhoods of S,
producing a manifold W with two boundary spheres F_0, F_1 and n annular
meridian tubes gluing the two. By the van Kampen argument (analogous to
Lemma 9.3), pi_1(F_0) -> pi_1(W) and pi_1(F_1) -> pi_1(W) are both
isomorphisms onto Z. W is irreducible (Schoenflies in M, tubes and F_1
on the exterior of any sphere). Then the surface-classification and
peripheral-preserving Dehn-Nielsen form of the mapping-class theorem
identifies a homeomorphism F_0 -> F_1 preserving labels; coning that
homeomorphism across W gives a homeomorphism W = F_0 x [0,1].
Transporting back: (M, S) = P x [a, b] with labelled strands. QED
9.4 Star replacement
Setup. Use centered Euclidean coordinates in source and target of a
local homeomorphism h, with h(0) = 0. Let
D = {v_1, ..., v_n} subset S^2, n >= 3,
R_i = {t v_i : t >= 0},
be distinct labelled directions, partitioned into forward F and
reverse E directions; each of F, E has at least two elements. Choose
narrow round cones C_i about R_i, and slightly wider C_i^+ with
pairwise disjoint angular closures. Assume as germs at 0:
h(R_i) is contained in interior C_i (i in F),
h^{-1}(R_i) is contained in interior C_i (i in E).
For each nonempty distorted system assume nested round spheres with
radii tending to zero in its own space, each meeting every arc of
that system exactly once, with the arc passing from one side of the
sphere to the other. Impose one of the following angular tests.
Test 1. Finite based generating loop system (gamma_1, ...,
gamma_{n-1}) for S^2 \ D, chosen outside the closed angular
caps of all cones. At arbitrarily small common source radii r,
the parametrized loops s |-> h(r gamma_a(s)) /
|h(r gamma_a(s))| are all close to gamma_a within a fixed
simultaneous homotopy tolerance in S^2 \ D; the tolerance
keeps image loops outside cone caps.
Test 2. Only forward directions, cyclically ordered on a
source equator. At arbitrarily small source radii, the
parametrized radial projection of the image equatorial circle
is uniformly close to the original parametrization, with
tolerance small compared with separations of successive
marked points.
PROPOSITION 9.5 (replacement at a star center). Under either of these
tests, there are neighborhoods of 0 in source and target and a
homeomorphism H between them, agreeing with h on the same angular
data away from a small neighborhood of 0, carrying every labelled ray
to itself as a set as a germ, and equal to the identity on a smaller
closed ball.
Proof. Eight stages.
(1) Clean sections. Choose round sections with radii tending to zero and
use Lemma 9.1 to clean the crossings of each distorted arc with the
chosen sections. The supports are inside narrow angular sectors and
do not interact with other sectors. After cleaning, all intersections
of a distorted arc with its successive sections are standard transverse
crossings.
(2) Strand is boundary-parallel. The portion of a distorted arc between
two successive round sections lies inside a topological ball (the
annular sector between the two shells, intersected with one sector). It
is a proper interval meeting the boundary exactly at the two successive
crossings. By Lemma 9.3 the proper arc is boundary-parallel. The same
conclusion applies to any forward (or reverse) distorted arc.
(3) Straighten strands. Straighten each strand inside its annular
sector, matching maps on the cap disks and identity on the sector sides.
The standard ball-diameter boundary homeomorphism respecting the two
endpoints extends by coning and fixes the diameter as a set. At the
outermost section make a transition inside the same sector and extend
by the identity outside a slightly larger neighborhood. The infinitely
many annular straightenings glue to a homeomorphism at the center with
continuous inverse, because their supports are balls with radii tending
to zero.
(4) Separately for forward and reverse. Apply stage (3) target-side for
the forward system F, and source-side for the reverse system E. The
sectors are disjoint, so target-side changes leave the reverse rays
fixed and vice versa.
(5) Combine. Pre and post-compose accordingly: a homeomorphism H
carrying every labelled ray of the combined system to itself as a set,
as a germ. The generator test passes: source loops and image
directions have positive margins from the supports. In the equatorial
case (Test 2), target changes give arbitrarily small angular
displacement (narrow sectors), so the parametric angular test persists
within its homotopy tolerance.
(6) Shell product. Choose a test radius r inside the exact-ray germ.
The locally flat sphere S_r = H(boundary B_r) meets each marked ray
exactly once in a standard topological crossing. A small round target
sphere lies inside H(B_r); together with S_r it bounds a shell to
which Lemma 9.4 applies. Its product comparison of marked boundary
spheres is homotopic to radial projection.
(7) Marked isotopy. In Test 1, the comparison of the outer boundary
map with the intended identity induces the identity on the chosen
based generators; the punctured-sphere mapping-class theorem makes
that comparison isotopic to the identity through maps fixing the
labelled marked points. The same theorem detects the orientation, and
the assumption n >= 3 excludes the two-puncture ambiguity. In Test 2,
the usual disjoint-arc isotopy test fixes these classes simultaneously.
In both cases the boundary identification is isotopic to the identity
fixing the labelled marked points.
(8) Assembly. Choose a smaller source sphere boundary B_s with intended
identity image in H(B_r). The product coordinates on source and target
annuli, combined with the marked-sphere isotopy, extend the outside
values of H across the annulus while respecting each labelled ray and
gluing to the identity on B_s. Set the map to identity on B_s, and to
h outside B_r. The result is a homeomorphism with the required
identity germ.
Support bound. The supports of the earlier sector straightenings and
of the final cutoff are arbitrarily small in both source and target;
the angular cone bounds and the strict margins in the tests give the
stated germ and neighborhood conclusions. QED
9.5 Relative germs
The assembly below uses a statement combining the above.
PROPOSITION 9.6 (relative germs). Let h : M -> N be a homeomorphism of
open 3-manifolds. Consider finitely many disjoint compact
configurations, each carrying:
(i) a marked regular crossing (surface + line, or curve +
surface);
(ii) a prescribed affine germ in C^1 charts near that
crossing;
(iii) a prescribed identity germ at a star center with
labelled rays, together with a designated smaller ball
B_s;
(iv) compact forward and inverse tolerances.
There is a small closed polyhedral neighborhood O of the union of all
these configurations such that h can be modified on a slightly larger
compact neighborhood to a homeomorphism h' that equals h on O and
equals the prescribed affine germs and star-center identities inside.
Proof. Apply Lemma 5.1 (relative PL approximation of a homeomorphism
of 3-manifolds) with K a polyhedral neighborhood of the prescribed
germs, producing a homeomorphism h_1 of the ambient manifolds
arbitrarily close to h in the uniform metric and equal to h on O.
Inside the prescribed affine and star-center germs, reinstall the
local models of Corollary 9.2 and Proposition 9.5; these are supported
away from O, so h_1 agrees with the local model inside the germ and
with h on O. The composite h' is the desired homeomorphism. All
supports are disjoint and small enough to fit inside the prescribed
compact neighborhoods. QED
10. TRACE BUDGETS AND SIMPLEX SQUEEZING
----------------------------------------
This section builds the derived-trace budget, which is the mechanism
that gives DERIVATIVE control in the assembly of Section 11 without
using any quantitative topological extension theorem.
Two pieces of bookkeeping are involved:
(a) a family of finitely many MEASURED CURVES (m = 1) and
MEASURED SURFACES (m = 2, only for p = 2) counted with
multiplicity, whose f-images are rectifiable;
(b) a TARGET TRIANGULATION that we may choose, sampled
generically, with the property that the number of times the
image trace crosses a complementary-dimensional skeleton is
controlled in expectation by the m-area of the trace times a
universal constant.
If the target parametrization T o G is chosen to concentrate near the
crossings, then the actual output m-area is at most this budget.
Averaging over the target sample gives an upper bound in terms of the
source trace energy - exactly what the radial blocks and kernel blocks
of Section 11 need.
Throughout this section, f : Omega -> Omega' is a fixed homeomorphism
in W^{1,p} (1 <= p <= 2), fixed after the compact jet selection of
Section 11.2. All constructions in Sections 10 and 11 use only this
f, never an auxiliary approximant.
10.1 Generic trace properties
We start with the existence of a suitable family of measured traces.
The point is that a.e. translation of a Lipschitz parametrization
template inherits the area formula and good multiplicity/regularity
properties from the source.
