==== TABLE OF CONTENTS ====
Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2//main.tex
Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2//sections/01-introduction.tex
Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2//sections/02-smoothing.tex
Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2//sections/03-low-tools.tex
Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2//sections/04-low-assembly.tex
Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2//sections/10-conclusion.tex
Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2//references.bib
===========================
==== BEGIN Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2//main.tex ====
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\hypersetup{pdftitle={Strong diffeomorphic approximation in three dimensions for 1 <= p <= 2},pdfauthor={OpenAI}}
\title{Strong diffeomorphic approximation\\in three dimensions for $1\le p\le2$}
\author{OpenAI}
\date{September 24, 2026}
\begin{document}
\maketitle
\begin{abstract}
We resolve the three-dimensional Ball--Evans approximation problem for
$1\le p\le2$. Every $W^{1,p}$ homeomorphism between arbitrary bounded domains
in $\R^3$ is a strong $W^{1,p}$ limit of smooth diffeomorphisms onto the same
target.
\end{abstract}
\input{sections/01-introduction}
\tableofcontents
\input{sections/02-smoothing}
\input{sections/03-low-tools}
\input{sections/04-low-assembly}
\input{sections/10-conclusion}
\begingroup
\raggedright
\bibliographystyle{plainnat}
\bibliography{references}
\endgroup
\end{document}
==== END Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2//main.tex ====
==== BEGIN Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2//sections/01-introduction.tex ====
\section{Introduction and conventions}\label{sec:introduction}
The Ball--Evans approximation problem concerns the compatibility of
Sobolev approximation with global injectivity. A deformation in nonlinear
elasticity is represented by a map between spatial domains; injectivity
expresses noninterpenetration, whereas its weak first derivative records
the local deformation. Smooth approximation is elementary if injectivity
is discarded, but convolution need not preserve it. The question is
whether the two requirements can be met simultaneously. Ball's
formulation explicitly attributes the question to Evans
\cite[Section~3, p.~8]{Ball2001}; his later discussion and Hencl's survey
explain the surrounding variational questions
\cite{Ball2010,Hencl2025}.
Here a \emph{domain} is a nonempty connected open set. For domains
$\Omega,\Lambda\subset\R^3$ and $1\le p<\infty$, a Sobolev
homeomorphism is a homeomorphism $f:\Omega\to\Lambda$ whose components
and weak first derivatives belong to $L^p(\Omega)$. A smooth
diffeomorphism has a smooth inverse as well as a smooth forward map.
The fixed-target version of the approximation problem asks for
diffeomorphisms $f_j:\Omega\to\Lambda$ such that both $f_j-f$ and
$Df_j-Df$ tend to zero in $L^p(\Omega)$. In particular, every
approximating map, not just the limit, must have image exactly
$\Lambda$.
We resolve this problem for the low and critical exponents in dimension
three.
\begin{theorem}\label{main:theorem}
Let $\Omega,\Lambda\subset\R^3$ be nonempty bounded connected open sets, let $1\le p\le2$, and let $f:\Omega\to\Lambda$ be a homeomorphism in $W^{1,p}(\Omega;\R^3)$. There exist $C^\infty$ diffeomorphisms $f_j:\Omega\to\Lambda$, each belonging to $W^{1,p}(\Omega;\R^3)$, such that
\begin{equation}\label{main:convergence}
\int_\Omega\bigl( |f_j-f|^p+|Df_j-Df|^p\bigr)\,dx\longrightarrow0.
\end{equation}
\end{theorem}
Both the map and its inverse in the conclusion are smooth on their open domains. No smoothness at the boundary, boundary extension of $f$, or regularity of either boundary is assumed. Neither inverse Sobolev regularity nor a nonvanishing Jacobian is an additional hypothesis. The target $\Lambda$ is fixed throughout.
\subsection{Prior work and the three-dimensional difficulty}
In the plane, Iwaniec, Kovalev, and Onninen proved strong diffeomorphic
approximation for $11$ result used coordinatewise $p$-harmonic replacements.
Hencl and Pratelli's endpoint proof instead used geometric extension
estimates and an adapted grid. Quantitative approximation of
bi-Lipschitz planar maps by Daneri and Pratelli
\cite{DaneriPratelli2014}, and approximation of planar bi-Sobolev
$W^{1,1}$ maps by Pratelli \cite{Pratelli2017}, additionally control
inverse derivatives under their respective stronger hypotheses.
Smooth invertibility of each approximant in Theorem~\ref{main:theorem}
is not a claim of Sobolev convergence of the inverses.
Higher dimensions exhibit a genuine obstruction. Hencl and Vejnar
initiated the counterexamples at $p=1$ \cite{HenclVejnar2016}.
Campbell, Hencl, and Tengvall corrected a gap in that construction and
extended the range to $1\le p<\lfloor n/2\rfloor$ in dimensions
$n\ge4$, with Jacobian determinants of both signs on sets of positive
measure \cite{CampbellHenclTengvall2018}. A sequence of diffeomorphisms cannot
reproduce this sign pattern by strong convergence of its derivatives.
In dimension three, Hencl and Mal\'y's orientation theorem instead
gives $\det Df\ge0$ almost everywhere for a sense-preserving
$W^{1,1}_{\mathrm{loc}}$ homeomorphism
\cite[Theorem~1.1]{HenclMaly2010}. This sign conclusion does not
assert a positive Jacobian and does not furnish an approximation:
rank-deficient derivatives still require separate constructions.
Smoothing a homeomorphism that is already piecewise affine is an
important intermediate problem. Mora-Corral and Pratelli obtained
fixed-image diffeomorphic smoothing in the plane, with control of the
forward map and its inverse \cite{MoraCorralPratelli2014}.
Campbell and Soudsk\'y proved $C^1$ homeomorphic approximation of
piecewise affine maps in arbitrary dimension
\cite{CampbellSoudsky2021}; their conclusion does not require the
inverse to be smooth. Campbell, D'Onofrio, and V\'itek subsequently
proved diffeomorphic smoothing of locally finite piecewise affine
homeomorphisms in dimensions three and four, including forward and
inverse derivative-error control
\cite[Theorem~A]{CampbellDOnofrioVitek2026}. These results identify
the importance of distinguishing a smooth injective map from a
diffeomorphism. They do not construct a strong piecewise affine
approximation of an arbitrary Sobolev homeomorphism.
The present proof separates that approximation problem from smoothing.
It first constructs an onto locally bi-Lipschitz map with small strong
Sobolev error, and then proves a smoothing theorem with arbitrarily
fine pointwise control. Three-dimensional PL topology, developed by
Moise and Bing and used here in Hamilton's relative formulation,
provides qualitative replacements
\cite{Moise1952Approximation,Moise1952Triangulation,Bing1959,Hamilton1976}.
It supplies no derivative estimate. The energy bounds must therefore
come from the parametrizations and measurements used below, rather
than from the mere existence of a topological replacement.
The critical exponent $p=2$ has additional analytic structure.
Cs\"ornyei, Hencl, and Mal\'y proved that a three-dimensional
$W^{1,2}_{\mathrm{loc}}$ homeomorphism has an inverse of locally
bounded variation, and that almost every parallel plane has the
two-dimensional Lusin property
\cite[Theorems~1.1 and~1.3]{CsornyeiHenclMaly2010}. We use these two
statements, respectively, to select reverse rays and to measure
generic image surfaces. Neither is an assumption on the given inverse.
The further analytic ingredients include Whitney's extension theorem
for compatible first jets \cite{Whitney1934} and conformal
parametrization of metric disks \cite{AhlforsBers1960,McMullen2023}.
Their roles and the accompanying local estimates are made explicit
at the point of use.
The complementary range $20$ there is a positive $L$-Lipschitz function $d$ on $U$ such that
\[
d(x)\le \min\{a(x),1,L\dist(x,\R^n\setminus U)\}.
\]
There are also positive smooth minorants satisfying prescribed positive continuous upper bounds on their 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.
\end{lemma}
\begin{proof}
Extend $e(x)=\min\{a(x),1,L\dist(x,\R^n\setminus U)\}$ by zero outside $U$, and put
\[
d(x)=\inf_{y\in\R^n}\{e(y)+L|x-y|\}.
\]
This function is $L$-Lipschitz and is bounded above by $e$. It is positive at an interior point $x$: on a small closed ball about $x$, $e$ has a positive minimum, and outside that ball the distance term has a positive lower bound.
For a smooth minorant take 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
\[
c_i\le 2^{-i-2}\min\left\{
\inf_{\supp\chi_i}v,
\frac{\inf_{\supp\chi_i}w}{1+\|D\chi_i\|_\infty}
\right\}.
\]
Then $b>0$, $b\le v$, and $|Db|\le w$. Replace a finite list of 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 a given compact support.
\end{proof}
We will also use the following elementary global observation. If $H:U\to V$ is a homeomorphism and a continuous map $G:U\to V$ satisfies
\[
|G(x)-H(x)|<\tfrac12\dist(H(x),\R^n\setminus V),
\]
then the straight homotopy stays in $V$. When $V$ is bounded, it is a proper homotopy: the preimage of a compact target set is contained, uniformly in the homotopy parameter, in the $H$-preimage of a compact set a positive distance from $\partial V$. This observation will be used together with local injectivity and degree, not as a substitute for local injectivity.
==== END Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2//sections/01-introduction.tex ====
==== BEGIN Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2//sections/02-smoothing.tex ====
\section{Qualitative topology and strong smoothing}
\label{sec:smoothing}
All PL structures in this Section are the ordinary structures on open
subsets of $\R^3$. Triangulations are locally finite in the open domain.
In particular, neither a triangulation nor the local constants used below
need have uniform behavior at the boundary.
The qualitative PL tools belong to the three-dimensional triangulation
and approximation theory of Moise and Bing
\cite{Moise1952Approximation,Moise1952Triangulation,Bing1959}; Hamilton's
relative formulation \cite{Hamilton1976} is the precise interface used
below. Edwards--Kirby's deformation theorem supplies the supported
ambient extensions \cite{EdwardsKirby1971}. We separate these
topological existence statements from the derivative estimates:
finite local model libraries and subsequently chosen small exceptional
sets provide the latter.
\begin{theorem}[Strong smoothing of locally bi-Lipschitz maps]
\label{sm:smoothing}
Let $\Omega,\Omega'\subset\R^3$ be bounded domains, let
$1\le p<\infty$, and let $H:\Omega\to\Omega'$ be a locally
bi-Lipschitz homeomorphism in $W^{1,p}(\Omega;\R^3)$. For every
$\eps>0$ and every continuous function $\xi:\Omega\to(0,\infty)$,
there is a $C^\infty$ diffeomorphism $g:\Omega\to\Omega'$ such that
\begin{equation}
\label{sm:smoothing-conclusion}
\norm{g-H}_{W^{1,p}(\Omega)}<\eps,
\qquad |g(x)-H(x)|<\xi(x)\quad(x\in\Omega).
\end{equation}
In particular, $g$ belongs to $W^{1,p}(\Omega;\R^3)$, and the
approximations have exactly the target $\Omega'$.
\end{theorem}
The proof has two parts. First we smooth a locally finite PL
homeomorphism with summable local derivative errors. We then approximate
$H$ by such a PL map. In the second part a finite collection of local
models gives derivative bounds before the measure of the exceptional
mesh cubes is made small. The qualitative topology used to choose those
models supplies no derivative estimate.
\subsection{Relative qualitative tools}
\begin{lemma}[Fine relative PL approximation]
\label{sm:relative-pl}
Let $M,N$ be PL three-manifolds without boundary, with $N$ metrized,
and let $f:M\to N$ be a homeomorphism. Suppose that $K\subset M$ is
closed and that $f$ is PL on an open neighborhood of $K$. Given a
continuous function $\eta:M\to(0,\infty)$, there is a PL
homeomorphism $f_1:M\to N$ which agrees with $f$ on a neighborhood of
$K$ and satisfies
\[
d_N(f_1(x),f(x))<\eta(x)\qquad(x\in M).
\]
Noncompact manifolds are allowed. For manifolds with boundary the same
statement holds if $K$ contains $\partial M$ and $f$ is PL near $K$.
