diff 99b3ba1..11c1f76 diffeomorphic_approx_3d_lp_le2_v2.txt
diff --git a/diffeomorphic_approx_3d_lp_le2_v2.txt b/diffeomorphic_approx_3d_lp_le2_v2.txt
index d129d59..2c0b336 100644
--- a/diffeomorphic_approx_3d_lp_le2_v2.txt
+++ b/diffeomorphic_approx_3d_lp_le2_v2.txt
@@ -1,19 +1,3 @@
-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
@@ -43,7 +27,7 @@ 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
+THEOREM A. 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
@@ -55,52 +39,6 @@ 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))
@@ -108,8 +46,8 @@ Continuous maps can fold. Consider
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.
+interesting: f may be far from differentiable, but only p-integrability
+in the Sobolev sense is required.
If injectivity is dropped, approximation is elementary: convolve f
with a smooth bump function, and both values and derivatives converge
@@ -121,6 +59,8 @@ 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.
+./ball-2001-singularities-computation-minimizers.pdf
+
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
@@ -132,12 +72,11 @@ 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 < p < infinity in R^3 calls for a separate construction.
-2. THE MAIN THEOREM AND CONVENTIONS
------------------------------------
+./iwaniec-kovalev-onninen-2011-diffeomorphic-approx.pdf
+./campbell-hencl-tengvall-2018-jacobian-signs.pdf
+./hencl-maly-2010-jacobians-sobolev-homeos.pdf
Throughout, a domain is a nonempty connected open set. All domains,
closures, and compactness statements are relative to the ambient open
@@ -145,7 +84,7 @@ 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.
+The following symbols appear throughout.
|A| Frobenius norm of a matrix A: sqrt(sum A_ij^2).
||A||_op operator norm.
@@ -175,7 +114,7 @@ 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
+reflection, 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
@@ -188,7 +127,7 @@ local injectivity.
Two theorems, besides Theorem A, organize the proof.
-THEOREM B (low-exponent approximation, Section 11). Let Omega, Omega'
+THEOREM B (low-exponent approximation). 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
@@ -197,7 +136,7 @@ bi-Lipschitz homeomorphism H : Omega -> Omega' with
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
+THEOREM C (smoothing). 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
@@ -207,13 +146,9 @@ diffeomorphism g : Omega -> Omega' with
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
+error, and use the triangle inequality. 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:
@@ -241,8 +176,13 @@ derivative estimates. The classical inputs are:
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.
+ Schoenflies, mapping class groups: the star replacement
+ described later.
+
+./hamilton-1976-triangulation-3-manifolds.pdf
+./papakyriakopoulos-1957-dehns-lemma-asphericity-knots.pdf
+./smale-1959.pdf
+./li-watts-2011-s2-diffeos-so3.pdf
These supply EXISTENCE of a homeomorphic replacement. They do not
estimate its derivative: a PL approximant might have arbitrarily large
@@ -257,7 +197,7 @@ slope. So the proof separates two things:
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
+has unbounded slope. The fix 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
@@ -265,8 +205,8 @@ 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
+topological extension theorem. The device is a
+BUDGET: 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
@@ -283,12 +223,8 @@ 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.
+Sketch of Theorem C (smoothing).
-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
@@ -334,8 +270,7 @@ 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)
-...................................
+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
@@ -373,28 +308,21 @@ 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
+Three completely topological facts must be recorded. 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.
+geometric shape from one local model to another. The subsequent
+arguments reduce the 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
+error bound there. Because the 3-manifolds in the 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
+For the statements needed, 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."
@@ -474,7 +402,7 @@ 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
+individual approximant. It is repeatedly used later as "after
making the value error smaller than (1/2) dist(h(x), target
complement), the smooth map is onto the target."
@@ -527,21 +455,18 @@ 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
+glue for every insertion: 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
+tolerance always means a positive continuous function on the open
+domain. When several such functions are imposed at once, replace a
finite list by their pointwise minimum; when there are countably many
-compact requirements, we allocate errors along a convergent positive
+compact requirements, 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
+What follows 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.
@@ -556,10 +481,10 @@ smooth and det Dg > 0 everywhere, g(Omega) = Omega', and
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.