LEMMA 10.1 (generic trace properties). Let f : Omega -> Omega' be a
homeomorphism in W^{1,p}_loc with 1 <= p <= 2. Fix finitely many
compact Lipschitz semialgebraic parametrization templates K_nu -> Omega
of dimension m in {1, 2} (with m = 2 only when p = 2). For almost
every translation of each template one has the following.
(a) The f-image of the translated template is countably
m-rectifiable. The m-dimensional area formula holds for it,
with the tangential derivative D_m f counted with parameter
multiplicity.
(b) The same holds on every lower (positive) stratum of a
finite semialgebraic stratification of the source
parametrization.
(c) At an intersection of the image trace with a complementary-
dimensional affine test stratum in general position, the source
trace lies in a regular m-dimensional stratum and has regular
full-rank parameter branches with locally constant multiplicity.
(d) Exceptional image values - parameter branch junctions and
intersections where the trace jumps in dimension - are avoided.
Proof. The m = 1 case: Sobolev slicing gives absolute continuity of f
along a.e. parallel line. This gives the 1D area formula and the Lusin
property. A smooth curve chart can be foliated by transverse translates
and straightened by a local ambient smooth diffeomorphism, giving the
same conclusion in that chart.
The m = 2 case: Csornyei-Hencl-Maly (2010, Theorem 1.3) prove that a
W^{1,2} homeomorphism in dimension three has the two-dimensional Lusin
property on a.e. parallel plane. Combined with Sobolev slicing and the
Lipschitz decomposition of a Sobolev map (Bojarski-Hajlasz 1993,
Theorem 3(1)), this gives the surface area formula. Straighten a smooth
graph by an ambient smooth diffeomorphism; apply the Csornyei-Hencl-
Maly theorem; Fubini for the translated graph templates closes.
Properties (c) and (d) follow by generic-position arguments: the set
of translations that miss them has measure zero, by countable
intersection over the locally finite family and the stratification. QED
10.2 Target meshes with lattice patches
The next lemma produces the target meshes used in Sections 11.4-11.6.
Their shape does not need to be uniform; only bounded local site counts
and upper size bounds are needed for the target-area budgets.
LEMMA 10.2 (target mesh with lattice patches). In an open subset V of
R^3, prescribe finitely many separated compact patches with buffers. In
a deep part of each patch one may require a rotated rectangular lattice
of side lengths d or eta d, where 0 < eta <= 1 is fixed. There are
locally finite triangulations of V agreeing there with the chain, or
Kuhn, triangulation of that lattice and with arbitrarily small
prescribed local upper size bounds.
For each local candidate, its sampling variables admit a physical
common translation t and normalized relative coordinates z whose JOINT
density satisfies
q(t,z) <= C d^{-3}, (10.1)
t in B(t_0, C d), z in Z subset B(0, C),
where dim Z and measure(Z) are bounded. Deterministic lattice relative
coordinates are held fixed; when there are no free relative coordinates
the integration means a unit point mass. The bounds are universal away
from enlarged lattice patches and may depend on eta inside them. Only
boundedly many candidates occur within a bounded number of local mesh
lengths. Formula (10.1) is used by JOINT integration, without
normalizing a conditional translation law after fixing z.
Proof. Choose a positive size function d(u) with a sufficiently small
absolute Lipschitz constant, satisfying the prescribed local upper
bounds and d(u) << dist(u, boundary V). The function d is constant
near each lattice patch, taking a fixed value compatible with the
prescribed deep lattice side length. Lemma 7.1 supplies such a
minorant.
Outside deep lattice regions take a maximal packing at scale d and
jitter each retained site in a ball of a fixed small fraction of its
local scale with uniformly bounded density. In a lattice patch use a
common uniform positional shift over one period. Truncate the
unshifted lattice lists in their buffers; the separated patches remain
many steps apart. The resulting sites cover V at distance at most C
d(u) and have bounded counts within that distance. A minimum separation
between every pair of sites is not needed.
Use their Delaunay subdivision, with a fixed pulling order (Kuhn) to
triangulate nonsimplicial cells. Localization is checked inside V:
an empty ball containing or passing through a point near u cannot have
radius much larger than d(u), because moving several mesh lengths from
that point toward the center stays inside V and gives a location well
inside the ball; the covering property then puts a site strictly
inside. The slow variation of d justifies using the same scale. Finite
restrictions of the cloud therefore stabilize near u; their ordinary
Delaunay subdivisions cover and agree there. This proves local
existence, local finiteness, and compatibility of the subdivisions.
Deep in a rectangular lattice, the Voronoi description shows that the
Delaunay cells are the rectangular boxes. Increasing lexicographic
pulling gives their Kuhn simplices. Small scale places any prescribed
deep compact set and its touching simplices inside this region.
Densities. Fix a candidate tuple of site indices before imposing the
Delaunay selection event. If the tuple contains lattice sites, all
belong to one shifted lattice. Write the physical shift as t and each
unstructured site as x_0 + t + d z_j, with x_0 a fixed anchor. The
original joint law of (shift, k unstructured sites) has density at
most C d^{-3(k+1)}. Replacing relative offsets by z_j has Jacobian
d^{3k}, so the joint density in (t, z_1, ..., z_k) is at most C d^{-3}.
If the tuple contains no lattice site, write one site as x_0 + t and
the other k-1 relative to it in units d. The Jacobian from raw
positions to (t, z_1, ..., z_{k-1}) is d^{3(k-1)}, giving the same
bound. The Delaunay selection indicator is bounded by 1 and discarded.
For each fixed z integrate the unnormalized translation slice, then
integrate z over the bounded normalized set Z. QED
Remark 10.2b (consequence). Formula (10.1) is the joint density of
translation and internal normalized coordinates. The single bound
q(t, z) <= C d^{-3} is what makes translation-averaging work: for any
bounded measurable function N(t, z) counting intersections, the
expectation satisfies
E N = int int q(t, z) N(t, z) dz dt <= C d^{-3} int int N(t,z) dz dt.
The particular form of N(t, z) is fixed by the source trace and by
the geometry of the tie strata. In 10.6 we compute the resulting bound
in terms of the source trace energy.
10.3 The tie budget
Let sigma be a tetrahedron in a target triangulation, with vertices
v_0, ..., v_3. Write lambda_0, ..., lambda_3 for its barycentric
coordinates.
DEFINITION 10.3 (tie budget). For a set I of m+1 vertices of sigma, the
DUAL STRATUM associated with I is
Z_{sigma, I}
= { x in relint sigma :
lambda_i = max_j lambda_j for i in I,
lambda_j < max_i lambda_i for j notin I }.
Write a source parametrization as P : K -> Omega and its measured image
as Gamma = f o P. For such a generic measured m-trace (Lemma 10.1),
its BUDGET B_m(Gamma) is the sum over its visits to these strata of
H^m( conv{ v_i : i in I } ),
with visits counted with parameter multiplicity. A weighted budget uses
in addition the supremum of a prescribed nonnegative continuous weight w
over sigma. A fixed linear change in the output may be applied when
computing the flat weights.
The rule is dimension-complementary: an m-dimensional trace is paired
with an (m+1)-vertex tie stratum of dimension 3 - m - 1 = 2 - m in the
tetrahedron, whose weight is the m-dimensional flat m-face on the
primal side.
Illustration with a triangle. Let v_0, v_1, v_2 be the vertices of a
triangle and lambda_i > 0 interior barycentric coordinates. For a > 0
consider
sum lambda_i v_i |-> [ sum lambda_i e^{a lambda_i} v_i ]
/ [ sum lambda_i e^{a lambda_i} ].
As a -> infinity, a point with one strictly largest coordinate moves
toward that vertex. A regular curve a small distance from a point of
lambda_0 = lambda_1 > lambda_2, crossing that tie once transversely
with endpoints in the two adjacent one-leader regions, instead runs
between v_0 and v_1. The concentration calculation below shows that
one such crossing contributes the edge length |v_1 - v_0| in the
limit. In a tetrahedron the same principle pairs a curve with a
two-leader surface (edge weight = length on the primal side), or a
surface with a three-leader line (triangle weight = area on the primal
side).
Why this is the right object. The budget counts how many times the
image of a source trace is squeezed against a lower-dimensional primal
face; the concentration map of 10.4 then realizes this count as actual
image m-area. Crucially, the expectation of the count over the target
sample (Lemma 10.6) is bounded by a universal constant times the
source trace energy; this is the analytic content that makes
derivative control possible.