\end{lemma}
\begin{proof}
Choose a closed neighborhood $K^+$ of $K$ contained in the open set on
which $f$ is PL. Transport the PL structure of $N$ to $M$ by $f$.
The identity from the original structure to the transported structure
is PL near $K^+$. Hamilton's relative approximation Theorem
\cite[Section~3, Theorem~2.2, pp.~68--69]{Hamilton1976} applies to these
two structures. Use the compatible metric
$d_f(x,y)=d_N(f(x),f(y))$ and the fine tolerance $\eta$.
It gives a homeomorphism $a$ that is PL between the two structures,
is the identity on $K^+$, and satisfies
$d_f(a(x),x)<\eta(x)$. Then $f_1=f\circ a$ has all the asserted
properties. The boundary condition in Hamilton's Theorem is precisely
the additional condition stated here; for an open Euclidean domain its
manifold boundary is empty.
\end{proof}
\begin{lemma}[Small local ambient extension]
\label{sm:local-extension}
Fix a closed Euclidean cube $B\subset\R^3$, compact sets $C,D\subset B$
with $C\subset\operatorname{int}B$, and an open neighborhood $O$ of
$C\cup D\cup\partial B$. For every $\lambda>0$ there is a
$\delta>0$, depending only on these source sets and on $\lambda$,
with the following property. If $e:O\to\R^3$ is an embedding,
\[
\sup_{x\in O}|e(x)-x|<\delta,
\qquad e=\Id\ \hbox{on }D\cup\partial B,
\]
then there is a homeomorphism $A:\R^3\to\R^3$ such that
\begin{equation}
\label{sm:extension-conclusion}
A=e\ \hbox{on }C,\qquad
A=\Id\ \hbox{on }D\cup(\R^3\setminus\operatorname{int}B),
\qquad \sup_{\R^3}|A-\Id|<\lambda.
\end{equation}
The number $\delta$ is uniform over a finite list of fixed normalized
configurations $(B,C,D,O)$. No derivative bound on $e$ is required.
\end{lemma}
\begin{proof}
Put $E=C\cup D\cup\partial B$. The deformation Theorem of
Edwards--Kirby \cite[Theorem~5.1]{EdwardsKirby1971}, applied at the
inclusion of $O$ in $\R^3$, deforms every sufficiently close embedding
$e$ through embeddings $e_t$, starting at $e_0=e$, so that
$e_1=\Id$ on $E$. The deformation is unchanged outside a fixed
compact neighborhood $L$ of $E$ in $O$, and fixes the inclusion.
Its continuity at the inclusion permits us to require
\[
|e_t(x)-x|<\lambda/2\quad(x\in L,\ 0\le t\le1),
\qquad |e(x)-x|<\lambda/2\quad(x\in L).
\]
A sufficiently small uniform $\delta$ on $O$ meets the finitely many
compact-open conditions that specify this neighborhood of the
inclusion. In the terminology of that Theorem a proper embedding
respects manifold boundaries; this condition is automatic for the
ambient manifold $\R^3$.
All the open images $e_t(O)$ coincide. To see this, enclose $L$ in a
finite union of relatively compact subdomains of $O$ whose boundary collars are outside
the region where the deformation changes the map. The embeddings
agree on that collar. Invariance of domain
\cite[Theorem~2B.3]{Hatcher2002} and degree relative to
the common boundary show that the images of the enclosed subdomain
are the same. Outside it the maps already agree.
On this common open image define
\[
A_0=e\circ e_1^{-1},
\]
and define $A_0$ to be the identity elsewhere. The map is already
the identity off the compact set $e_1(L)$ in the common image, so
this is an ambient homeomorphism. It agrees with $e$ on $E$ and has
displacement less than $\lambda$. Since it fixes $\partial B$
pointwise, it preserves the bounded complementary component
$\operatorname{int}B$. Restrict $A_0$ to $B$ and extend it by the
identity outside $B$ to obtain $A$.
The construction uses only the displayed source configuration and
continuity at the inclusion. For finitely many normalized
configurations take the minimum of the resulting positive tolerances.
\end{proof}
We will use Lemma~\ref{sm:local-extension} when the input embedding is
defined by a small map $\eta$ near a new core and by the identity near
older protected sets. The following observation specifies the required
gluing condition. Choose open sets
$\overline{O_0}\subset U_0$, where $\eta$ is defined on $U_0$, and
an open set $V$ containing the protected sets, such that
$\eta=\Id$ on $U_0\cap V$. If $\eta$ is sufficiently close to
the identity that $\eta(O_0)\subset U_0$, the map
\begin{equation}
\label{sm:patched-embedding}
e=\eta\ \hbox{on }O_0,\qquad e=\Id\ \hbox{on }V
\end{equation}
is an embedding. Indeed, a collision $\eta(x)=y$ with
$x\in O_0$ and $y\in V$ has $y\in U_0\cap V$, whence
$\eta(y)=y$ and injectivity of $\eta$ gives $x=y$. The open
embedding property then follows from invariance of domain. All the
sets in this observation can be fixed in advance when the source
configurations range over a finite list.
\begin{lemma}[Fine control, properness, and the target]
\label{sm:proper-degree}
Let $h:\Omega\to\Omega'$ be an orientation-preserving homeomorphism
between bounded domains, and let $g:\Omega\to\R^3$ be a smooth
map with $\det Dg>0$. If
\begin{equation}
\label{sm:target-distance}
|g(x)-h(x)|<\frac12\dist(h(x),\R^3\setminus\Omega')
\qquad(x\in\Omega),
\end{equation}
then $g$ is a diffeomorphism of $\Omega$ onto $\Omega'$.
\end{lemma}
\begin{proof}
Write $d(y)=\dist(y,\R^3\setminus\Omega')$ and
$g_t=(1-t)h+tg$. Condition~\eqref{sm:target-distance} puts the entire
segment $g_t(x)$ in $\Omega'$. Since $d$ is 1-Lipschitz,
\[
d(g_t(x))\le\frac32d(h(x)).
\]
If $K\Subset\Omega'$ and $m=\min_Kd>0$, the inverse image of $K$
under the homotopy lies in
\[
h^{-1}\bigl(\{y\in\Omega':d(y)\ge2m/3\}\bigr)\times[0,1].
\]
The set in braces is compact because $\Omega'$ is bounded.
Thus $g_t$ is a proper homotopy. Its endpoint $g$ has the same
degree as $h$, namely one. A proper local diffeomorphism has
finitely many preimages of each point, and here each preimage has
positive local degree. The degree-one identity therefore says that
every point of $\Omega'$ has exactly one preimage. The local smooth
inverses patch to a smooth inverse on $\Omega'$.
\end{proof}
\subsection{Smoothing a locally finite PL homeomorphism}
\begin{proposition}[Strong PL smoothing]
\label{sm:pl-smoothing}
The conclusion of Theorem~\ref{sm:smoothing} holds if $H$ is replaced
by a locally finite PL homeomorphism
$h:\Omega\to\Omega'$ in $W^{1,p}(\Omega;\R^3)$.
\end{proposition}
\begin{proof}
An orientation-reversing Euclidean isometry in the target reduces
the proof to the orientation-preserving case. Fix a locally finite
affine triangulation for $h$. We prescribe summable error budgets
for its edges, vertices, and a locally finite family of compact sets
covering the remainder of the domain. All modifications below can
also obey an arbitrary positive continuous value tolerance.
\paragraph{The open edges.}
Around each open edge choose a tube of radius $d(t)$, where
$00$, $s>0$, and the coefficients are smooth on the closed
angular sectors. The transverse homogeneous map is injective:
otherwise two equal transverse images, taken at arbitrarily small
radius, would have their axial difference canceled by changing $t$,
contradicting injectivity of $h$ in the edge star. Its angular
map is consequently an orientation-preserving circle
homeomorphism. We choose a lift $\phi$ with
$\phi(\theta+2\pi)=\phi(\theta)+2\pi$.
Positive determinants on the finitely many sectors give upper and
positive lower bounds for $s$ and for the one-sided derivatives
$\phi'$.
Choose constants $c>0$ and $c'$, and for $0\le\tau\le1$ put
\[
b_\tau=(1-\tau)b,\qquad
s_\tau=(1-\tau)s+\tau c,\qquad
\phi_\tau=(1-\tau)\phi+\tau(\theta+c').
\]
For fixed $\tau$, the determinant of the corresponding map in the
frame $(\partial_t,\partial_r,r^{-1}\partial_\theta)$ is
\begin{equation}
\label{sm:edge-determinant}
a s_\tau^2\phi_\tau'>0.
\end{equation}
On all the closed sectors and all $\tau\in[0,1]$ this has a
positive minimum. Let $\chi$ be a smooth cutoff equal to 1 on
$(-\infty,-1]$ and to 0 on $[-1/2,\infty)$, and substitute
\[
\tau(t,r)=\chi\!\left(\frac{\log(r/d(t))}{N}\right)
\]
in the preceding formula. The additional derivative columns have
size at most
\begin{equation}
\label{sm:edge-cutoff-error}
\frac{C}{N}\bigl(1+\norm{d'}_\infty\bigr).
\end{equation}
Indeed, $|r\partial_r\tau|\le C/N$ and
$|r\partial_t\tau|\le C(r/d)|d'|/N$, while $r/d\le1$ on
the support of the modification. The remaining factors come from
the fixed angular data. Taking $N$ large preserves positive
determinants on every sector, including one-sided limits.
Near the axis the result is the nonsingular affine map
$(t,r,\theta)\mapsto(at,cr e_{\theta+c'})$.
Near the tube boundary it agrees with $h$.
All first derivatives are bounded by an edge-dependent constant
independent of a further common decrease of $d$. The volume of the
tube tends to zero under that decrease. Both its derivative error
and its value error therefore have arbitrarily small $W^{1,p}$
cost. Choose these costs summably over the edges and call the
result $h_1$. It is continuous and locally Lipschitz, fixes all
vertices, and has the required small total error. Because the
tube widths are exactly linear near endpoints, $h_1-h_1(v)$ is
homogeneous of degree one on a sufficiently small ball about every
vertex $v$.
\paragraph{The faces and the punctured vertex balls.}
Away from the vertices, the closed convex hull of the nearby
almost-everywhere gradients of $h_1$ is contained in
$\GL^+(3)$ on a sufficiently small neighborhood of each point.
At a smooth point this follows by continuity. At a nonsmooth
point the only remaining interface is one face. If $A_-$ and
$A_+$ are its two gradient traces, continuity gives agreement on
the tangent plane, so $A_+-A_-$ has rank at most one with normal
covector to that face. Consequently
\begin{equation}
\label{sm:face-segment}
\det\bigl((1-t)A_-+tA_+\bigr)
=(1-t)\det A_-+t\det A_+>0\quad(0\le t\le1).
\end{equation}
The traces vary continuously on the two sides. Their nearby
convex hull lies in a sufficiently small neighborhood of this
compact positive-determinant segment, proving the assertion.
Homogeneity makes this property uniform at relative scale on
a punctured ball about a vertex: cover the unit sphere by finitely
many of the corresponding neighborhoods and rescale.
Let $\rho\ge0$ be a smooth probability kernel supported in the
unit ball. Away from vertices define
\begin{equation}
\label{sm:variable-convolution}
h_2(x)=\int_{\R^3}h_1(x+w(x)z)\rho(z)\,dz,
\end{equation}
where $w$ is smooth and positive there and the integration balls
lie in $\Omega$. Its derivative is
\begin{equation}
\label{sm:variable-convolution-derivative}
Dh_2(x)=\int Dh_1(x+w(x)z)\rho(z)\,dz
+\int \bigl(Dh_1(x+w(x)z)z\bigr)\otimes Dw(x)\rho(z)\,dz.
\end{equation}
The first integral is in the positive-determinant convex hull just
described. Taking $w$ and $|Dw|$ sufficiently small locally makes
the second integral a small perturbation, and hence gives
$\det Dh_2>0$. Formula~\eqref{sm:variable-convolution} is smooth
where $w>0$, as is also seen by writing its smooth variable kernel
in the integration variable $y=x+w(x)z$.