+hand; at no point is an existence theorem invoked 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.
+open set Omega, T has the following features used 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
@@ -577,13 +502,11 @@ Because 1 <= p < infinity and h in W^{1,p}, the expression
is finite. The sum of tube measures is summable along edges and
vertices.
-We may reduce to the orientation-preserving case by precomposing with
+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
@@ -626,7 +549,7 @@ Jacobian
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
+phi is a translation per sector), this gives
det D h_tau = a s_tau^2 > 0 on every closed sector and every tau.
@@ -649,7 +572,7 @@ where
tau(t, r) = chi( log( r / d(t) ) / N ),
-with N a large absolute constant to be chosen. Here and below we
+with N a large absolute constant to be chosen. Here and below
write b_tau(theta) and s_tau(theta) for the constant-per-sector
functions described above.
@@ -693,12 +616,12 @@ 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
+finitely many edges meeting any compact set, 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:
+The map h_1 has three properties used in the sequel:
(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
@@ -710,9 +633,7 @@ The map h_1 has three properties we use in Section 6.4:
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
+Keep the notation introduced above. 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
@@ -747,14 +668,12 @@ 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
+homogeneous of degree 1 about v (property (c) above). 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
@@ -793,7 +712,7 @@ At a vertex v of the triangulation, choose
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
+(property (c) above), the degree-1 homogeneity survives the
averaging: for small r = |x - v| and unit u,
h_2(v + r u) - h_1(v)
@@ -814,11 +733,9 @@ 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.
+punctured ball B_v = B(v, R_v), by the same construction.
Write F = h_2 - h_2(v) : B_v -> R^3.
Euler's identity. Because F is homogeneous of degree 1 and smooth on
@@ -892,8 +809,8 @@ 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)
+agrees with h_2 there. So FA extends by h_2 to all of Omega minus
+the vertex balls B_v, and a continuous map results (indeed smooth)
on Omega.
Value and derivative cost over one vertex ball. Inside B_v minus
@@ -912,8 +829,6 @@ 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)
@@ -938,23 +853,20 @@ 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
+The next part 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
+What follows 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
+Given a small patch of H, 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.
@@ -973,10 +885,10 @@ 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
+to DH. On a bad cube the slope cannot be bounded of a PL replacement in
this way.
-Rather than shrinking the mesh first and then replacing badly, we use
+Rather than shrinking the mesh first and then replacing badly, use
two features of a fine dyadic mesh with only finitely many normalized
shapes:
@@ -990,24 +902,21 @@ shapes:
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
+derivative bounds M_P are fixed is the mesh shrunk, 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,
+Order of choices. 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.
+insertion tolerance a by backward recursion. Then, INDEPENDENT of
+any final mesh, prepare the finite PL library and its derivative bound
+M_P. Then choose a fine mesh and execute the two rounds of insertions.
+Only then do the bad-cube measures enter. 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
@@ -1046,8 +955,6 @@ 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
@@ -1125,11 +1032,9 @@ 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
+library of PL replacements finite.
-The next lemma is the technical heart of the section. It inserts a new
+The next lemma is the technical heart. 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
@@ -1185,13 +1090,13 @@ a bounded neighborhood, A_0 is a small perturbation of the identity:
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
+outside B_Q(3), though its support is formally extended 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
+each half-shrunk older box, 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
@@ -1205,9 +1110,7 @@ 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
+At every insertion the older protected cores a small fixed
amount, so that protection is lost slowly enough to allow recursion
through many stages.
@@ -1220,14 +1123,14 @@ subsequent stage, decrease every older protected patch's radius by
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
+core throughout, and at every insertion half of the current is usable
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
+B_Q(5) the update
|G(A(x)) - H(x)| <= |G(A(x)) - H(A(x))| + K_P |A(x) - x|.
@@ -1248,8 +1151,6 @@ 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
@@ -1257,7 +1158,7 @@ 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
+piecewise affine map L : Omega -> R^3. L is not asserted to be
globally injective.
A cube Q with coarse index P = P(Q) is called good if there is an
@@ -1310,12 +1211,10 @@ 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.
+measure tends to zero as the local mesh upper bound tends to zero.
+postpone the summability of bad-cube errors.