The map whose concentration we exploit is defined as follows.
Concentration map. On each tetrahedron sigma, define a map T_a : sigma
-> sigma, called the CONCENTRATION or SQUEEZE map, with parameter a > 0
and a function b(lambda) chosen so that the map is onto and invertible.
The leading behavior for large a is
T_a(lambda_0, ..., lambda_3)_i ~= lambda_i e^{a lambda_i} /
sum_j lambda_j e^{a lambda_j}
with a small nonlinear correction controlled by b(lambda) ~= 1 / log(a + e).
On each closed simplex sigma, T_a is C^1 and has a positive lower bound
on the Jacobian determinant. Over a small simplex the parameter b is
bounded below; on larger simplices the value becomes smaller, but the
finest cells are small, so a remains bounded below.
The precise shape of the concentration map is in Lemma 10.4.
Let us pin down the concentration (squeeze) map. Fix a tetrahedron
sigma with vertices v_0, ..., v_3 and barycentric coordinates
lambda_0, ..., lambda_3 in the closed simplex. Define
Pi_a(lambda)_i = lambda_i exp(a lambda_i) / Z_a(lambda),
Z_a(lambda) = sum_j lambda_j exp(a lambda_j),
for a > 0. The map Pi_a is not automatically onto the closed simplex;
its image fills the simplex except near the vertices. To make Pi_a a
homeomorphism of sigma and to make its inverse Lipschitz, multiply by a
correction b(lambda) > 0:
T_a(lambda) = b(lambda) Pi_a(lambda), 0 < b(lambda) <= 1.
A concrete formula for b is
b(lambda) = 1 / (1 + log(a + e) sigma(1 - max_i lambda_i)), (10.2)
where sigma = sigma(lambda) is a small oscillation factor. On a closed
simplex where some lambda_i has a positive lower bound, b is bounded
below by an absolute constant, so T_a is a small perturbation of the
projection onto the leading vertex and is a homeomorphism.
Invertibility for fixed a. Given output ratios z_i = T_i / T_max in
(0, 1], for a trial D >= 0 let r_i be the unique solution of
z_i = r_i exp(-D(1 - r_i)), 0 <= r_i <= 1. (10.3)
The function on the right is strictly increasing in r_i. Each r_i is
nondecreasing in D, with zero and maximal ratios fixed at 0 and 1. Put
R = sum_i r_i and lambda_i = r_i / R. The trial value of a is a = DR,
an increasing function of D with increase at least D. The outside
ratios are exactly the r_i, so b at this trial inverse is nondecreasing
in D; the prescribed value exp(1/b) - e is nonincreasing. So
DR = exp(1/b(lambda)) - e (10.4)
has a unique solution D by continuity, with sign at D = 0 dominated by
the right side and positive sign for large D. This proves bijectivity.
On an individual closed simplex b is bounded below, so the solution
has bounded D. Equation (10.3) has uniformly Lipschitz inverse in this
bounded range, including at zero output coordinates. The scalar
equation in (10.4) has increase in D at least the increase in the left
side, so its root depends Lipschitz-continuously on the output. Since
T_max >= 1/4, forming the output ratios is also Lipschitz. The forward
formula is Lipschitz on that simplex, and its inverse is Lipschitz.
Asymptotics for large a. In a compact set of the simplex at a fixed
positive distance from the (m+1)-fold leader ties, at most m
coordinates are near max_i lambda_i; the others are exponentially small
in a. The active coordinates sum to one up to exponentially small
errors, so their image lies in a face of dimension at most m - 1; the
m-Jacobian there is zero. The m-Jacobian of T_a is therefore dominated
by the image of the counted tie strata.
10.4 The simplex squeeze
LEMMA 10.4 (simplex squeeze). Let G parametrize finitely many compact
m-traces (m in {1, 2}) by finite closed-piece C^1 formulas. Suppose
their intersections with the Z_{sigma, I} are regular transverse hits
away from higher ties and simplex boundaries. Every counted hit is in
the interior of each counted regular m-dimensional parameter branch;
parameter boundaries and lower-dimensional strata avoid the counted
dual strata. Then
int J_m D(T_a o G)
-> sum_{counted hits} H^m( conv{ v_i : i in I } ), (10.5)
as a -> infinity uniformly on the compact traces, with the error
depending only on the traces, the fixed list of configurations, and
1/a.
Proof. Fix a compact trace set K away from the (m+1)-fold leader ties.
For a large enough that b(lambda) < 1/2 on K (uniformly, because b is
small in lambda away from the simplex boundary on K), the concentration
estimate above applies. The m-Jacobian of T_a there is exponentially
small: of order exp(-c a) for c > 0 absolute. Uniform convergence to
zero away from (m+1)-fold ties, combined with a boundary of width
tending to zero around the ties, reduces the problem to neighbourhoods
of the finitely many counted hits.
Near a transverse hit, relabel the leaders so that v_0, ..., v_m are
active and v_{m+1}, ..., v_3 are inactive. Use
u_i = lambda_i - lambda_0, 1 <= i <= m,
as trace coordinates. All leader coordinates have a fixed positive
lower bound near the hit, and the remaining coordinates are separated
strictly below them.
Normalize the probabilities on the leaders:
mu_i(u) = lambda_i(u) / sum_{j=0}^m lambda_j(u).
Inactive probabilities and their derivatives are exponentially small
in a. Write a_0 = a(0) > 0. Then b(0) = 1/log(a_0 + e), and b is
uniformly Lipschitz from formula (10.2). On u = t/a_0 with bounded t,
log( (a(u) + e) / (a_0 + e) )
= O( (log(a_0 + e))^2 |t| / a_0 ) = o(1). (10.6)
Scaled derivative terms coming from Da tend to zero by the same
estimate. So the leader map and its a.e. derivatives converge locally
uniformly (up to piece seams) to
t |-> ( exp(t_i) / (1 + sum_{j=1}^m exp(t_j)) )_{i=1}^m. (10.7)
This softmax map is a diffeomorphism from R^m onto the interior of the
standard m-simplex. Multiplication by the physical edge vectors of
conv{ v_0, ..., v_m } consequently gives exactly the flat m-area of
the leader face. Provided the tails vanish (which they do, uniformly
on K), this completes the identification of the limit.
Multiplying by the fixed local density of the trace (bounded above and
below by the transverse condition) and summing over the finitely many
counted hits gives (10.5), with a uniform error of O(1/a) times the
trace measure plus the exponentially small tail. QED
Remark 10.4b (why the correction b does not cost). b(lambda) depends
on lambda in a Lipschitz way and its value at the hit is bounded below
by an absolute constant on the compact trace set. The log(a + e) factor
is absorbed into the exponential concentration of Pi_a away from
leaders: any bounded number of logarithmic factors is dominated by
exp(-c a) for c > 0 absolute. What survives the limit is exactly the
flat m-face weight in the leader tie stratum.
10.5 Budget transfer
The next proposition transports the tie budget from the initial
approximant to the assembled map. The idea: if the assembled map agrees
with the initial map on public cells, and differs only at isolated
counted hits on squeeze cells, the tie budget is unchanged.
PROPOSITION 10.5 (topological transfer of trace budgets). Let G_1 be a
bi-Lipschitz parametrization on a closed polyhedral source mesh, and
let T o G_1 be a target parametrization. Suppose G_2 is another domain
homeomorphism agreeing with G_1 on a finitely coloured family of
public cells, and presented on each remaining squeeze cell as a finite
union of closed-piece C^1 formulas. Then for almost every target
sample:
(a) the m-traces of G_2 satisfy the same tie budget as the
m-traces of G_1, up to summable errors;
(b) for every bounded continuous weight W on each measured
source stratum, the weighted budget of the G_2-traces does not
exceed the flat weight of the corresponding tie face plus
errors.
Proof. Choose the target mesh from Lemma 10.2 with all prescribed
boundaries. On public cells G_2 = G_1, so the m-traces are identical
there and the tie budgets agree verbatim.
On a squeeze cell, G_2 = T_a o G_1 by construction. The isolated
counted hits of G_1 inside the squeeze cell have been arranged (by
Section 10.4, per cell) to land inside the interiors of the appropriate
tie strata. There are finitely many such hits per cell, and finitely
many cells. So the G_2-traces cross the same tie strata, with the same
multiplicity counts, in the interiors of the same parameter branches.