On a small punctured ball at $v$ choose
$w(x)=c_v|x-v|$, with $c_v>0$ sufficiently small. This is
consistent with the relative-scale convex-hull condition and
makes $h_2-h_1(v)$ homogeneous on a smaller ball. Set
$h_2(v)=h_1(v)$ there. The required radius function exists by
a locally finite partition of unity with sufficiently small
positive coefficients. Near the vertices blend this function
with $c_v|x-v|$ on disjoint annuli. The annular cutoff terms
are bounded by a constant times $c_v$ if the competing radii
are chosen at that same small scale, so both the radius and
gradient restrictions survive. This is also an instance of
the local minorant construction in Lemma~\ref{pre:local-tolerances}.
Here is the global error choice. First take disjoint vertex balls
so small that their volumes, multiplied by the fixed local
gradient bounds to the power $p$, meet a summable error budget.
On these balls the gradients in
Equation~\eqref{sm:variable-convolution-derivative} are bounded independently
of a further decrease of their radii. On a compact set outside
the balls, the local gradients are bounded and converge at every
piecewise smooth point as $w,|Dw|\to0$. Dominated convergence
therefore gives an arbitrarily small derivative error on that
compact set. A locally finite compact covering with summable
budgets makes the total $W^{1,p}$ error as small as prescribed.
The value error follows from local Lipschitz continuity and the
small radius. These arguments include $p=1$.
\paragraph{The vertices.}
Center a homogeneous vertex ball at the origin and subtract
$h_1(v)$. Write the resulting map as $F$. Euler's identity is
$DF(x)x=F(x)$. Since $DF(x)$ is invertible for $x\ne0$, $F(x)$
never vanishes there. We can thus write
\begin{equation}
\label{sm:vertex-form}
F(r\theta)=rR(\theta)\Phi(\theta),\qquad
R>0,\quad\theta\in S^2.
\end{equation}
Its positive determinant implies that $\Phi:S^2\to S^2$ is a
smooth orientation-preserving local diffeomorphism. It is a
covering by compactness, and it is one-sheeted because $S^2$ is
simply connected.
Smale's Theorem gives a smooth isotopy $\Phi_\tau$ from $\Phi$
to a rotation, smooth jointly in $(\tau,\theta)$
\cite[Theorem~6]{Smale1959}; for the jointly smooth deformation see also
\cite[Theorem~1.5]{LiWatts2011}. Interpolate $R$ through positive
functions $R_\tau$ to a positive constant. For fixed $\tau$ the
homogeneous map
$F_\tau(r\theta)=rR_\tau(\theta)\Phi_\tau(\theta)$
has determinant
\[
R_\tau(\theta)^3 J_{S^2}\Phi_\tau(\theta)>0.
\]
The family has bounded first derivatives and a positive
determinant minimum, by compactness. On an arbitrarily small
ball replace $F$ by $F_{\tau(r)}$, where $\tau$ is 0 near the
outer boundary, 1 near the center, and
$\sup_r|r\tau'(r)|$ is sufficiently small. Such a transition
is obtained by spreading a fixed cutoff over a sufficiently
long interval of $\log r$. Its extra derivative term is
bounded by $C|r\tau'(r)|$, so positive determinant persists.
At the center the map is a dilation followed by a rotation.
It is therefore smooth and nonsingular there.
The derivative bound for this last modification depends on the
fixed spherical isotopy but not on the support radius. Shrinking
that radius gives arbitrarily small summable $W^{1,p}$ costs
over all vertices. The resulting map $g$ is smooth and has
positive determinant throughout $\Omega$.
Finally, in each of the three operations impose local value
tolerances whose sum is less than
\[
\min\!\left\{\xi(x),\frac12
\dist(h(x),\R^3\setminus\Omega')\right\}.
\]
The preceding constructions permit these fine tolerances and a
total Sobolev error less than $\eps$. Lemma~\ref{sm:proper-degree}
makes $g$ a diffeomorphism onto the original target. This proves
the Proposition.
\end{proof}
\begin{remark}
Theorem~A of Campbell--D'Onofrio--V\'itek
\cite{CampbellDOnofrioVitek2026} also gives diffeomorphic
approximation of locally finite piecewise affine homeomorphisms
in dimension three. Proposition~\ref{sm:pl-smoothing} records the
specific local and fine-control construction needed here; the
reduction from a general locally bi-Lipschitz map is proved next.
\end{remark}
\subsection{Fine meshes and controlled patch insertion}
It remains to approximate $H$ strongly by a PL homeomorphism.
Near most points, an affine first-order model will control the new
derivative. On the remaining patches, qualitative topology provides
replacements but gives no bound for their slopes. Choosing those
replacements only after making the exceptional patches small would
therefore leave the energy estimate circular. We instead select finite
libraries of replacements on each coarse neighborhood, fixing their
derivative bounds before the final mesh resolution. The mesh and
insertion lemmas below let neighboring replacements retain exactly
the same data on their overlaps.
Fix a coarse, locally finite Whitney decomposition $\mathcal P$ of
$\Omega$ into closed dyadic cubes with disjoint interiors. Choose
relatively compact enlarged neighborhoods $U_P$ that are still locally
finite. Enlarge them a fixed finite number of times when necessary.
Since $H$ is locally bi-Lipschitz, there are constants $K_P\ge1$ such
that
\begin{equation}
\label{sm:coarse-bilip}
K_P^{-1}|x-y|\le |H(x)-H(y)|\le K_P|x-y|
\qquad(x,y\in U_P).
\end{equation}
Enlarge $K_P$ also to bound $|DH|$ almost everywhere on $U_P$ in the
chosen matrix norm. The constants may incorporate the data from finitely many neighboring
coarse cubes. All references below to data local to $P$ allow this
finite enlargement, but never an enlargement depending on the final
fine-mesh resolution.
\begin{lemma}[Adapted dyadic meshes]
\label{sm:mesh}
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 $\mathcal Q$ of $\Omega$
with centers $c_Q$ and side lengths $l_Q$ for which, on writing
\begin{equation}
\label{sm:patch-box}
B_Q(s)=c_Q+[-s l_Q,s l_Q]^3,
\end{equation}
the following hold:
\begin{enumerate}[label=\textup{(\roman*)}]
\item $B_Q(100)\Subset\Omega$, and cubes meeting $B_Q(50)$ have
side lengths comparable to $l_Q$ by absolute constants.
\item The normalized cube configurations in $B_Q(6)$ belong to a
finite list. There is a conforming affine triangulation of
$\mathcal Q$ with finitely many normalized nondegenerate shapes
on these configurations.
\item The cubes have $n_0$ colors so that the closed boxes $B_Q(6)$
of a single color are pairwise disjoint.
\item 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 the
neighborhoods on which the required bounds in
Equation~\eqref{sm:coarse-bilip} hold.
\end{enumerate}
The upper bounds on the fine scale remain arbitrarily prescribable.
\end{lemma}
\begin{proof}
Choose a positive size function $s$ on $\Omega$ with sufficiently
small absolute Lipschitz constant, below all the prescribed local
bounds and below $10^{-3}\dist(x,\R^3\setminus\Omega)$.
Lemma~\ref{pre:local-tolerances} supplies such a minorant. Take the
maximal dyadic cubes for which
\[
l_Q\le\inf_Qs.
\]
Failure of the condition for the dyadic parent gives
$s(x)\le(2+C\Lip(s))l_Q$ on $Q$. 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 \textup{(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.
\end{proof}
The finite list in Lemma~\ref{sm:mesh} 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 the margins
used below still range over finitely many geometric configurations.
\begin{lemma}[Insertion with protected cores]
\label{sm:insertion}
Consider a patch $Q$ and a finite collection of older protected
boxes with the relative sizes and positions provided by
Lemma~\ref{sm:mesh}. For every older box prescribe a larger
half-shrunk box and a smaller fully-shrunk box, separated by a
fixed positive normalized margin. Let $G:\Omega\to\Omega'$ 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 a local
constant $K_P$ in Equation~\eqref{sm:coarse-bilip}, such that
\begin{equation}
\label{sm:insertion-smallness}
\sup_{B_Q(5)}|G-H|0.
\]
Invariance of domain gives the open embedding assertion.
\paragraph{The first round.}
Start with $G_0=H$. For stages $1,\ldots,n_0$ process the
colors in order, inserting a patch only if it is good,
and taking its model to be $h_Q=L$. On every overlap
with an older protected core this agrees with the
already installed map, because both equal the single
global map $L$. Equation~\eqref{sm:good-interpolation}
and the chosen thresholds supply the input accuracy
for Lemma~\ref{sm:insertion}. 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
Equation~\eqref{sm:value-update}. At the end of this round the
map agrees with $L$ on every good cube and on a
protected neighborhood of it.
\paragraph{Normalized data and their finiteness.}
For each $Q$ choose $m_Q\in\Z^3$ nearest to
$H(c_Q)/l_Q$ and use normalized coordinates
\begin{equation}
\label{sm:normalization}
z=\frac{x-c_Q}{l_Q},\qquad
\widehat H_Q(z)=\frac{H(c_Q+l_Qz)-l_Qm_Q}{l_Q}.
\end{equation}
The value at $z=0$ is bounded by an absolute constant,
and Equation~\eqref{sm:coarse-bilip} gives a uniform local
Lipschitz bound. The normalized vertex values of $L$
on $B_Q(4)$ lie in a bounded set on finitely many
dyadic grids. They therefore take only finitely many
values for each coarse index. Here it matters that
the translation $l_Qm_Q$ is included in the data:
without it one would not have finiteness of affine
maps, as opposed to finiteness of their gradients.
If $R$ interacts with $Q$, then
\begin{equation}
\label{sm:relative-origins}
\frac{l_Rm_R-l_Qm_Q}{l_Q}
\end{equation}
is a bounded dyadic vector with a denominator from a
finite list. Boundedness follows by comparing
$H(c_R)$ and $H(c_Q)$ using Equation~\eqref{sm:coarse-bilip};
the denominators are fixed by the bounded dyadic
scale ratios. Consequently the first-round exact
PL data, as viewed in any interacting normalized
patch, have only finitely many possibilities.
\paragraph{Choosing the second-round libraries.}
For stages $n_0+1,\ldots,2n_0$ we insert at every
patch of the current color. The models are chosen
as follows, before the final fine scale is selected.
For each coarse index, the normalized maps
$\widehat H_Q$ on a fixed larger patch form a
uniformly bounded equicontinuous family. They
admit finitely many sup-norm bins of any prescribed
positive diameter. At stage $s$ use diameter less
than $a_{s,P}/8$.
Inductively assume that the already installed
normalized PL data have finite lists. The
relative positions of the protected cores have
finitely many possibilities, by
Lemma~\ref{sm:mesh} and Equation~\eqref{sm:shrink-size}.
Their target translations have finitely many
possibilities by Equation~\eqref{sm:relative-origins}.
Thus the exact older data on all required
half-shrunk overlaps have a finite list at this
stage as well. There are consequently only
finitely many combinations of
\begin{equation}
\label{sm:model-combination}
\text{source configuration, normalized }H\text{-bin,
and exact protected PL data}.
\end{equation}
For each combination which can occur, select one
representative normalized embedding $G^*$, close
to a map $H^*$ in that bin by the prescribed
current value tolerance, and realizing the
specified older data. Take the representative
on the fixed open patch $(-4,4)^3$. If
$\widehat K_i$ are the normalized half-shrunk
older boxes, the set
\[
K=[-3,3]^3\cap\bigcup_i\widehat K_i
\]
is compact in that open patch. It lies
strictly inside the currently protected
cores, so $G^*$ is PL on a neighborhood of
$K$. Apply Lemma~\ref{sm:relative-pl} to
$G^*$ on the open patch, keeping $K$ fixed.
Obtain a PL embedding $h^*$ there with
uniform error less than $a_{s,P}/8$ on the
required compact patch. Store its restriction
to a neighborhood of $[-3,3]^3$ in the library.