-7.6 Finite PL libraries before resolution selection
-
-This is the step that removes circularity. We prepare a finite library
+This is the step that removes circularity. Prepare a finite library
of local PL replacements and fix their derivative bounds M_P BEFORE
choosing the fine resolution of the mesh.
@@ -1362,7 +1261,7 @@ 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^*.
+PL embedding is denoted 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
@@ -1387,8 +1286,6 @@ Finite derivative bound M_P. By induction on s:
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).
@@ -1402,7 +1299,7 @@ G : Omega -> Omega' satisfying
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
+M_P. Until the final resolution choice, 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
@@ -1413,13 +1310,13 @@ 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
+invariants follow from the update inequality. 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
+Here the finiteness prepared above 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
@@ -1449,7 +1346,7 @@ 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,
+fixed above 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
@@ -1457,15 +1354,15 @@ 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
--------------------------------------
+finite PL homeomorphism g, and (together with the smoothing theorem) Theorem C.
-Two analytic lemmas are used in Section 11. Neither is topological:
+Two analytic lemmas are used in the assembly. 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.
+./ahlfors-bers-1960-riemann-mapping-variable-metrics.pdf
+
Definitions. Let Q = [-1/2, 1/2]^2. For a bi-Lipschitz embedding h : Q
-> R^d, put
@@ -1475,15 +1372,13 @@ 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
+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,
@@ -1528,7 +1423,7 @@ 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
+solution is unique up to a conformal map of the target, normalize to
the rectangle R.
Boundary regularity. Because g = M^2 I in a two-sided neighborhood of
@@ -1579,7 +1474,7 @@ in R^2 x {0}, so W_j S_j -> 1 and W_j, S_j -> 1. By (8.7) this forces S_j
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}
+L^2 energy up to 1 + o(1), giving (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
@@ -1595,19 +1490,17 @@ 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:
+Two uses in the assembly:
(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.
+ what the actual derivative energy is compared 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
@@ -1639,7 +1532,7 @@ parametrization gives exactly
int_e |DF|^p = L^p H^{1-p},
-the m = 1 bound in the form used in Section 11.6.
+the m = 1 bound in the form used later.
Case m >= 2. Fix 0 < alpha < 1 and 0 < delta < 1 and put
@@ -1689,7 +1582,7 @@ 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).
+Combining gives (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
@@ -1707,11 +1600,8 @@ form (8.1) of the disk estimate. After rescaling Q to the parameter cell,
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:
+the analytic part to the qualitative topology:
(a) cleaning an isolated surface/line crossing to a regular
transverse crossing with a rigid affine germ nearby;
@@ -1721,9 +1611,9 @@ Section 11 to the qualitative topology of Section 5:
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 derivative estimates all come from earlier analysis.
-The classical inputs we use are listed once and for all:
+The classical inputs are listed once and for all:
locally flat Schoenflies and extensions of homeomorphisms
from a sphere (Brown 1960, 1962);
@@ -1740,8 +1630,6 @@ 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
@@ -1828,15 +1716,13 @@ 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
+on the planar annulus, giving 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
@@ -1869,9 +1755,7 @@ 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
+One general product recognition is needed, used in Stage 6 of the star
replacement.
LEMMA 9.4 (punctured product sphere). Let M be homeomorphic to S^2 x I,
@@ -1901,8 +1785,6 @@ 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
@@ -2013,8 +1895,6 @@ 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
@@ -2049,11 +1929,8 @@ 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
+What follows builds the derived-trace budget, which is the mechanism
+that gives DERIVATIVE control in the assembly without
using any quantitative topological extension theorem.
Two pieces of bookkeeping are involved:
@@ -2062,7 +1939,7 @@ Two pieces of bookkeeping are involved:
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
+ (b) a TARGET TRIANGULATION that may be freely chosen, 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
@@ -2072,16 +1949,13 @@ 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.
+of the assembly 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
+Throughout, f : Omega -> Omega' is a fixed homeomorphism
+in W^{1,p} (1 <= p <= 2), fixed after the compact jet selection that follows. All subsequent constructions 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 starting point is 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.