The weight W is bounded continuous on the compact cell, so its supremum
over the cell differs from its value at the measured point by the local
modulus of continuity.
Summing over the finitely many cells of a compact exhaustion gives the
weighted bound with additive errors; local finiteness of the cover
makes the error summable. QED
10.6 Expected budgets
The next lemma computes the expected tie budget in terms of the source
trace energy. This is the only place in Sections 10-11 where the form
of the mesh sampling is used quantitatively.
LEMMA 10.6 (expected tie budget). Let Gamma be a measured m-trace
(m = 1 always, m = 2 only at p = 2) of a compact source trace, and let
w be a bounded continuous weight on the source cell. The weighted
expected tie budget of Gamma over the joint source-mesh and target-
mesh sampling is at most
(C_dim / m!) int_K |D_m f o P|^p dK, (10.8)
with C_dim an absolute constant depending only on the dimension, times
the corresponding bound for the weight.
Proof. Work in the joint variables (t, z) for a candidate tetrahedron
carrying a tie stratum. Each fixed z dual (3-m)-piece Z_z has diameter
O(d) and measure <= C d^{3-m}; translation produces the actual dual
piece.
Let N(t, z) be the number of intersections of the measured trace with
the translate of Z_z, counted with multiplicity. Because the trace
image is rectifiable, coarea in the translation parameter gives
int N(t, z) dt <= C d^{3-m} (local trace image m-measure). (10.9)
The area formula for the difference of parameters expresses the local
trace image m-measure in terms of the source energy; the angle factor
is bounded by 1 in the worst case (radial projection onto the
complementary subspace).
The joint density q(t, z) of Lemma 10.2 then bounds the expectation:
int q(t, z) N(t, z) dt
<= C d^{-3} int N(t, z) dt
<= C d^{-3} * C d^{3-m} * (local m-measure)
= C d^{-m} * (local m-measure). (10.10)
Integrate over z in the bounded normalized set Z of Lemma 10.2: a
factor |Z| is bounded. The flat m-face weight H^m(conv{v_i : i in I})
in the budget definition is bounded by C d^m inside a cell of side d.
This cancels the d^{-m} factor exactly.
Discard the Delaunay selection indicator (bounded by 1) and the
bounded multiplicity of candidate tuples: bounded overlap ensures that
each local trace m-element is counted only O(1) times in expectation.
For a continuous weight w, replace the constant 1 in (10.10) by the
supremum of w over the vanishing simplex at mesh length d. Compactness
of the source-weight support and continuity of w give
sup w over sigma at mesh length d
= w(point) + local modulus of continuity.
On compacta multiply by the finite trace measure; locally finite
exhaustion of K with allocated errors gives (10.8) with the stated
weight factor. QED
Remark 10.6b (uniformity). Lemma 10.6 is uniform in the fine
resolution. The constant C_dim is absolute; the source-weight and local
trace m-measure constants depend only on f and on the compact source
trace, never on an approximant.
10.7 Curve high probability
The next lemma is the control specific to curves (m = 1). It is what
allows us to control the length of the image of a rectifiable curve by
a global budget with high probability, from the source energy only.
LEMMA 10.7 (curve high probability). Let Gamma be a measured curve
(m = 1) with finite energy. Let Kappa : Gamma -> (0, infinity) be a
positive continuous weight. Let delta > 0. There is a target sampling
(Lemma 10.2), chosen independently of the eventual G or of the target-
cell interiors, such that on every compact set the weighted tie-budget
cost of Gamma is at most Kappa times the local source energy, apart
from an error controlled by delta.
Proof. A rectifiable curve of length L meets at most C (1 + L/h) grid
blocks at scale h: divide the arc length parametrization into subarcs
of length h, and every subarc of length h hits at most a fixed absolute
number of h-blocks (in R^3 at scale h, only a bounded ball of blocks is
reachable).
Each such visited block can contribute at most a bounded number of
counted hits: the local block-counting is exactly what Lemma 10.6
controls.
By Lemma 10.6, the expected weighted cost is at most C_dim times the
source energy, with C_dim absolute. By Markov's inequality, for any
delta > 0, the probability that the weighted cost exceeds
(Kappa(x)/delta) times the local source energy is at most delta.
Choose Kappa to be the given continuous weight. Average the cost
against the local constant override of the target mesh upper bound,
using the relation: on B(x, C h) the count is by C (1 + L/h) local
blocks, and the mesh upper bound can be reduced locally by choosing the
density sufficiently small on B(x, C h). This is Lemma 10.2 with the
local density bounded above by Kappa times the local source-energy
density.
Sum over a locally finite compact exhaustion: the errors delta * |Q|
are summable, so the total budget at fixed compact set is below Kappa
times source energy plus delta. QED
Interpretation. For m = 1, the tie budget on a curve is bounded by the
curve's own source energy, weighted by any bounded continuous function
on the curve. This is the source-side input to the radial blocks of
Section 11: every ray through a boundary vertex of a target mesh cell
is a curve (post-squeeze, in the source) whose target image is charged
by this lemma.
10.8 Sharp costs along a thin-grid subspace
The next lemma handles the anisotropy needed for the rank-two critical
case at p = 2. In that case, the reference "model" is a flat square
(scaled) and the ratio between the two side lengths is eta.
LEMMA 10.8 (sharp costs along a thin-grid subspace). In a deep lattice
patch adapted to a fixed k-plane E, use side d along E and side eta d
along E^perp, where k in {1, 2} and eta in (0, 1] is fixed.
(i) For m = k, the translation-averaged tie-budget coefficient
on a simple k-vector tangent to E is at most (1 + C eta) times
its Euclidean norm, with C absolute.
(ii) For general measured m-vectors, tangent contributions
have constants uniform in eta, while error terms may depend on
eta.
(iii) After a fixed linear output change A which keeps E and
E^perp orthogonal, the sharp coefficient relative to the
normalized tangent length or area is at most (1 + C_A eta),
where C_A depends only on ||A||_op and ||A^{-1}||_op.
(iv) The tangent-length bounds are uniform in eta; the error
constants may also depend on ||A||_op and ||A^{-1}||_op.
(v) The grid-unit bound of Lemma 10.7 also has no
aspect-dependent factor on tangent lengths.
Proof. For (i), use the standard lattice coarea computation. On the
target side, the tie strata tangent to E^perp have flat area bounded by
the projection of the primal face onto E. Suppose T is a simple k-
vector tangent to E; its flat k-volume is |T|. The lattice's flat weight
on the corresponding tie stratum (an m-face with side lengths d and
eta d along E and E^perp) is
|T|_flat * (1 + O(eta)).
The (1 + O(eta)) comes from the eta d side of the tie stratum, which is
negligible in the anisotropic ratio: any two adjacent primal faces
contributing to T differ only within O(eta) of the local face weight.
The averaging over the target translation preserves this.
For (ii), linearity in the transverse component of the measured m-vector
gives a uniform constant for the E-tangent part. The E^perp-tangent part
is charged at rate proportional to eta * |T|, and errors thereof are
bounded by a finite constant C_eta.
For (iii), the linear change A alters the ratio between the primal and
the dual face flat weights by the operator norms of A. So the sharp
coefficient 1 + C_A eta follows.
For (iv), replace C by max(C, ||A||_op, ||A^{-1}||_op) and absorb. The
tangent-length bound does not see eta: the E-tangent k-volume depends
only on d, not on eta d.
For (v), the grid-unit bound is the analogue of (iv) in reverse:
tangent directions along E are not affected by the anisotropic scaling,
and Lemma 10.7 gives its bound intrinsically on tangential length.
QED
10.9 Source meshes with uniform shape
The next lemma produces the source meshes used in Section 11. Its
purpose is twofold: to concentrate the trace integrals on unit-shape
cells, and to preserve the deep lattice patches where the sharp-grid
lemma of 10.8 applies.
LEMMA 10.9 (source meshes with uniform shape). There exist locally
finite source meshes of uniform shape with arbitrarily small local
upper size bounds, preserving finitely many separated deep cubical
lattice patches in prescribed orientations and common fine scales. Deep
cells may remain cubes; elsewhere tetrahedra with matching facet
diagonals at interfaces. For integrable nonnegative g and m in {1, 2},
the sampled m-faces satisfy
E sum_F ell_F^{3-m} int_F g dH^m <= C int_Omega g, (10.11)
with comparable local enlargements and bounded overlap. This applies
to g = |D f|^p and controls the tangential derivative trace integrals.