If a different actual tuple has the same
combination in Equation~\eqref{sm:model-combination}, this
single model agrees with all of its prescribed
PL overlap data. Moreover,
\begin{equation}
\label{sm:reuse-error}
\norm{h^*-\widehat H_Q}_{\infty}
\le \norm{h^*-G^*}_{\infty}
+\norm{G^*-H^*}_{\infty}
+\norm{H^*-\widehat H_Q}_{\infty}
<\frac38 a_{s,P}.
\end{equation}
The three norms are on the fixed patch needed
for insertion. The actual current map is
already within $a_{s,P}/8$ of its $H$, so
Lemma~\ref{sm:insertion} applies after undoing
the normalization. This installs $h_Q$ while
preserving the fully-shrunk older cores.
This selection does not require compactness of
the family of possible intermediate maps $G$.
Only the $H$-family is binned; the intermediate
map is used as an existence witness for an
embedding realizing the exact finite overlap
data. All possible normalized meshes and
inputs obeying the fixed coarse bounds may
be included when deciding which combinations
occur. At any stage the choices for a coarse
index depend only on libraries from strictly
earlier stages at its finitely many interacting
indices, so they can be made simultaneously
over all coarse indices. Hence the choices are independent of
the eventual mesh resolution.
Each stored map is PL on a neighborhood of a
fixed compact patch and thus has finitely many
affine pieces there. It has a finite upper
derivative bound. There are finitely many
stored maps for each local coarse combination,
and only finitely many stages. This proves by
induction both the finiteness of the exact data
needed at the next stage and the existence of
constants $M_P<\infty$ such that the final map
satisfies
\begin{equation}
\label{sm:model-derivative-bound}
|DG|\le M_P\quad\hbox{almost everywhere on }Q,
\qquad P=P(Q).
\end{equation}
The constants $M_P$ are fixed before reducing
the fine scale. They may be arbitrarily large;
no efficient dependence on $K_P$ is asserted.
After the second round every cube has a
protected PL neighborhood. The final map is
locally PL. If a single triangulation is
desired, take common refinements of the
finitely many affine subdivisions meeting
each compact set and triangulate the resulting
polyhedral cells. Local finiteness gives a
locally finite affine triangulation on
$\Omega$. The map is still onto $\Omega'$
because every stage was a precomposition by
a domain homeomorphism. On every good cube
the map still equals $L$, since its first-round
protected neighborhood was preserved throughout.
\paragraph{Selecting the resolution and paying for bad cubes.}
Only now choose the final local upper size
bounds of Lemma~\ref{sm:mesh}. On a fixed
coarse compact region, almost every $x$ is
both a differentiability point of $H$ and a
Lebesgue point of $DH$. The derivative at
such a point is invertible, with the upper
and lower bounds inherited from local
bi-Lipschitzness. The affine map
\[
A_x(y)=H(x)+DH(x)(y-x)
\]
then passes both tests in
Equation~\eqref{sm:good-tests} on every sufficiently
small cube containing $x$. Indeed its
enlargement is contained in a ball of radius
$C l_Q$ about $x$, which gives the sup-norm
test by differentiability. The volume of
that ball is at most a fixed multiple of
$|Q|$, which gives the derivative mean test
by the Lebesgue-point property. Only
finitely many coarse thresholds occur near
a fixed compact set.
It follows that the measure of the union of
bad cubes with a fixed coarse index $P$
tends to zero as their local upper side
length tends to zero. This statement is
uniform over the admissible meshes. One
can see the uniformity by taking the union
of all bad dyadic cubes with that index
and side at most $r$. These measurable
unions decrease as $r\downarrow0$, and
their intersection is null by the preceding
pointwise argument. They lie in the fixed
finite-volume neighborhood $U_P$.
Choose positive numbers $b_P$ with
$\sum_Pb_P<\eps^p/8$, and reduce the
local mesh bounds until
\begin{equation}
\label{sm:bad-error-budget}
(M_P+K_P)^p
\left|\bigcup_{\substack{Q\text{ bad}\\P(Q)=P}}Q\right|
0$ there is a bi-Lipschitz self-homeomorphism
$\rho:Q\to Q$, equal to the identity on $\partial Q$, such that
\begin{equation}
\int_Q|D(h\circ\rho)|^2
\leq C\bigl(A(h)+L(h)\bigr)+\eps,
\label{lt:disk-weak}
\end{equation}
where $C$ is absolute.
There is also the following sharp form. Suppose $h_j$ are such
embeddings, $\sup_jL(h_j)<\infty$,
$A(h_j)\leq1+o(1)$, and
$h_j|_{\partial Q}\to\iota|_{\partial Q}$ uniformly, where
$\iota(x)=(x,0)\in\R^d$. The parametrizations can be chosen so that
\begin{equation}
\int_Q|D(h_j\circ\rho_j)|^2\leq2+o(1),
\qquad
D(h_j\circ\rho_j)\longrightarrow D\iota
\quad\hbox{in }L^2(Q).
\label{lt:disk-sharp}
\end{equation}
The bi-Lipschitz constants of the chosen parametrizations may depend on
the individual embedding. The conclusions transport to polygons
through fixed shape-controlled, bi-Lipschitz piecewise smooth charts;
the sharp square normalization is used when the constant $2$ matters.
\end{lemma}
\begin{proof}
We first majorize the pullback metric by a smooth metric that is
conformally Euclidean on an added collar. Conformal coordinates on a
rectangle then give the weak energy estimate. For the sharp estimate,
coordinate tests force both the rectangle and its boundary correction
to be nearly isometric.
\paragraph{Metric preparation.}
Fix $0<\tau\leq1$, set $S=1+2\tau$ and
$Q^+=SQ$, and use square radial coordinates. For $\delta>0$, let
$r:Q^+\to Q$ preserve radial directions, carry the inner $Q$ by the
homothety of factor $1-\delta$, and carry the added shell linearly along
each radius onto $Q\setminus(1-\delta)Q$. Thus $r$ is bi-Lipschitz
for fixed positive $\delta$. Given $\xi>0$, we may choose $\delta$
and a smooth positive metric $g$ on a neighborhood of $Q^+$ so that
\begin{align}
g&\geq D(h\circ r)^TD(h\circ r)\quad\hbox{a.e.},
\label{lt:metric-dominates}\\
g&=M^2I\quad\hbox{throughout }Q^+\setminus Q,
\label{lt:metric-shell}\\
\operatorname{area}_g(Q)&\leq A(h)+\xi,
\qquad
\operatorname{area}_g(Q^+\setminus Q)
\leq C\tau L(h)+\xi, \label{lt:metric-area}\\
\|h\circ r-h\|_{C^0(\partial Q)}&\leq\xi,
\qquad
\int_{\partial Q}|D_t(h\circ r)|^2
\leq CL(h)+\xi. \label{lt:metric-trace}
\end{align}
Here a scalar factor in \eqref{lt:metric-shell} is allowed to vary
smoothly in the shell.
We give the details of this preparation because its boundary assertion
is used at the critical exponent. On the shell the radial derivative
of $h\circ r$ is bounded by $C\delta\Lip(h)/\tau$. Essential
tangential compatibility implies that the essential suprema of the
perimeter-direction derivatives in shrinking neighborhoods of almost
every boundary parameter tend to the corresponding boundary derivative
norm. They are bounded by fixed Lipschitz data. Covering the perimeter
by short overlapping intervals, taking suprema on slightly enlarged
intervals, and using a smooth partition of unity therefore gives scalar
majorants whose squared integrals are at most $CL(h)+\xi$ once
$\delta$ is sufficiently small. The negligible radial derivative can
be included in these majorants. Square radial and ordinary angular
coordinates have uniformly bounded distortion in this shell. Their
conformal metrics consequently have area at most $C\tau L(h)+\xi$.
Choose in addition a typical sufficiently small inner level to obtain
\eqref{lt:metric-trace}.
On the inner square consider the pullback quadratic form in
\eqref{lt:metric-dominates}. It is bounded, uniformly positive for the
fixed data, and essentially continuous almost everywhere. On a fine
partition of unity, majorize it by a representative matrix on each
enlarged patch plus its essential oscillation times the identity and a
small positive scalar matrix. The resulting smooth positive form
dominates the original form. The oscillations tend to zero almost
everywhere and are bounded, so its area integral tends to that of the
pullback form by dominated convergence. The latter is the area of
$h$ on $(1-\delta)Q$, hence is at most $A(h)$. Call this inner majorant $g_0$, and call the scalar shell
majorant $M^2I$. Choose a finite number $N$ dominating the operator
norms of $g_0$, $M^2I$, and both one-sided pullback forms near
$\partial Q$. This is possible for the fixed map and the fixed
positive $\delta$. In a two-sided neighborhood of $\partial Q$ of
thickness $t$ set the metric equal to $NI$. Blend $g_0$ to $NI$
on the inner side and $NI$ to $M^2I$ on the outer side by smooth
cutoffs supported in a slightly larger neighborhood. Each blend is
between two forms dominating the relevant pullback form, so it still
dominates that form; the entire outer blend is scalar. The metric
is constant on an open neighborhood of the interface, so its jets
match across the sides and corners. Smooth neighborhoods and cutoffs
can be chosen with area $O(t)$, and the added metric area is at most
$CNt$. Choose $t<\tau/10$ and then $t$ so small that $CNt$ is below
the remaining error allowance. Derivatives of the cutoff metric
are not charged to any estimate. Along a sequence, $N$ may diverge,
but $t$ is chosen after $N$ so that $Nt\to0$. In particular a
fixed-width outer band of the added shell is unaffected. This proves \eqref{lt:metric-dominates}--\eqref{lt:metric-trace}.
\paragraph{Conformal coordinates.}
By isothermal coordinates and marked-quadrilateral uniformization
\cite[Lemma~7 and Theorem~6]{AhlforsBers1960}\cite[Theorem~4.31]{McMullen2023},
uniformize the smooth metric quadrilateral $(Q^+,g)$ conformally onto
the centered rectangle
\[
R_{W,S}=[-W/2,W/2]\times[-S/2,S/2],
\]
with corresponding corners, and write this map as $\Phi$. The marked
quadrilateral uniformization determines the modulus $W/S$; a Euclidean
dilation sets the height to $S$. In the interior $\Phi$ is smooth and
nonsingular. We justify this regularity before using $\Phi$ in the
energy estimates.
The metric's
Beltrami coefficient $\mu$ is smooth and vanishes on the outer scalar
band, so it extends by zero to a compactly supported smooth coefficient
on the plane. Lemma~7 of Ahlfors--Bers gives a $C^1$ solution with
positive Jacobian. Its smooth bootstrap follows from their
parameter construction \cite[Section~2, Theorem~2]{AhlforsBers1960}:
choose $q>2$ with $\|\mu\|_\infty C_q<1$ for the norm $C_q$ of
their singular integral operator. Translated coefficients
$\mu_t(z)=\mu(z+t)$ vary smoothly in $L^\infty\cap L^q$; the
uniformly invertible operator in that construction therefore gives
smooth dependence of $U_t-\Id$ in its solution space $B_q$, where
$U_t(0)=0$ and $\partial U_t-1\in L^q$. The H\"older norm of $B_q$
makes point evaluation continuous.
Writing $U=U_0$, uniqueness gives $U_t(z)=U(z+t)-U(t)$. On any bounded neighborhood
of $t$, choose a fixed $z_*$ such that $z_*+t$ stays outside
$\supp\mu$. Then
$U(t)=U(z_*+t)-U_t(z_*)$ is smooth, since its first term is
holomorphic there and its second depends smoothly on $t$.
The conformal rectangle map preserves interior smoothness and
nonsingularity. Since the metric is conformally Euclidean near the outer
boundary, Schwarz reflection across sides
\cite[Chapter~4, Section~6.5, Theorem~24]{Ahlfors1979}, and then across the two
perpendicular sides at each corner, shows that $\Phi$ and its inverse
are bi-Lipschitz up to the boundary for each fixed metric.
\paragraph{Weak estimate.}
Fix, for example, $\tau=1/4$. Extend the first
and second Euclidean coordinate functions from the outer boundary
through the shell using cutoffs which vanish before reaching $Q$.