@@ -2127,9 +2001,10 @@ 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
+./csornyei-hencl-maly-2010-homeos-sobolev-wnminus1.pdf
+./bojarski-hajlasz-1993-pointwise-sobolev.pdf
-The next lemma produces the target meshes used in Sections 11.4-11.6.
+The next lemma produces the target meshes used later.
Their shape does not need to be uniform; only bounded local site counts
and upper size bounds are needed for the target-area budgets.
@@ -2212,11 +2087,9 @@ 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
+the geometry of the tie strata. The resulting bound is computed shortly
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.
@@ -2271,7 +2144,7 @@ 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.
+The map whose concentration is exploited 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
@@ -2288,7 +2161,7 @@ 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
+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
@@ -2343,8 +2216,6 @@ 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
@@ -2414,8 +2285,6 @@ 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
@@ -2442,8 +2311,7 @@ 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
+counted hits of G_1 inside the squeeze cell have been arranged (by the squeeze construction, 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.
@@ -2455,10 +2323,8 @@ 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
+trace energy. This is the only place where the form
of the mesh sampling is used quantitatively.
LEMMA 10.6 (expected tie budget). Let Gamma be a measured m-trace
@@ -2520,11 +2386,8 @@ 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
+allows controlling 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
@@ -2564,13 +2427,10 @@ 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
+on the curve. This is the source-side input to the radial blocks that follow: 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.
@@ -2630,9 +2490,7 @@ 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
+The next lemma produces the source meshes used later. 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.
@@ -2678,7 +2536,7 @@ 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:
+Classical consequence. A fact used repeatedly below:
the m-dimensional area formula, applied to a bi-Lipschitz image of an
m-dimensional source-mesh face, gives
@@ -2688,10 +2546,7 @@ m-dimensional source-mesh face, gives
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
+The following analysis assembles the earlier results 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'
@@ -2699,13 +2554,7 @@ 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
+First comes the analytic composition fact used throughout. Then radial blocks are constructed, compact jets and kernel blocks are selected, sections are parametrized, uncontrolled cells are completed, and all choices are ordered to prove Theorem B.
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
@@ -2739,8 +2588,6 @@ 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
@@ -2824,6 +2671,8 @@ 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).
+./simon-2014-introduction-gmt.pdf
+
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
@@ -2861,7 +2710,7 @@ 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
+star-modification neighborhoods. 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.
@@ -2905,8 +2754,6 @@ 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
@@ -2915,7 +2762,7 @@ 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).
+a local C^1 diffeomorphism b (the radial-block construction 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
@@ -2960,14 +2807,12 @@ 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
+forces correct derivative. More precisely, if destination is constrained
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
@@ -3040,8 +2885,6 @@ 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
@@ -3102,14 +2945,6 @@ Substituting the preceding face estimates gives (11.11). Allocate
the intermediate errors so their converted sum is below the
prescribed cell error. QED
-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.
@@ -3120,7 +2955,7 @@ 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
+and plane tiles. 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,
@@ -3147,7 +2982,7 @@ 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
+Derivative comparison. The controlled cells are separated into three
types.
Full rank. On a full-rank core, H = b_i except on the small
@@ -3168,7 +3003,7 @@ 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
+data are fixed, combining the three cases with the outside-core
integral and obtain
int_Omega |DH - Df|^p <= C_p eps_0 + o(1). (11.12)
@@ -3193,12 +3028,9 @@ 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 THE MAIN THEOREM
------------------------------
-
-We now combine Theorems B and C to prove Theorem A.
+Combining Theorems B and C proves Theorem A.
-STATEMENT. Let Omega, Lambda be bounded domains in R^3, let 1 <= p <=
+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
@@ -3206,16 +3038,16 @@ 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).
+Proof (two stages).
-Stage one (Theorem B, Sections 5-11). For every eps > 0 and every
+Stage one (Theorem B). 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
+Stage two (Theorem C). 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
@@ -3238,10 +3070,8 @@ 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
+Dimension three, 2 < p < infinity. A separate construction gives the
+same strong approximation in this range. Together with Theorem A it
covers all 1 <= p < infinity in dimension three.
Dimension four and higher. Campbell, Hencl and Tengvall construct
@@ -3252,83 +3082,3 @@ 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?
-
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-END OF PROOF AND REFERENCES.
-