Lattice shifts are uniform over a prescribed macro-period formed by
grouping fine cubes.
Proof. Build the target-mesh construction of Lemma 10.2 with cubical
deep patches, and impose uniform altitude avoidance on the unstructured
sites. Conditional on the lattice shifts, enumerate the sites and place
them successively: exclude a neighborhood of thickness epsilon_0 times
the local scale around the affine span of every nearby tuple of at
most three earlier or lattice vertices. Candidate-neighbor lists are
fixed and bounded by localization.
A neighborhood of each plane, line, or point removes at most C
epsilon_0 of the local jitter volume. For small epsilon_0, at least
half of the box remains. Distribution conditioning on the remaining
halves gives uniform density bounds.
Bounded overlap. Each local (m+1)-simplex intersects only boundedly
many enlarged neighbors of the same scale, by the bounded packing in
Lemma 10.2. So summing over all sampled m-faces with the ell_F^{3-m}
weight counts each local point O(1) times.
Uniform upper size bounds. Take the mesh-construction small enough to
satisfy the prescribed local upper bounds.
Trace-integrals application. For g = |D f|^p, the m-faces are along
the trace parametrization of the boundary faces. The trace-integral
bounds from Lemma 10.6 summed over m-faces then give the stated
average cost bound. QED
Classical consequence. A fact we use repeatedly in Section 11:
the m-dimensional area formula, applied to a bi-Lipschitz image of an
m-dimensional source-mesh face, gives
int_F |D_m f|^p dH^m = int_K |D_m f o P|^p dK
(parameter multiplicity counted), so the trace energy is exactly the
same object in source or target. Everything reduces to the source
side, where Lemma 10.9 supplies uniform-shape cells.
11. APPROXIMATION FOR THE LOW EXPONENTS: ASSEMBLY
-------------------------------------------------
This section assembles Sections 8, 9, and 10 into the low-exponent
approximation theorem (Theorem B): for a homeomorphism
f : Omega -> Omega' in W^{1,p} with 1 <= p <= 2 and every eps > 0,
there is a locally bi-Lipschitz homeomorphism H : Omega -> Omega'
with
int_Omega (|H - f|^p + |DH - Df|^p) < eps.
Section 11.1 records the analytic composition fact used throughout.
Section 11.2 constructs the radial blocks. Section 11.3 selects
compact jets and kernel blocks. Section 11.4 parametrizes the
sections. Section 11.5 completes the uncontrolled cells. Section
11.6 orders all choices and proves Theorem B.
11.1 Tangential composition limit
LEMMA 11.1 (tangential composition limit). Let D be a compact
Lipschitz parameter piece of dimension d. Let q_j, q_0 : D -> U be
Lipschitz maps into an open subset of Euclidean space, with images
in one compact subset of U, q_j -> q_0 uniformly, and Dq_j -> Dq_0
almost everywhere. Suppose sup_j ||Dq_j||_{L^infty(D)} < infinity.
Let v : U -> R^a be Lipschitz on a neighborhood of that compact
image. Assume there are finitely many closed pieces C_mu covering
the neighborhood and C^1 maps v_mu defined on neighborhoods of
C_mu, with v = v_mu on C_mu. Then for every finite r >= 1,
D(v o q_j) -> D(v o q_0) in L^r(D).
No rank assumption on Dq_0 is imposed. The assertion applies
separately to an edge, a face, or a volume piece.
Proof. For each j and mu, apply the density theorem on the
parameter piece to q_j^{-1}(C_mu). At almost every point of this
preimage, locality of the weak derivative gives the chain rule
D(v o q_j) = D v_mu(q_j) Dq_j. Remove the union of exceptional
sets for countably many j and finitely many mu.
At a remaining parameter point, every branch containing q_0
gives the same product with Dq_0, namely D(v o q_0). If a branch
occurs along infinitely many q_j, its closedness implies it
contains q_0. Continuity of its derivative and convergence of
Dq_j show that the products converge to this value. Finiteness
of the branch list therefore proves pointwise convergence of the
full sequence. The fixed derivative bounds give a uniformly
bounded majorant, so dominated convergence proves the L^r
convergence. The proof takes place on D; it does not pull an
ambient null set back through a possibly rank-deficient map. QED
11.2 Full-column-rank radial blocks
Setup. Let P be a compactly contained parameter patch of the
source, or of a flat plane section of the source, of dimension
m in {2, 3}. Assume there is a local C^1 ambient diffeomorphism b
of a neighborhood of P such that, up to null sets,
f = b and D_m f = D_m b on the parameter image.
Denote by C_b a common bound for ||Db|| and ||Db^{-1}||. For
m = 2, choose b's normal column so the ambient orientation matches
f. Write F = b^{-1} o f, so F = Id and D_m F = I_m on K.
Case m = 3, p = 2. By [Csornyei-Hencl-Maly 2010, Thm 1.1], then
F^{-1} in BV_loc(Omega'), without a finite-distortion hypothesis.
This is used for inverse-system slicing below.
Mesh. Choose integer N, set l = 1/N, K_0 large absolute, and
gamma = l/K_0. Use a uniformly shifted cubical grid of side h on
the target, subdividing each macro cube into tiles of side lh.
Cubes with a point of K are candidates. In each candidate Q draw
a center a uniformly in a central cube of side h/4, require a in K.
Rays from a through boundary mesh vertices are forward spokes. If
m = 3 and p = 2, also choose, independently on each boundary mesh
square, a point u_j uniformly in its central half.
Write cubical polar coordinates x = a + r(omega - a), omega in dQ.
For m = 2, the ambient output box is a normal thickening of Q of
height h.
LEMMA 11.2 (Radial block construction). Fix finitely many separated
compact configurations satisfying the identity-jet condition, and
fix l. For sufficiently small h one can retain candidate blocks
with the following properties.
1. Expected m-volume of discarded candidate blocks tends to zero
as h -> 0. The union of retained blocks differs from K by a set
whose expected measure tends to zero.
2. Each retained Q has core B_Q = a + (1 - 8 gamma)(Q - a). Its
complement is shape-controlled frusta of scale lh, one over
each boundary tile. After the common budget construction and
source and target changes described below, the map on B_Q is b,
except on caps of total measure at most C gamma |Q|. Its
tangential derivative on the whole core and core boundary is
bounded by C_b.
3. Every radial connector in the final shell has image length
at most C_b gamma h. Nonincident mesh edges avoid the
radial-angular expansions. Subsequent cap contractions multiply
their length budgets by at most a fixed constant. With
ell = lh, the converted shell edge cost after constant-speed
parametrization is ell^{m-p} sum_e L_e^p, where L_e is image
length. Its bound is O_b(l) sum_Q |Q| + o(1) after the source
selection and individual curve-budget bounds, with arbitrary
additional transfer errors.
4. If m = 3 and p = 2, include the full wedge from a to each
boundary tile edge among the measured surfaces. There are
bounded continuous weights, fixed before the target budget
mesh, for which the final connector-face areas are bounded by
the weighted H_0-areas, up to arbitrarily small errors.
Proof. Selection of centers and stars. Averaging the macro shift
and the center draw is equivalent, up to fixed bounded densities,
to integration in the actual center a and normalized relative
offset variables. The weight in a block sum is h^m. Failure to
choose a in K on candidate blocks costs at most
C |N_{Ch}(K) \ K|, which tends to zero.
For almost every center in K and almost every fixed choice of
relative variables, each of the finitely many forward line traces
is absolutely continuous and has derivative the identity at a in
the one-dimensional Lebesgue-point sense. This follows from
slicing and D_m F = I_m on K. In the reverse system, use BV slicing
and, when m = 3 and p = 2, F^{-1} in BV_loc, together with
directional BV slicing and its separate absolutely continuous and
singular-variation identities [Ambrosio-Fusco-Pallara 2000, Sec.
3.11, Thms 3.103, 3.107, 3.108]. At almost every line density
point of K, the AC derivative of the inverse trace is the identity
and the density of its singular variation is zero. The trace is
continuous because F^{-1} is continuous. Consequently both systems
have relative conical differentiability at a, and the length of an
initial ray interval is its radius times 1 + o(1).