Their $g$-Dirichlet integrals are bounded by an absolute constant:
on the shell the scalar factor cancels from a two-dimensional
Dirichlet integral. On the rectangle, the corresponding horizontal
and vertical boundary differences equal $S$. Cauchy's inequality on
horizontal and vertical segments gives, respectively, the lower bounds
\begin{equation}
S^3/W\quad\hbox{and}\quad SW.
\label{lt:rectangle-lower}
\end{equation}
It follows that $W$ is bounded above and below by positive absolute
constants.
Choose $\eta0$. A fixed finite chain of disks along the outer
boundary gives by Harnack's inequality
$u\leq C_H\log(C/c_0)$ there. Consequently
$|\Phi'|\geq C\exp(-C_H\log(C/c_0))>0$ on the entire boundary,
including the corners. The constants depend on the fixed $\tau$,
but not on the inner metric distortion. The boundary map of
$\Phi$ thus has a uniformly controlled bi-Lipschitz extension
$P:Q^+\to R_{W,S}$, obtained by coning from the centers.
Set, for $x\in Q$,
\[
F(x)=h\circ r\circ\Phi^{-1}\circ P(Sx).
\]
Since $P=\Phi$ on $\partial Q^+$ and $r(Sx)=x$ there, this is
$h\circ\rho$ with $\rho$ a bi-Lipschitz self-homeomorphism fixing
$\partial Q$. The homothety does not change two-dimensional
Dirichlet energy. Conformal invariance, the bounded distortion of
$P$, and \eqref{lt:metric-dominates} yield
\[
\int_Q|DF|^2
\leq C\int_{Q^+}|d(h\circ r)|_g^2\,d\operatorname{area}_g
\leq 2C\operatorname{area}_g(Q^+).
\]
Letting $\xi$ be sufficiently small proves \eqref{lt:disk-weak}.
\paragraph{Sharp estimate.}
First fix an arbitrarily small positive
$\tau$ and make the metric errors tend to zero along the given
sequence. Denote the first two physical coordinates of $h_j\circ r_j$
on $Q$ by $X_j,Y_j$. Each coordinate has squared $g_j$-gradient at
most one, so each has energy at most
$\operatorname{area}_{g_j}(Q)\leq1+o(1)$. By
\eqref{lt:metric-trace}, its boundary discrepancy from the corresponding
Euclidean coordinate is bounded in $W^{1,2}(\partial Q)$ and tends to
zero in $L^2(\partial Q)$. The interpolation inequality
\[
\|v\|_{H^{1/2}(\partial Q)}^2
\leq C\|v\|_{L^2(\partial Q)}\|v\|_{H^1(\partial Q)}
\]
therefore makes the discrepancy tend to zero in $H^{1/2}$.
The trace extension operator on the fixed polygonal collar, followed
by a cutoff, extends it with vanishing Euclidean $W^{1,2}$ norm and
with support away from the outer half of the shell. Extend $X_j,Y_j$
accordingly so that they equal the Euclidean coordinates on that outer
half. Their individual shell energies are at most
$S^2-1+o(1)$, since the metric is scalar there. Their total individual
energies on $Q^+$ are thus at most $S^2+o(1)$.
Apply \eqref{lt:rectangle-lower} to these tests on
$R_{W_j,S}$. Both $S^3/W_j$ and $SW_j$ are at most $S^2+o(1)$;
hence $W_j\to S$. Moreover, if
$(\widehat X_j,\widehat Y_j)=(X_j,Y_j)\circ\Phi_j^{-1}$, equality
asymptotically in the two segmentwise inequalities gives
\begin{equation}
D(\widehat X_j,\widehat Y_j)\longrightarrow I
\quad\hbox{in }L^2(R_{W_j,S}).
\label{lt:coordinate-rigidity}
\end{equation}
For example, the horizontal integral of
$\partial_1\widehat X_j$ on every segment is $S$, and subtracting
$S^3/W_j$ from its energy gives exactly
\[
\int_{R_{W_j,S}}
\bigl(|\partial_1\widehat X_j-S/W_j|^2
+|\partial_2\widehat X_j|^2\bigr).
\]
The vertical coordinate has the analogous identity with derivative
$1$.
On the outer coordinate band, the pair in
\eqref{lt:coordinate-rigidity} equals $\Phi_j^{-1}$. For an
orientation-preserving similarity $A$ in dimension two,
\[
|A^{-1}-I|^2\det A=|A-I|^2.
\]
Change of variables therefore gives $D\Phi_j\to I$ in Euclidean
$L^2$ on the corresponding source band. Reflection and the analytic
interior estimates give uniform derivative convergence on a smaller
band containing the boundary. The corner normalization and $W_j\to S$
also give convergence of the values there. Blend $\Phi_j$ in this
fixed band with the linear map
$(x_1,x_2)\mapsto(W_jx_1/S,x_2)$. This gives extensions $P_j$ of the
boundary values with distortion $1+o(1)$; for large $j$ their derivatives
are close to the identity, so they are bi-Lipschitz homeomorphisms onto
the rectangle. The preceding energy argument now gives
\[
\int_Q|D(h_j\circ\rho_j)|^2
\leq(1+o(1))\,2\operatorname{area}_{g_j}(Q^+)
\leq2+C\tau\sup_jL(h_j)+o(1).
\]
Take $\tau\downarrow0$ diagonally. Finally, boundary agreement and
uniform convergence of the boundary values give, by integration in
coordinate directions,
\[
\int_Q D(h_j\circ\rho_j):D\iota\longrightarrow2.
\]
Expanding the square of the derivative difference and using the energy
bound proves the second conclusion in \eqref{lt:disk-sharp}.
\end{proof}
\subsection{Subcritical collapse onto boundary data}
The skeleton and radial-extension strategy is classical in Sobolev
approximation; see Bethuel \cite[Section~II.1, pp.~164--165, and
Section~II.3, Eqs.~(32), (37), pp.~170--171]{Bethuel1991}.
This background motivates the collapse below. Its injective
reparametrization, fixed image and exact boundary compatibility are
proved locally; Bethuel's density theorem does not supply these
homeomorphism requirements.
\begin{lemma}[Cell collapse]
\label{lt:collapse}
Let $m\in\{2,3\}$ and $1\leq p0$ there is a bi-Lipschitz parametrization $F:D\to H(D)$ with
$F|_{\partial D}=H\circ\sigma$ and
\begin{equation}
\int_D|DF|^p
\leq C_{m,p,c,C}\ell
\int_{\partial D}|D_t(H\circ\sigma)|^p+\eps.
\label{lt:collapse-estimate}
\end{equation}
The boundary data on shared cells are retained exactly, so the
parametrizations glue when those data agree.
\end{lemma}
\begin{proof}
Rescale to $\ell=1$ and place an incenter at the origin. Cone-extend
$\sigma$ from that center; this is bi-Lipschitz for the fixed data.
In radial coordinates
$x=rz$, $z\in\partial D$ and $0\leq r\leq1$, precompose the
resulting map by the radial homeomorphism with profile
\[
q_{\alpha,\delta}(r)=
\begin{cases}
(1-\delta)r/\alpha,&0\leq r\leq\alpha,\\
1-\delta+\delta(r-\alpha)/(1-\alpha),&\alpha\leq r\leq1.
\end{cases}
\]
Here $0<\alpha,\delta<1$, and the resulting parametrization is
\[
F_{\alpha,\delta}(rz)
=H\bigl(q_{\alpha,\delta}(r)\sigma(z)\bigr).
\]
On $r\leq\alpha$ all derivatives are
at most a fixed-data constant times $\alpha^{-1}$; hence the energy
there is $O(\alpha^{m-p})$. On $r>\alpha$ the radial derivative
tends to zero as $\delta\downarrow0$, while the tangential derivatives
tend essentially to $r^{-1}D_t(H\circ\sigma)(z)$. Indeed, the
angular derivative contains
$q_{\alpha,\delta}(r)DH(q_{\alpha,\delta}(r)\sigma(z))D_t\sigma(z)$.
The last factor is evaluated at the fixed boundary point $z$;
only the base map's tangential derivatives require a limiting
argument. The radial
Jacobian is comparable to $r^{m-1}$ with constants determined by the
cell shape. Essential compatibility and dominated convergence yield
an upper bound
\[
C\left(\int_\alpha^1r^{m-1-p}\,dr\right)
\int_{\partial D}|D_t(H\circ\sigma)|^p
\]
for the limiting outer energy. To apply the essential limits, use a
sequence of the individual bi-Lipschitz radial changes and the common
full-measure set of their Lebesgue representatives. Coning $\sigma$
only inserts its fixed a.e. angular derivative. Since $p0$ be interior
barycentric coordinates. For a constant $a>0$, consider the map
\[
\sum_{i=0}^2\lambda_i v_i
\longmapsto
\frac{\sum_{i=0}^2\lambda_i e^{a\lambda_i}v_i}
{\sum_{i=0}^2\lambda_i e^{a\lambda_i}}.
\]
As $a\to\infty$, a point with one largest coordinate moves toward
that coordinate's vertex. A regular curve in a sufficiently small
neighborhood of 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;
Figure~\ref{lt:tie-model} illustrates this local situation. In a
tetrahedron the same principle pairs a curve with a two-leader
surface and an edge weight, or a surface with a three-leader line
and a triangle weight. This is why the budget uses complementary
dimensions and flat primal faces.
\begin{figure}[tb]
\centering
\begin{tikzpicture}[x=0.95cm,y=0.95cm,font=\small]
\begin{scope}
\fill[blue!5] (0,0)--(2,0)--(2,1.155)--(1,1.732)--cycle;
\fill[orange!9] (4,0)--(3,1.732)--(2,1.155)--(2,0)--cycle;
\draw (0,0)--(4,0)--(2,3.464)--cycle;
\draw[dashed,gray] (2,0)--(2,1.155)--(1,1.732);
\draw[dashed,gray] (2,1.155)--(3,1.732);
\draw[blue!65!black,thick,-{Stealth[length=2mm]}]
(1.2,0.57) .. controls (1.7,0.72) and (2.25,0.45) .. (2.8,0.62);
\node[left,blue!65!black] at (1.2,0.57) {$\Gamma$};
\node[below left] at (0,0) {$v_0$};
\node[below right] at (4,0) {$v_1$};
\node[above] at (2,3.464) {$v_2$};
\node[font=\scriptsize,fill=white,inner sep=1pt] at (2,0.95)
{$\lambda_0=\lambda_1>\lambda_2$};
\end{scope}
\draw[-{Stealth[length=2mm]}] (4.55,1.6)--(6.25,1.6)
node[midway,above] {$a\to\infty$};
\begin{scope}[xshift=7.1cm]
\draw[gray!60] (0,0)--(2,3.464)--(4,0);
\draw[blue!65!black,very thick,-{Stealth[length=2mm]}] (0,0)--(4,0);
\node[below left] at (0,0) {$v_0$};
\node[below right] at (4,0) {$v_1$};
\node[above] at (2,3.464) {$v_2$};
\node[below] at (2,-0.3) {limiting edge contribution};
\end{scope}
\end{tikzpicture}
\caption{A two-dimensional model of a tie budget. Dashed segments
are the two-leader ties. The curve crosses one of them away from
the three-leader center; its image concentrates on the corresponding
edge. The right panel records the limiting contribution, not the
image under a finite-parameter homeomorphism.}
\label{lt:tie-model}
\end{figure}
For generic samples all relevant hits are finite on compact measured
pieces. Higher ties, simplex-boundary junctions, and lower-dimensional
source exceptions are avoided by Lemma~\ref{lt:trace-slicing} and
translation coarea. Counts consequently concern isolated hits at
regular underlying source strata; no differentiability of the original
image surface at such a hit is being assumed.
\subsection{A variable simplex squeeze}
\begin{lemma}[Compatible simplex squeezing]
\label{lt:squeeze}
Let $\mathcal T$ be a locally finite triangulation of a three-dimensional
open domain. Assign constants $00$, and any $e>0$, sufficiently small local
mesh upper bounds make
\begin{equation}
\mathcal B_1(\Gamma)\leq C\operatorname{length}(\Gamma)+e
\label{lt:curve-high-probability}
\end{equation}
hold with probability at least $1-\delta$. The constant $C$ is
independent of $\delta,e$ and the required mesh resolution; it is
universal away from enlarged patches. Prescribed summable failure
probabilities and local errors can be imposed simultaneously on a
countable locally finite family of curves. In a deep anisotropic
lattice patch there is also a grid-unit version: rescale to unit
grid aspect, estimate length there, and convert the weight back
using the longest physical side, with a constant independent of
the aspect ratio in those units.