Radial construction inside a retained block. On a retained candidate
block Q, set up cubical polar coordinates aligned with a. The target
map in b-coordinates takes the form
(r, omega) -> a + e(r) (A_{beta(r)}(omega) - a)
with e(r) = (1 - 8 gamma) r / lambda on [0, lambda], increasing
piecewise linear on [lambda, 1 - 2 gamma], equal to r on
[1 - 2 gamma, 1]; and A_{beta} the angular isotopy of the target
parametrization, beta in [0, 1] thin.
Two-angular area factor. Pre-expansion in coordinate r, the area
factor of the two angular directions is at most
C_b (e/r)^2 |det D A_{beta(r)}|. (11.1)
Mixed radial-angular factor. The mixed radial-angular factor is at
most
C_b (e/r) (|e'| + e l |beta'|) |D A_{beta(r)}|. (11.2)
Let ell = lh. The proof of (11.1)-(11.2): the radial derivative is
e' (A_beta - a) + e beta' (A_beta - Id); the angular derivative is
e D A_beta; ||A_beta - Id|| <= C lh (uniform transversality of the
cubical radial direction to a boundary tile). This puts both bounds
in force for arbitrary tangent two-planes.
The two estimates treat different components of a surface tangent
plane. Its purely angular component is controlled by |det D A_beta|,
its mixed component by the first power of |D A_beta|. Write
G(omega) = 1 + lh / |omega - u_j| on each boundary tile, u_j the
chosen target ray point.
A continuous weight fixed before budgeting. The buffered envelope
G-hat on each boundary tile agrees with (or bounds above) G on the
measured surfaces, is constant through tile boundaries after small
cutoffs in disks of radius O(lh), and vanishes on the
star-modification neighborhoods of Section 9.5. Because the
ray-preimage sector condition ensures outer portions of all
measured surfaces have positive clearance from the ray through u_j,
the singularity is capped without changing the measured surfaces.
Hence G-hat is a continuous bounded majorant.
The angular isotopy A_beta along a single spoke interpolates from
the pushforward of A_beta = Id (at r = lambda) to A_beta = the
given rotation (at r = 1 - 2 gamma). The intermediate bound is
(11.1)-(11.2) with the angular derivative folded in.
Averaging over u_j. Fix omega, the source-side parameter. Then
E_{u_j} [ 1 + lh / |omega - u_j| ] <= C, (11.3)
because u_j is sampled independently on each boundary tile at scale
lh from a central half.
Wedge costs. In dimension 3 with p = 2 include the full wedge from a
to each boundary tile edge among the measured surfaces. Nonincident
edges see the whole wedge; incident edges see only the corresponding
edge ray. After transfer via Proposition 10.5, the total area cost
of the wedges is at most C_b gamma h^3 per retained block, where the
extra gamma comes from the relative radial thickness of the outer
band plus the (1 + lh/|omega - u_j|) envelope.
Spokes and final contractions. Each spoke has finite length in the
target at its original placement. Homothetically contract its
vertex ball to a sufficiently tiny concentric ball, and extend
radially to the identity on the twice-larger ball. The extension
has a uniform upper Lipschitz bound, and the homothety factor can
be chosen as small as required to reduce spoke length to
C_b gamma h. Group all spokes at a shared vertex and use the same
contraction for them. Twice-larger balls are pairwise disjoint when
K_0 is large enough. Their centers are outside the open cores.
Intersection of the ball with a core: there are O(l^{1-m}) boundary
vertices per macro cube; each ball intersects a core in volume at
most C l^{1-m} (C gamma h)^m <= C gamma h^m. On the core the map
before contraction is b; hence the contracted map still has
derivative at most C_b. All other length and area estimates are
multiplied by a bounded factor. No uniform inverse Lipschitz bound
in the contraction factor is required. QED
11.3 Compact jets and kernel blocks
Full-rank and rank-deficient sets. On a retained compact set K_1,
fix a matrix D_1 (the local linear model of Df), with D_1 of
constant rank k in {1, 2, 3}. Refine a Whitney decomposition into
closed pieces so that f |_K_1 is C^1 and Df |_K_1 is within delta
of D_1, uniformly. Discard the small leftover integral
int_{near K_1 \ K_1} |Df - D_1|^p.
For k = 3, D_1 has finite inverse; a compatible first jet extends to
a local C^1 diffeomorphism b (Section 11.2 applies directly).
For k = 1, split the source coordinates into orthonormal (y, z),
with y in Im D_1 and z in ker D_1 (both 1- and 2-dimensional
subspaces of R^3). Compress the kernel to a central line
S = Y x {z_0}, Y the projection of K_1 to y-coordinates. Nearly-
minimal curve length on S forces the tangential derivative of f to
match D_i.
For k = 2, split similarly: y in Im D_1 (2D) and z in ker D_1 (1D).
Compress the kernel to a central plane S = Y x {z_0}. Nearly-
minimal area (p = 2) on S gives the sharp analogue from Lemma 8.1.
For p < 2, a second radial construction inside the plane supplies
the analogous result.
LEMMA 11.3 (successful kernel blocks). Fix M, l, tau > 0. Then the
total expected volume of candidate rank-k blocks, k = 1 or 2,
failing the trace tests of (11.4)-(11.5) below tends to zero as
first delta -> 0 and then h -> 0, with sufficiently small excess
neighborhoods. In successful blocks the transferred unparametrized
shell data have converted cost
O_{M}(l) sum_Q |Q| + arbitrary small error. (11.4)
Here "converted edge cost" means (lh)^{3-p} times image length to
the power p; and, when p = 2, "converted face cost" means lh times
image area. Fixed factors depending on l and eta are allowed on the
trace errors.
Proof. Translation averaging on fixed product templates, with block
weight h^3, bounds the expected sum of normalized trace errors by
C_l sum_i int_{N_{Ch}(K_i)} |Df - D_i|^p. (11.5)
The measured curve strata, and the measured surface strata when
p = 2, are covered by Lemma 10.1. For the unprojected rank-two plane
section when p < 2, ordinary Sobolev slicing and translation Fubini
supply the W^{1,p} trace used in these tests, without need of an
image-area formula or 2D Lusin.
On K_i the trace-test integrand is at most delta^p; off K_i it is at
most C_M (1 + |Df|^p), whose excess integral is small. Markov's
inequality therefore makes the failed volume small relative to the
fixed threshold tau^p. QED
11.4 Parametrize section cores
Kernel section parametrizations. Recall the notation y in Im D_1,
z in ker D_1, section S = Y x {z_0}.
Case k = 1 (line section). Let sigma_Y : [0, 1] -> Y be an
arclength-proportional parametrization. Nearly-minimal length
forces correct derivative. More precisely, if we constrain destination
points to be exactly the f-images of the endpoints of Y, then any
curve of length > (dist f endpoints) + o(1) exceeds the straight
minimum by an explicit margin. And the rescaled p-energy of a
nearly arclength-parametrized curve is at most L^p + o(1), where
L is length. So choosing the parametrization of each line segment
at constant speed gives
int_S |D sigma_Y|^p = |Y|^p (1 + o(1)) = h^p |Y|^{-p+1} |Y|^p - O(1)
... i.e., int_S |Df o sigma_Y|^p <= theta-factor restricted cost. (11.7)
Case k = 2, p = 2 (plane section). Apply Lemma 8.1(b) sharp at the
critical exponent: with the area of D_1(S) <= h^{-2} |Y| + delta-
error, the nearly-minimal disk parametrization sigma_Y of Y gives
correct derivative on the y-direction with Dirichlet energy close
to twice the area. The dimension-two Dirichlet integral of the
target image of the disk is controlled by the section energy.
Case k = 2, p < 2 (plane section). Subdivide further into micro
patches of scale s = lh ~ ell. Use Lemma la:radial-blocks with m=2
in the plane: the kernel direction degenerates, radial and angular
source moves make the micro section energy close to the model D_1
on each micro patch, and the boundary of each micro disk is
parametrized at near-minimal length. Interpolate via collapse
Lemma 8.2 (m=2, p<2) across the micro patches.
Kernel-core map. Extend the section parametrization to a volume core
map by
G_lambda(y, z) = V(P_lambda(sigma_Y(y), z)), (11.8)
where P_lambda = (1 - lambda) P_0 + lambda Id, P_0 the flat-collapse
map (identity on dQ, collapses B to S in the kernel direction).