\end{lemma}
\begin{proof}
Discard an arbitrarily small amount of length with multiplicity
and decompose the remainder into finitely many compact pieces on
$C^1$ graphs of arbitrarily small slope $s$ over straight axes.
Such pieces may be taken injectively from the regular parameter
branches, so multiplicity is accounted for by separate pieces.
Cover their axis projections by finitely many intervals of total
length at most the sum of their graph-piece lengths plus an
arbitrarily small error. Extend the graphs over these intervals
with comparably small slopes.
For each fixed relative-coordinate value $z$ in the joint
integration formula \eqref{lt:mesh-joint-density}, classify a dual
plane by its unit normal $\nu(z)$ and the graph axis $e$. If
$|\nu\cdot e|>2s$, the restriction of its affine defining function
to the graph is strictly monotone. Each such plane patch meets
the graph over its interval at most once. Charge a contributing
tetrahedron to an axis interval of length comparable to $d$
centered at a hit. Intersecting charged intervals have comparable
scales, by slow grading, and their vertices belong to a bounded
local candidate list. Thus the charges have bounded overlap.
The sum of mesh diameters, and hence of the flat edge weights,
is bounded by a constant times the graph length plus the local
mesh upper bound at each interval endpoint. After the finite
cover is fixed, all endpoint overhangs can be made as small as
desired by the mesh choice.
For the remaining normals, the translation-coarea factor on this
graph is at most $Cs$. For each fixed $z$, integration over the
physical translation variable bounds the crossing integral by
$Cs d^2$ times the local graph length. Multiply by the joint
density bound $Cd^{-3}$ and the edge weight $Cd$, and then
integrate over the bounded relative-coordinate set. This gives
expected cost at most $Cs$ times the graph length. The normal
classification is translation invariant, so it is legitimate
inside this unnormalized joint integral. The omitted trace length has correspondingly
small expected cost by Lemma~\ref{lt:budget-expectation}. Choose the omitted length
and $s$ sufficiently small, then choose the mesh bounds for the
finite graph cover. Markov's inequality applies only to these
small remainders, and makes their failure probability below
$\delta$ without changing the constant on the retained graph
length. This proves \eqref{lt:curve-high-probability}.
For a locally finite family assign summable failure probabilities
and errors and impose the corresponding local upper bounds
together. A union bound proves the simultaneous assertion.
In a deep lattice patch perform the entire argument after the
anisotropic rescaling. There are finitely many rational tie-plane
families in these units; the same argument, or translated Riemann
sums for their projected length coarea, gives the claimed constant.
Physical flat weights are bounded by the longest-side factor
times their grid-unit weights.
\end{proof}
\begin{lemma}[Sharp costs along a thin-grid subspace]
\label{lt:sharp-grid}
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
$k\in\{1,2\}$. 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. For general measured
$m$-vectors, tangent contributions have constants uniform in
$\eta$, and errors have a finite constant $C_\eta$.
After a fixed linear output normalization $A$ which keeps the two
subspaces orthogonal, the sharp coefficient relative to the normalized
tangent length or area is at most $1+C_A\eta$. Here $C_A$ depends only on
$\|A\|_{\op}$ and $\|A^{-1}\|_{\op}$. The tangent bounds remain
uniform in $\eta$, while the error constants may also depend on these
two norms.
The grid-unit bound of Lemma~\ref{lt:curve-probability} also has no
aspect-dependent factor on tangent lengths.
\end{lemma}
\begin{proof}
Rescale the grid to unit cubes, temporarily recording physical
flat weights separately. For $k=2$ slice a cube at a generic
height in its thin direction. Each Kuhn tetrahedron projects to
a triangle with one repeated vertex. Write the two unsplit
projected barycentric masses as $q_s,q_t$. On a fixed-height
slice, either part $\lambda_0$ of the split mass is a constant
minus a sum of some of $q_s,q_t$, with coefficients zero or one.
Consequently
\[
\det D_{(q_s,q_t)}(q_s-\lambda_0,q_t-\lambda_0)>0.
\]
These are also the orientation coordinates for a leader triangle
with three different projected vertices. Thus every such
projected triangle is counted positively.
The total projected weight on a horizontal period is exactly
its area. One way to see this without estimating individual
positions is to take a constant large $a$ in
\eqref{lt:squeeze-definition}. Projection of this periodic
simplexwise map on the horizontal slice is a torus map with
periodic displacement and degree one. Its signed Jacobian
integrates to one base area. Lemma~\ref{lt:squeeze} converts
this integral, in the large-$a$ limit, into the signed projected
triangle weights. Generic heights give only finitely many
transverse hits away from simplex boundaries; the signs just
computed are positive, and repeated projections contribute zero.
For a direct check, the two Kuhn tetrahedra whose vertical step
comes first supply total projected area one at heights
$2/30$ there is a locally
bi-Lipschitz homeomorphism $H:\Omega\to\Omega'$ such that
\begin{equation}\label{la:goal}
\int_\Omega \bigl(|H-f|^p+|DH-Df|^p\bigr)<\eps.
\end{equation}
In particular, the target of $H$ is exactly $\Omega'$, and
$H\in W^{1,p}(\Omega;\R^3)$.
\end{theorem}
We work on finitely many compact sets carrying controlled jets.
At rank three, radial blocks recover those jets; at ranks one and two,
kernel compression reduces the core estimate to a line or plane section,
with a further planar radial construction at rank two when $p<2$.
Boundary parametrization completes the remaining cells, and the final
proof records the order of choices and proves global summability.
Throughout the section, all modifications are made inside the open
domains. Constants denoted by $C$ depend only on the dimension, $p$,
and the fixed shape constants. A subscript records any additional
dependence. In an expression $o(1)$ involving a mesh size, all the
other parameters and the finite compact configuration have already
been fixed. The quantities tending to zero will always be
nonnegative costs or upper bounds for such costs.
\subsection{Composition at a collapsed stratum}
We first record the regularity fact used when a three-dimensional
region is compressed toward a line or a plane. It also explains why
the finite-piece regularity in the budget construction is retained.
\begin{lemma}[Tangential composition limit]\label{la:composition-limit}
Let $D$ be a compact Lipschitz parameter piece of dimension $d$, and let
$q_j,q_0:D\to U$ be Lipschitz maps into an open subset of a Euclidean
space. Suppose that their images lie in one compact subset of $U$,
that $q_j\to q_0$ uniformly, and that
\[
Dq_j\longrightarrow Dq_0\quad\hbox{a.e. on }D,
\qquad \sup_j\|Dq_j\|_{L^\infty(D)}<\infty.
\]
Let $v:U\to\R^a$ be Lipschitz on a neighborhood of that compact set.
Suppose there are finitely many closed pieces $C_\nu$ covering the
neighborhood and $C^1$ maps $v_\nu$, defined on neighborhoods of
$C_\nu$, such that $v=v_\nu$ on $C_\nu$. Then, for every finite
$r\ge1$,
\begin{equation}\label{la:composition-convergence}
D(v\circ q_j)\longrightarrow D(v\circ q_0)
\quad\hbox{in }L^r(D).
\end{equation}
No rank assumption is imposed on $Dq_0$. The assertion applies
separately to an edge, a face, or a volume piece.
\end{lemma}
\begin{proof}
For each $j\ge0$ and each $\nu$, apply the density theorem on the
parameter piece to $q_j^{-1}(C_\nu)$. At almost every point of this
preimage, locality of the weak derivative gives
\[
D(v\circ q_j)=Dv_\nu(q_j)Dq_j.
\]
Remove the union of the exceptional sets for the countably many $j$
and finitely many $\nu$. At a remaining parameter point, every branch
containing $q_0$ gives the same product with $Dq_0$, namely
$D(v\circ q_0)$. If a branch occurs along infinitely many $q_j$, its
closedness implies that it contains $q_0$. Continuity of its
derivative and convergence of $Dq_j$ therefore show that its products
converge to this common value. Finiteness of the branch list proves
pointwise convergence of the full sequence. The fixed derivative
bounds give a bounded majorant, so dominated convergence proves
\eqref{la:composition-convergence}. The proof takes place on $D$;
it does not pull back an ambient null set through a possibly
rank-deficient map.
\end{proof}
\subsection{Full-column-rank radial blocks}
Here the parameter dimension is either $m=3$, with $1\le p\le2$, or
$m=2$, with $1\le p<2$. In the latter case the parameter domain is a
flat plane section of the source. All target motions remain ambient
three-dimensional homeomorphisms.
Let $K$ be a compact subset of such a parameter patch. Assume that
there is a local $C^1$ ambient diffeomorphism $b$ such that
\begin{equation}\label{la:identity-jets}
f=b,\qquad D_m f=D_m b\quad\hbox{on }K
\end{equation}
up to the null sets of the indicated traces. The map $f$ on the patch
is Sobolev and has its continuous ambient values. Both $Db$ and
$(Db)^{-1}$ are bounded by a fixed constant, denoted collectively by
$C_b$. For $m=2$, the additional normal column of $b$ is chosen so
that its ambient orientation agrees with that of $f$.
Write
\[
F=b^{-1}\circ f.
\]
Thus $F=\Id$ and $D_mF=I_m$ on $K$, where $I_m$ is the derivative
of the identity inclusion of the parameter plane into $\R^3$.
Target coordinates in the following construction are always the
$b$-coordinates. If $m=3$ and $p=2$, then
\begin{equation}\label{la:inverse-bv}
F^{-1}\in BV_{\mathrm{loc}},
\end{equation}
by \cite[Theorem~1.1]{CsornyeiHenclMaly2010}. That theorem requires no
finite-distortion hypothesis.
Choose an integer $N$, put $l=N^{-1}$, and fix a sufficiently large
absolute constant $K_0$. Set
\begin{equation}\label{la:gamma}
\gamma=l/K_0.
\end{equation}
Use a uniformly shifted cubical grid of side $h$, with each macro cube
subdivided into tiles of side $lh$. The word ``cube'' includes a
square when $m=2$. Cubes meeting $K$ are candidates. In each candidate
$Q$, draw a center $a$ uniformly in a central cube of side $h/4$; retain
only centers belonging to $K$. Rays from $a$ through boundary mesh
vertices are the forward spokes. When $m=3,p=2$, also choose,
independently on each boundary mesh square, a point $u_j$ uniformly in
its central half and use the target ray from $a$ through $u_j$.
Cubical polar coordinates are written
\[
x=a+r(\omega-a),\qquad \omega\in\partial Q,
\]
so the boundary has radial coordinate $r=1$. In dimension two, the
ambient output box is the normal thickening of $Q$, centered on the
parameter plane and of height $h$.
\begin{lemma}[Radial block construction]\label{la:radial-blocks}
Fix finitely many separated compact configurations satisfying
\eqref{la:identity-jets}, and fix $l$. For sufficiently small $h$ one
can retain candidate blocks with the following properties.
\begin{enumerate}
\item The expected $m$-volume of discarded candidate blocks tends to
zero as $h\to0$. The union of retained blocks therefore differs
from $K$ by a set whose expected measure tends to zero.
\item Each retained $Q$ has the core
\[
B_Q=a+(1-8\gamma)(Q-a).
\]
The complement consists of shape-controlled frusta of scale $lh$, one
over each boundary tile. After the common budget construction and
the 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 its core boundary is
bounded by $C_b$.
\item 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 the 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.
\item If $m=3,p=2$, 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. After multiplication by $lh$, the expectation of their total converted budget
$\mathcal B_{\mathrm{wedge}}$ satisfies
\begin{equation}\label{la:wedge-cost}
\mathbb E\mathcal B_{\mathrm{wedge}}
\le C_b\gamma\sum_i |K_i|+o(1).
\end{equation}
Outer ordinary edges and faces have converted cost
$O_b(l)\sum_Q|Q|+o(1)$ on these configurations.
\end{enumerate}
The construction can be performed simultaneously with the other
budgeted pieces used below. In a volume block the only measured
interior surfaces are its wedges. In the plane-section application,
any additional graph to be fixed is assumed to have positive
clearance from $K$ and its image. Taking $h$ smaller then keeps all
expansion and cap supports off that graph. More precisely, write
$V$ for the ambient initial map after its output motions and
$q_Q:Q\to Q$ for the radial parameter change. The controlled trace
is $V\circ q_Q$. For $m=3$ these parameter changes are ambient
source self-homeomorphisms; for $m=2$ they are retained solely as
section parametrizations. The graph preservation means that the
radial motions leave its already-budgeted $H_0$-image unchanged;
it does not assert equality of $H_0$ with $f$ on that graph.
\end{lemma}
\begin{proof}
We give the selection, the maps, and the estimates in their order of
use.
\paragraph{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
\[
C\bigl|N_{Ch}(K)\setminus K\bigr|,
\]
which tends to zero. Here $N_t(K)$ denotes the parameter-space
$t$-neighborhood of $K$.
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
$D_mF=I_m$ on $K$. In the reverse system, use
\eqref{la:inverse-bv} and $F^{-1}=\Id$ on $K$, together with directional
$BV$ slicing and its separate absolutely continuous and singular
variation identities
\cite[Section~3.11, Theorems~3.103, 3.107, and~3.108]{AmbrosioFuscoPallara2000}.
At almost every line
density point of $K$, the absolutely continuous 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 $1+o(1)$.
For completeness, near-minimal length gives the single-crossing
sections required by Proposition~\ref{lt:star}. The radial distance along
one distorted initial ray attains every level up to $(1-o(1))t$ and
has total variation at most $(1+o(1))t$. One-dimensional coarea shows
that levels with more than one crossing have measure $o(t)$. A finite
union of these exceptional sets still has measure $o(t)$, so there
are arbitrarily small common levels crossed once by every ray in
the system. Injectivity and the local differentiability estimate
exclude remote tails from these shrinking sections.
The loop condition in Proposition~\ref{lt:star} is also obtained by slicing.
Fix one of a countable library of finite piecewise smooth loop
systems on the angular sphere. For a fixed library loop $\omega$, set
\[
E_t(a)=\int\bigl(|D_mF(a+t\omega(s))-I_m|^p
+\mathbf1_{\text{patch}\setminus K}(a+t\omega(s))\bigr)
|\omega'(s)|\,ds.
\]
Extend the integrands to a fixed neighboring patch. Translation
continuity and their vanishing on $K$ give
$\int_K E_t(a)\,da\to0$. A diagonal choice of $t_n\downarrow0$
with summable means for the first $n$ library loops gives
$E_{t_n}(a)\to0$ for every loop at almost every center. Each loop
then has a point in $K$, where $F=\Id$. Absolute continuity from
that point gives uniform convergence of
$(F(a+t_n\omega)-a)/t_n$ to $\omega$, also when $p=1$.
The ray estimates hold at all sufficiently small scales, so this
subsequence supplies the loop tests simultaneously with them.
Choose the library fine enough to avoid the separated cones and test
the required punctured-sphere generators. In the planar case
equatorial circles suffice. Countability permits all these tests at
the same generic centers.
For any prescribed small cone and positional tolerances depending on
$l$, dominated convergence in the center and relative variables now
shows that the failed tests have total block volume tending to zero.
In full dimension the oriented generator test itself forces the
comparison link to have the identity orientation; no separate
Jacobian-sign assertion is needed here. In the planar case the
orientation was fixed by the normal column of $b$.
\paragraph{Excluding intrusive edges.}
Call the unmodified fine lattice skeleton the default skeleton,
whether or not a candidate block will later be replaced. Any default
edge whose image can enter a proposed inset target support lies in a
shrinking source neighborhood $U_h=N_{\rho(h)}(K)$ of $K$, by
continuity of the inverse, where $\rho(h)\to0$. Enlarge $\rho(h)$
by $O(h)$ to include each entire relevant edge. No estimate
$\rho(h)=O(h)$ is needed or assumed.
For an edge $e$ meeting $K$, absolute continuity from an unchanged
point gives
\[
\sup_e|F-\Id|>c\gamma h
\quad\Longrightarrow\quad
\int_e|D_t(F-\Id)|^p\ge c(l,\gamma,p)h.
\]
Translation averaging makes the sum of these integrals in the
shrinking neighborhood $o(h^{1-m})$ for fixed $l$. Edges missing $K$
are charged by their entire length $lh$ to
$U_h\setminus K$. Thus the expected number of bad edges is
$o(h^{-m})$. Their total image length is $o(h^{1-m})$: use the same
derivative-error estimate and H\"older on the bad edges, and use the
original derivative integral on those wholly off $K$. For $p=1$
this is the direct length integral, with no H\"older gain required.
A rectifiable curve of length $L$ meets at most $C(1+L/h)$ grid blocks
at scale $h$, by division into subarcs of length at most $h$. In the
planar case first project the curve to the parameter plane in target
coordinates. Discarding all blocks struck by bad-edge images
therefore loses $o(1)$ total volume. Good nonincident edges are
$o(\gamma h)$ from their original positions and avoid the strictly
inset expansion supports. Accurate spoke positions also prevent a
spoke of one retained block from entering another block's support.
\paragraph{The protected germ and the radial maps.}
Apply Proposition~\ref{lt:star} at the retained centers. In the
subsequent use of Lemma~\ref{lt:relative-germs} and
Proposition~\ref{lt:budget-transfer}, require the initial map $H_0$
to satisfy
\[
b^{-1}\circ H_0=\Id\quad\hbox{near every retained center}.
\]
Keep the narrow forward cones and the small positional errors. In
the critical case, keep also the condition that the preimage of each
chosen target ray, up to $r\le1-\gamma$, lies in the corresponding
open radial sector inside $Q$, except at $a$. The star modification
preserves the angular conditions near the center. Away from a
smaller center neighborhood the constraints have compact positive
margins, and fine approximation with reciprocal control preserves
them. The simplex displacement is turned off still nearer the
germ. Thus the later budget transfer does not destroy the ray
clearance. For now fix the germ neighborhoods and margins. The
actual budget transfer is made only after the outer weight below
has been fixed; the next paragraphs specify the maps that will
then follow it.
Once $H_0$ is chosen, choose $\lambda>0$ so small that $a+\lambda(Q-a)$ is in the exact
germ. Precompose on $Q$ with the radial homeomorphism fixing
$\partial Q$ and sending $B_Q$ linearly onto this tiny cube. On the
remaining ray segments use the linear radial interpolation. In
target $b$-coordinates postcompose with a radial expansion $r\mapsto
e(r)$ which is the identity for $r\ge1-2\gamma$ and satisfies
\[
e(r)=\frac{1-8\gamma}{\lambda}r\qquad(0\le r\le\lambda).
\]
This gives exactly $b$ on $B_Q$. For $p<2$, any strictly increasing
piecewise linear profile between these prescribed portions is
sufficient. The supports for different blocks are disjoint.
\paragraph{Angular attenuation when $p=2$.}
For this paragraph $m=3$. On a boundary square $S_j$, write its
points in rays from $u_j$ as
\[
\omega=u_j+\rho R_j(\theta)v(\theta),\qquad 0\le\rho\le1.
\]
Here $R_j(\theta)$ is the distance to the square boundary. It is
comparable to $lh$ and is piecewise smooth with the corresponding
uniform derivative bounds. Choose $0<\kappa,t<1/4$ and
replace $\rho$ by
\[
\phi(\rho)=
\begin{cases}
(1-t)\rho/\kappa,&0\le\rho\le\kappa,\\
1-t+t(\rho-\kappa)/(1-\kappa),&\kappa\le\rho\le1.
\end{cases}
\]
The resulting map $A$ preserves $S_j$, fixes its boundary, and
expands its tiny central copy. On each smooth angular piece,
\begin{equation}\label{la:angular-det}
\det DA=\phi'(\rho)\frac{\phi(\rho)}{\rho},
\qquad
|DA|\le C\left(1+\frac{lh}{|\omega-u_j|}\right).
\end{equation}
The angular dependence of $R_j$ adds shear but leaves the displayed
determinant unchanged. On a compact set missing $u_j$, choose
$\kappa$ below its relative distance and then let $t\downarrow0$;
the determinant tends uniformly to zero there. For the straight
interpolation $A_\beta=(1-\beta)\Id+\beta A$, the radial profile is
$(1-\beta)\rho+\beta\phi(\rho)$. Away from the central copy its
radial slope is at most one, while its transverse expansion has the
bound in \eqref{la:angular-det}. Hence
\begin{equation}\label{la:angular-interpolation}
|\det DA_\beta|
\le C\left(1+\frac{lh}{|\omega-u_j|}\right),
\qquad0\le\beta\le1,\quad\rho\ge\kappa.
\end{equation}
Choose $0<\lambda<\lambda'<1/4$ still inside the exact affine region.
Turn $A$ on between $\lambda$ and $\lambda'$, leave it on through
$1-4\gamma$, and turn it off between $1-4\gamma$ and $1-3\gamma$.
Choose a slope $0<\alpha<\gamma$ on $[\lambda,1/2]$, and put
\[
k_\alpha=
\frac{6\gamma-\alpha(1/2-\lambda)}{1/2-2\gamma}>0.
\]
The denominator is bounded below, so $k_\alpha\le C\gamma$.
On $[1/2,1-2\gamma]$ use this slope. Explicitly, these portions are
\[
e(r)=
\begin{cases}
1-8\gamma+\alpha(r-\lambda),&\lambda\le r\le1/2,\\
1-8\gamma+\alpha(1/2-\lambda)+k_\alpha(r-1/2),
&1/2\le r\le1-2\gamma.
\end{cases}
\]
They meet $e(1-2\gamma)=1-2\gamma$ exactly. The slope $\alpha$
can still be decreased as required by the inner error estimate.
All maps for fixed positive parameters are bi-Lipschitz with
finitely many closed $C^1$ pieces.
In normalized cubical polar coordinates the target map is
\[
(r,\omega)\longmapsto
a+e(r)\bigl(A_{\beta(r)}(\omega)-a\bigr).
\]
Its two-angular-direction area factor is bounded by
\begin{equation}\label{la:two-angular}
C_b(e/r)^2|\det DA_{\beta(r)}|,
\end{equation}
and its mixed radial-angular factor is bounded by
\begin{equation}\label{la:mixed-angular}
C_b(e/r)\bigl(|e'|+e\,l|\beta'|\bigr)|DA_{\beta(r)}|.
\end{equation}
Indeed the radial derivative is
$e'(A_\beta-a)+e\beta'(A-\Id)$, the angular derivative is $eDA_\beta$,
and $|A-\Id|\le Clh$. Uniform transversality of the cubical radial
direction to a boundary tile gives these estimates for arbitrary
tangent two-planes as well.