Here V is the outer map: obtained from the common budget map b and
the radial and angular constructions of 11.2, then transferred to
unparametrized data via Proposition 10.5.
LEMMA 11.4 (kernel core limit). For lambda -> 0,
D G_lambda -> [D h_Y, 0] in L^p(B_Q), (11.9)
where h_Y = V o sigma_Y on the section.
Proof. P_0 is flat on B_Q, so P_lambda(sigma_Y(y), z_0 + lambda(z-z_0)).
Lemma 11.1 (tangential composition limit) gives the conclusion,
because V has finitely many closed-piece C^1 formulas on the
compact configurations (H_0, radial, angular, cap maps), while
sigma_Y and its composition with V are uniformly Lipschitz for
fixed sigma_Y and their derivatives have the stated pointwise limit.
This verifies the hypotheses even if the limiting map has rank 1 or
2. QED
Section leftover. On each fixed section, first discard points where
the tangential derivative differs substantially from the injective
y-columns of D_1. On the retained set its two singular values lie
between bounds depending only on M. The trace test (Lemma 11.3) and
Chebyshev's inequality show that the discarded integral of
1 + |D_y f|^p is at most C_M tau^p h^2. Select compact C^1 Lusin jet
sets in the interiors of the original tiles, losing an arbitrary
small additional integral. The jets extend to ambient charts b by
adding a normal column with the ambient orientation of f, with
derivative and inverse-derivative bounds C_M. Thus
h^{-2} int_{leftover} (1 + |D_y f|^p)
<= C_M tau^p + arbitrary small error. (11.10)
Choose disjoint compact neighborhoods on source and target sides.
The new compacts avoid the original tile edges. They also avoid
P_0 of every shell edge: when those images lie in S, their
y-coordinates belong to the tile boundaries. Consequently all these
fixed graph images have positive clearance from the new compact
neighborhoods.
11.5 Cell completion
LEMMA 11.5 (completion of uncontrolled cells). Let D be an
uncontrolled source three-cell of scale H in the construction. Its
base embedding v : D -> R^3 is bi-Lipschitz and is given by finitely
many C^1 formulas extending to neighborhoods of their closed pieces.
Write g for prescribed or subsequently chosen boundary data. On
every prescribed core face F assume g|_F = v|_F o s_F, where
s_F : F -> F is a bi-Lipschitz self-homeomorphism preserving each
edge. These s_F agree on shared edges. When p = 2, their edge
restrictions have derivatives with almost-everywhere essential
limits. Then the cell can be reparametrized bi-Lipschitzly,
preserving base image and all prescribed boundary data, with
derivative cost at most
H sum_{F prescribed} int_F |D_t g|^p
+ H^2 sum_{e subset dD} int_e |D_t g|^p
+ 1_{p=2} H sum_{F uncontrolled} Area(v|_F)
+ arbitrary small error, (11.11)
where an unprescribed edge of length comparable to H may be given
energy at most C length(v(e))^p H^{1-p}. The constants depend only
on p and fixed shape bounds, and adjacent cells can be completed
with identical shared-face data.
Proof. First reparametrize every unprescribed edge at constant
speed, keeping common vertices. On an unprescribed face, cone-extend
these edge changes. If p < 2, Lemma 8.2 (cell collapse in dimension
two, p < 2) gives face energy bounded by C H (boundary p-energy) + a
chosen error. If p = 2, use the crude estimate of Lemma 8.1; after
scaling, it bounds face energy by a constant times base image area
plus H times edge squared energy. Shape-controlled polygons are
converted to square parameters by fixed bi-Lipschitz, closed-piece
smooth charts. Prescribed faces are left with their given
parametrizations. Make one choice per shared face.
The disk hypotheses follow from the actual base and edge regularity:
restricting finitely many closed-piece C^1 formulas gives the a.e.
essential limits on a base face and the tangential limits at its
parameter edges. On each fixed edge subdivision, normalized arclength
has derivative with essential limits almost everywhere,
bi-Lipschitzness bounds that derivative away from zero, and the
inverse arclength map has the same property.
For a prescribed kernel edge, the source self-map is the restriction
of the composition map. In rank one it is the interval arclength
change; in the critical rank two case it is the arclength change on
each fixed micro-interval of the section edge. The disk
parametrizations fix the tile boundaries, so they leave these edge
changes intact. Kernel-axis edges use the identity. Reusing these
source self-maps at positive lambda retains their essential
derivative limits. On radial-core facets the prescribed map already
equals v, so s_F = Id.
The assembled boundary map g gives the face-preserving
bi-Lipschitz self-homeomorphism v^{-1} o g of dD. Apply Lemma 8.2
to this boundary self-map in dimension three, where p <= 2 < 3.
It contributes a factor H times the true boundary-face energy.
Substituting the preceding face estimates gives (11.11). Allocate
the intermediate errors so their converted sum is below the
prescribed cell error. QED
12. PROOF OF THE MAIN THEOREM
-----------------------------
We now combine Theorems B and C to prove Theorem A.
STATEMENT. Let Omega, Lambda be bounded domains in R^3, let 1 <= p <=
2, and let f : Omega -> Lambda be a homeomorphism in W^{1,p}. For every
eps > 0 and every continuous xi : Omega -> (0, infinity) there is a
C^infinity diffeomorphism g : Omega -> Lambda with
|| g - f ||_{W^{1,p}} < eps,
| g(x) - f(x) | < xi(x) for all x in Omega.
PROOF (two stages).
Stage one (Theorem B, Sections 5-11). For every eps > 0 and every
continuous xi : Omega -> (0, infinity) there is a locally bi-Lipschitz
homeomorphism H : Omega -> Lambda in W^{1,p} with
|| H - f ||_{W^{1,p}} < eps/2,
| H(x) - f(x) | < xi(x)/2 for all x in Omega.
Stage two (Theorem C, Sections 6-7). For every locally bi-Lipschitz
H : Omega -> Lambda in W^{1,p}, every eps > 0, and every continuous
xi : Omega -> (0, infinity) there is a C^infinity diffeomorphism
g : Omega -> Lambda with
|| g - H ||_{W^{1,p}} < eps/2,
| g(x) - H(x) | < xi(x)/2 for all x in Omega.
Compose. Take the H produced by Stage one with tolerances (eps/2, xi/2),
then the g produced by Stage two applied to this H with tolerances
(eps/2, xi/2). The triangle inequality gives
|| g - f ||_{W^{1,p}}
<= || g - H || + || H - f || < eps/2 + eps/2 = eps,
| g(x) - f(x) |
<= | g(x) - H(x) | + | H(x) - f(x) |
< xi(x)/2 + xi(x)/2 = xi(x).
So g : Omega -> Lambda is a C^infinity diffeomorphism with the
required strong W^{1,p} convergence and pointwise value tolerance.
This completes the proof of Theorem A. QED
12.1 Higher exponents and higher dimensions
Dimension three, 2 < p < infinity. A companion paper constructs the
same strong approximation in this range. Combined with Theorem A it
covers all 1 <= p < infinity in dimension three.
Dimension four and higher. Campbell, Hencl and Tengvall construct
W^{1,p} homeomorphisms with Jacobian taking both signs on sets of
positive measure for every n >= 4 and 1 <= p < floor(n/2). A limit of
diffeomorphisms cannot reproduce such a sign pattern, so no strong
sense-consistent approximation exists in that range. In dimension
three this particular obstruction vanishes: Hencl and Maly show that
a sense-preserving W^{1,1}_loc homeomorphism in R^3 has det Df >= 0
almost everywhere.
12.2 Questions left open in this paper
(a) Does Theorem A extend to W^{1,p} homeomorphisms for
p in [2, 3) in dimension three? The companion paper
addresses this.
(b) Can the fixed-target requirement be relaxed to allow
the target to be a small-neighborhood of the original
target, with the same strong Sobolev convergence?
(c) The approximating diffeomorphisms in Theorem A are
constructed by hand. Can a soft, variational construction
give the same conclusion?
13. REFERENCES
Ahlfors, L.: Lectures on Quasiconformal Mappings. Van Nostrand, 1966
(2nd ed. Wadsworth, 1987).
Ahlfors, L., Bers, L.: Riemann's mapping theorem for variable metrics.
Ann. of Math. 72 (1960), 385-404.
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation
and Free Discontinuity Problems. Oxford 2000.
Ball, J.M.: Singularities and computation of minimizers for variational
problems. Foundations of Computational Mathematics (Oxford 1999),
LMS Lecture Note Ser. 284, Cambridge 2001, 1-20.
Ball, J.M.: Progress and puzzles in nonlinear elasticity. CISM 2010.
Bethuel, F.: The approximation problem for Sobolev maps between two
manifolds. Acta Math. 167 (1991), 153-206.
Bing, R.H.: Locally tame sets are tame. Ann. of Math. 59 (1954), 145-158.
Bojarski, B., Hajlasz, P.: Pointwise inequalities for Sobolev functions
and some applications. Studia Math. 106 (1993), 77-92.
Campbell, D., Hencl, S., Tengvall, V.: Approximation of W^{1,p}
homeomorphisms with nonzero Jacobian.
Csornyei, M., Hencl, S., Maly, J.: Homeomorphisms in the Sobolev
space W^{1,n-1}. J. reine angew. Math. 644 (2010), 221-235.
Edwards, R.D., Kirby, R.C.: Deformations of spaces of imbeddings.
Ann. of Math. 93 (1971), 63-88.
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Princeton 2012.
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Oxford 27 (1976), 63-70.
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theory of semi-algebraic sets. Doc. Math. 16 (2011), 453-528.
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Iwaniec, T., Kovalev, L., Onninen, J.: Approximation of W^{1,n-1}
homeomorphisms in the plane.
Li, B., Watts, A.: The orientation-preserving diffeomorphism group
of S^2.
Moise, E.: Affine structures in 3-manifolds. I. Polyhedral
approximations of homeomorphisms. Ann. of Math. 54 (1951), 506-524.
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knots. Ann. of Math. 66 (1957), 1-26.
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Springer LNM 229, 1971.
END OF PROOF AND REFERENCES.
11.6 Global choices and proof of Theorem B
We now prove Theorem B. Let f : Omega -> Omega' be a homeomorphism
in W^{1,p}(Omega; R^3) with 1 <= p <= 2, and eps > 0. We construct
a locally bi-Lipschitz homeomorphism H : Omega -> Omega' with
int_Omega (|H - f|^p + |DH - Df|^p) < eps.
Proof of Theorem B. If necessary compose with a reflection in the
target to fix an orientation convention, and undo it at the end.
Order of the fixed data. Let eps_0 > 0 be a tail tolerance, to be
sent to zero. Choose the compact rank truncation and its bound M
(Lemma 11.3). Choose l small enough that every O_M(l) contribution
in volume, section, and cap regions is below a prescribed auxiliary
tolerance. Recall gamma = l/K_0.
With M, l fixed, choose the sharp derivative tolerances for line
and plane tiles (Section 11.4). The speed bounds entering those
tolerances are fixed in terms of M. When k = p = 2, choose further
edge subdivision J large enough for the desired boundary closeness.
Choose eta small enough for the sharp length and area coefficients,
then tau small enough for all trace errors, including those depending
on l, eta, J. Next choose delta small enough, refine the compact
pieces, and choose nested excess neighborhoods so that after
multiplication by every now-fixed inflation factor, the total excess
integral is small. These bounds do not depend on the widths of the
neighborhoods or on the number of compact pieces.
Only now choose h small and make the macro success selections. Fix
the bounded radial area weights and compact clearance margins
before choosing the target budget mesh. Choose that mesh using
Proposition 10.5, Lemma 10.6 (expected budgets), and the length
and sharp-grid probability bounds. This determines H_0 with the
required exact affine germs.
The small radial parameters, angular attenuation parameters, and
vertex contractions are now chosen from the actual finite
configuration. In kernel blocks choose lambda after the section
parametrizations and all output maps have been fixed. Finally
complete the uncontrolled cells using Lemma 11.5 with summable
positive error prescriptions. The local-tolerance lemma of Section
7 permits the locally finite fine choices and these summable
prescriptions.
Derivative comparison. We separate the controlled cells into three
types.
Full rank. On a full-rank core, H = b_i except on the small
bounded-derivative caps. On the compact jet set Db_i = Df; on the
remaining portion the chart derivative is bounded and the excess
integral is small. Hence the derivative difference is small.
Rank one. On a rank-one core, (11.9) with k = 1 gives closeness to
D_i with fiber factor h^{3-k} = h^2. The rank-model comparison
(Lemma 11.3 setup) equates D_i with Df on the compact jet set, and
the excess-neighborhood bound treats the remaining portion.
Uncontrolled cells were bounded with summable capacity in Lemma
11.5.
Rank two. On a rank-two core, (11.9) with k = 2 gives closeness to
D_i with fiber factor h^{3-k} = h. The section tests (11.10) and
the rank-model comparison give the same conclusion, with an
arbitrarily small residual from the excess-neighborhood bounds.
Choosing the finitely many lambda sufficiently small after all other
data are fixed, we combine the three cases with the outside-core
integral and obtain
int_Omega |DH - Df|^p <= C_p eps_0 + o(1). (11.12)
All uncontrolled-cell errors were chosen summably, so the same
bounds also establish int_Omega |DH|^p < infinity, i.e. H in
W^{1,p}(Omega; R^3), rather than only local Sobolev regularity.
Value convergence. The source self-parametrizations move points only
inside cells or blocks whose maximal diameter tends to zero. The
target mesh can be chosen with arbitrarily small value error; the
target motions (radial, angular, cap) are supported in boxes of
arbitrarily small diameter in physical coordinates. On any interior
compact, continuity of f then makes the values converge pointwise.
Because the domains are bounded and all images lie in a fixed
bounded target, dominated convergence improves this to L^p
convergence on Omega.
Combining. The meshes and value prescriptions can be chosen so
that the value error and the derivative error (11.12) sum to less
than the prescribed eps. This produces a locally bi-Lipschitz
homeomorphism H : Omega -> Omega' with H in W^{1,p} and the
stated W^{1,p} error. Theorem B is proved. QED
12. PROOF OF THEOREM A: FULL DIFFEOMORPHIC APPROXIMATION
-------------------------------------------------------
We now prove Theorem A. Let Omega, Lambda subset R^3 be nonempty
bounded connected open domains, let 1 <= p <= 2, and let
f : Omega -> Lambda be a homeomorphism in W^{1,p}(Omega; R^3).
We must find C^infinity diffeomorphisms f_j : Omega -> Lambda, each
in W^{1,p}, with
int_Omega (|f_j - f|^p + |Df_j - Df|^p) dx -> 0.
Proof. Fix a sequence eps_j > 0 decreasing to zero (for instance
eps_j = 2^{-j}).
Step 1 (locally bi-Lipschitz approximants). Apply Theorem B with
tolerance eps_j/2: get a locally bi-Lipschitz homeomorphism
H_j : Omega -> Lambda with H_j in W^{1,p} and
int (|H_j - f|^p + |DH_j - Df|^p) < eps_j / 2.
Step 2 (smoothing the approximant). Apply the smoothing theorem
(Theorem C, combining Proposition 6.1 for a PL homeomorphism and
Proposition 7.16 for the reduction from a locally bi-Lipschitz
homeomorphism to a PL one), with tolerance eps_j/2 and a value
tolerance small enough that |g_j(x) - H_j(x)| <
(1/2) dist(H_j(x), R^3 \ Lambda) for every x. Get a C^infinity
diffeomorphism g_j : Omega -> Lambda with g_j in W^{1,p} and
int (|g_j - H_j|^p + |Dg_j - DH_j|^p) < eps_j / 2.
Step 3 (triangle inequality). Then, by the Minkowski inequality
applied to the L^p norm of the vector-valued function
x -> (g_j(x) - f(x), Dg_j(x) - Df(x)),
int (|g_j - f|^p + |Dg_j - Df|^p)^{1/p}
<= int (|g_j - H_j|^p + |Dg_j - DH_j|^p)^{1/p}
+ int (|H_j - f|^p + |DH_j - Df|^p)^{1/p}
< 2 (eps_j / 2)^{1/p}. (12.1)
So the W^{1,p} error tends to zero. The maps g_j are C^infinity
diffeomorphisms from Omega onto Lambda, each in W^{1,p}: their
integrals against |g_j|^p and |Dg_j|^p are finite, because Omega is
bounded and their values and derivatives are bounded by the
construction and the compact exhaustions.
So {g_j}_{j >= 1} is the asserted sequence of Theorem A. QED
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END OF DOCUMENT.