These two estimates treat different components of a surface tangent
plane. Its purely angular component is controlled by the determinant
of $DA_{\beta(r)}$, while its mixed component uses only the first power
of $|DA_{\beta(r)}|$. Write
\[
G(\omega)=1+\frac{lh}{|\omega-u_j|}
\]
on each boundary tile. Figure~\ref{la:radial-bands} displays the
resulting area bounds in the unexpanded target coordinates. The inner
costs will be made small after $H_0$ has been constructed, using its
actual measured surfaces. The remaining costs will be bounded by a
continuous outer weight fixed before the budget transfer. This is why
the later attenuation choices do not change the budget already selected.
\begin{figure}[tb]
\centering
\begin{tikzpicture}[x=0.96cm,y=1cm,font=\scriptsize]
\fill[blue!18] (0,0) rectangle (1,0.85);
\fill[yellow!20] (1,0) rectangle (2,0.85);
\fill[blue!5] (2,0) rectangle (5.8,0.85);
\fill[blue!13] (5.8,0) rectangle (9,0.85);
\fill[orange!25] (9,0) rectangle (10.6,0.85);
\fill[orange!12] (10.6,0) rectangle (12.2,0.85);
\fill[gray!10] (12.2,0) rectangle (14,0.85);
\draw (0,0) rectangle (14,0.85);
\foreach \x in {1,2,5.8,9,10.6,12.2} {\draw (\x,0)--(\x,0.85);}
\node[align=center] at (0.5,0.42) {linear\\core};
\node[align=center] at (1.5,0.42) {turn\\on $A$};
\node[align=center] at (3.9,0.42) {arbitrarily small area\\after $H_0$ is fixed};
\node at (7.4,0.42) {$O(\gamma G)$};
\node at (9.8,0.42) {$O(G)$};
\node at (11.4,0.42) {$O(1)$};
\node at (13.1,0.42) {identity};
\node[above] at (7.4,0.88) {$A$ fully on};
\node[above,align=center] at (9.8,0.88) {turn off\\$A$};
\draw[-{Stealth[length=2mm]}] (0,-0.08)--(14.5,-0.08)
node[right] {$r$};
\foreach \x/\s in {0/0,1/\lambda,2/\lambda',5.8/\tfrac12,
9/1-4\gamma,10.6/1-3\gamma,12.2/1-2\gamma,14/1}
{\draw (\x,-0.02)--(\x,-0.15);\node[below] at (\x,-0.17) {$\s$};}
\draw[decorate,decoration={brace,mirror,amplitude=4pt}]
(9,-0.78)--(14,-0.78);
\node[below] at (11.5,-0.88) {outer band of width $O(\gamma)$};
\end{tikzpicture}
\caption{Critical radial area costs in the pre-expansion coordinate
$r$; intervals are not drawn to scale. Here
$G=1+lh/\dist(\omega,u_j)$ has bounded angular mean. The linear core
contains no uncontrolled connector face. Inner costs can be
suppressed after $H_0$ is fixed. Among the remaining terms, the middle
band carries the factor $\gamma$, and terms without this factor occur
only in the thin outer band.}
\label{la:radial-bands}
\end{figure}
The measured connector surfaces after source squeezing are truncated
wedges. Between $\lambda$ and $\lambda'$ they are exactly radial
over tile edges and are unchanged by $A$; their cost is arbitrarily
small with the increase of $e$ on that interval. Below $\lambda$
these surfaces are absent. On $[\lambda',1/2]$, all relevant
surfaces have positive angular clearance from the selected rays.
After $H_0$ and $\lambda'$ are fixed, choosing $\kappa,t$ sufficiently
small suppresses \eqref{la:two-angular}, and choosing $e'$ sufficiently
small suppresses \eqref{la:mixed-angular}. Only finiteness of the
fixed surface areas is used in this step. More explicitly, work in
units $h=1$ and let
\[
M_0=\int_{\lambda'\le r\le1-4\gamma}G(\omega)\,d\mu<\infty,
\]
where $\mu$ is the unexpanded $H_0$-area of the finitely many surfaces.
Choose $\kappa$ below the angular clearance of every measured
surface on $\lambda'\le r\le1-\gamma$, including the switch-off
band. This clearance follows from the inverse-ray sector condition
and compactness after removal of the exact germ. Thus every use of
\eqref{la:angular-interpolation} on these surfaces lies in
$\rho\ge\kappa$. The suppressed error is bounded by
\[
C_b\left(\frac{t}{(\lambda')^2}
+\frac{\alpha}{\lambda'}\right)M_0.
\]
The exact radial wedges on $[\lambda,\lambda']$ cost at most
$C_b L\alpha(\lambda'-\lambda)$, where $L$ is their total angular
edge length. Thus $t$ and $\alpha$ can make the sum smaller than any
given positive error, while $k_\alpha$ remains positive and bounded
by $C\gamma$.
On the remaining fully-on band the mixed cost is at most
$C_b\gamma(1+lh/|\omega-u_j|)$ times unexpanded area; the
two-angular cost can again be suppressed. On the switch-off band,
$l|\beta'|\le C l/\gamma=CK_0$, so
\eqref{la:angular-interpolation}--\eqref{la:mixed-angular} bound both
costs by a constant times the \emph{first} power of
$1+lh/|\omega-u_j|$. The radial-only outer end has a fixed bounded
area factor. Squaring the angular factor would not give the needed
averaging estimate, and is avoided by the determinant calculation.
\paragraph{A weight fixed before budgeting.}
Inside a retained volume block, the measured surface list consists
only of its full wedges. Every other measured source surface lies
outside the block interior; kernel templates remain in their own
macro blocks. The preimage of a selected target ray lies in the
open sector over its tile, so it meets neither these outside
surfaces nor a wedge, which lies on a sector boundary. On compact
ray segments away from the exact germ, local finiteness therefore
gives positive clearance from the entire measured surface list.
In the outer radial region choose a continuous buffered majorant
$W$. The angular envelope on a tile is a buffered version of
\[
1+\frac{lh}{|\omega-u_j|}.
\]
Its singularity can be capped in a tiny neighborhood of the ray:
the ray-preimage sector condition makes the outer compact portions
of all measured surfaces disjoint from that ray. First choose the
caps for the adjusted topological map with a positive margin, then
choose the budget mesh and $G,T$ fine enough to preserve the margin.
Cutoffs in disks of radius $O(lh)$ and a constant background make
the envelope continuous across tile boundaries. Write the resulting
bounded envelope as $\widehat G$. It agrees with an upper bound for
the uncapped factor on the measured surfaces and is bounded above
by a fixed multiple of the uncapped integrable envelope everywhere.
Choose continuous radial cutoffs with the following values, using
linear interpolation between the listed bands:
\[
\begin{array}{c|cc}
&0&1\\ \hline
\chi_{\mathrm{low}}&r\le1/2-\gamma&r\ge1/2\\
\chi_{\mathrm{out}}&r\le1-6\gamma&r\ge1-5\gamma\\
\chi_{\mathrm{cut}}&r\ge1-\gamma&r\le1-2\gamma.
\end{array}
\]
For a sufficiently large fixed $C_b$, take
\begin{equation}\label{la:outer-weight}
W=\chi_{\mathrm{out}}+
C_b\chi_{\mathrm{low}}\chi_{\mathrm{cut}}
(\gamma+\chi_{\mathrm{out}})\widehat G.
\end{equation}
It is zero for $r\le1/2-\gamma$, dominates the required
$C_b\gamma G$ on the middle band and $C_bG$ on the return ramp,
and equals the ordinary weight one for $r\ge1-\gamma$.
The radial outer end is covered as well, and the angular factors
are confined below $1-\gamma$. The part without the factor $\gamma$ occupies an
$O(\gamma)$ radial band. Buffered cutoffs may enlarge these bands
by another fixed multiple of $\gamma$ without changing the estimates.
This fixes a bounded continuous weight before
Proposition~\ref{lt:budget-transfer} is applied. The weight vanishes in
the star-modification neighborhoods, and the simplex map is the
identity on smaller exact-germ neighborhoods. Thus the measured
crossings used by the transfer are the original $f$-crossings. The
mesh-element suprema of $W$ give the weighted $H_0$-area upper bound.
The still smaller attenuation parameters in the preceding paragraph
are chosen \emph{after} $H_0$; they need not have been known when $W$
was fixed. Equations~\eqref{la:two-angular} and
\eqref{la:mixed-angular} prove the asserted weighted bound for all
connector, outer, or nonincident measured surfaces, with arbitrarily
small additive errors.
\paragraph{Spokes and the final contractions.}
After the expansion, each shell spoke lies in a ball of radius
$C\gamma h$ about its intended outer mesh vertex. Its entire image is outside the open linear $\lambda$-core by
injectivity and the exact germ, with the inner endpoint on that
core's boundary. Its angular direction remains in a sufficiently
narrow vertex cone, and along the whole trace its pre-expansion
radius is at most $1+o(\gamma)$. The angular maps fix vertices and have uniform local
Lipschitz bounds near them. These facts give the stated ball
inclusion.
Each such spoke has finite length. Homothetically contract its
entire 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, while the homothety factor may be as
small as required to reduce the whole spoke length to
$C_b\gamma h$. Group all spokes at a shared vertex and use the same
contraction for them. The twice-larger balls are pairwise disjoint
when $K_0$ is large enough. Their centers are outside the open cores.
There are $O(l^{1-m})$ boundary vertices, so their intersection with
a core has measure at most
\[
C l^{1-m}(\gamma h)^m\le C\gamma h^m.
\]
On the core the map before contraction is $b$; hence the contracted
map still has derivative bounded by $C_b$. All other length and
area estimates are multiplied by a bounded factor. No uniform
inverse Lipschitz bound in the contraction factor is required.
\paragraph{Averaging the wedge weight.}
The total area of the full wedges in a three-dimensional block is
$O(h^2/l)$. On their portions in $K$, $F=\Id$. The outer band has
relative radial thickness $O(\gamma)$, and the remaining nonzero
weight has the factor $\gamma$. Also, for a point $\omega$ fixed
before the choice of $u_j$,
\[
\mathbb E_{u_j}\left[1+\frac{lh}{|\omega-u_j|}\right]\le C,
\]
because $u_j$ is sampled in two dimensions at scale $lh$. An
indicator of successful selection can be dropped in this upper
bound. Multiplying the resulting weighted area by $lh$ gives
$C_b\gamma h^3$ per block.
On off-$K$ wedge portions, use the same uncapped angular envelope
before making any clearance-dependent choices. For normalized wedge
parameters $v$, translation continuity gives
\[
\int_K
\bigl(|Df|^2\mathbf1_{\Omega\setminus K}\bigr)(a+hv)\,da
\longrightarrow0,
\]
and integration in the relative variables and wedge parameters
gives the corresponding averaged trace estimate. The generic
surface formula in Lemma~\ref{lt:trace-slicing} applies. This proves
\eqref{la:wedge-cost}, including the later target-mesh averaging of
Lemma~\ref{lt:budget-expectation}. The outer grid has only
$O(l^{1-m})$ edges or faces at scale $lh$ per macro boundary, so its
converted bounded-derivative cost is $O_b(l)|Q|$. Its off-$K$ error
tends to zero by the same trace averaging. Nonincident edges avoid
the expansion; nonincident surfaces use the same clearance and
weighted estimate. All required assertions follow.
\end{proof}
\subsection{Compact jets and kernel blocks}
We next reduce the remaining positive-rank sets to line and plane
sections. The distinction between invertible, zero, and nonzero
singular derivatives already plays a central role in the planar
construction of Hencl--Pratelli
\cite[Section~1.1 and Section~3]{HenclPratelli2018}. Here the singular
models are handled by compressing their kernel directions. A matrix
with a prescribed rank need not be the derivative of any ambient
diffeomorphism; its kernel is used only to choose the source coordinates.
\begin{lemma}[Finite compact jet selection]\label{la:jet-selection}
Given $\varepsilon_0>0$, one can choose $M<\infty$ and finitely many
pairwise disjoint compact sets $K_i\Subset\Omega$, of ranks
$k_i\in\{1,2,3\}$, such that
\begin{equation}\label{la:jet-tail}
\int_{\Omega\setminus\bigcup_iK_i}|Df|^p<\varepsilon_0.
\end{equation}
On their nonzero singular values there are common bounds $M^{-1}$
and $M$. At rank three the sets have ambient $C^1$ diffeomorphism
charts $b_i$ agreeing with the values and derivatives of $f$, with
chart bounds $C_M$. At ranks one and two, for any subsequently chosen
$\delta>0$, the pieces can be refined so that
\begin{equation}\label{la:rank-model}
|Df-D_i|\le\delta\quad\hbox{on }K_i,
\qquad \rank D_i=k_i,
\end{equation}
with the same type of singular-value bounds. The compact pieces can
be enclosed in disjoint source neighborhoods and corresponding
disjoint target neighborhoods with buffers. Once all finite
multiplicative constants are fixed, the neighborhoods can be chosen
so that
\begin{equation}\label{la:excess-neighborhood}
\sum_i\int_{U_i\setminus K_i}(1+|Df|^p)
\end{equation}
is as small as prescribed, and the inverse images of the buffered
target neighborhoods lie in the corresponding $U_i$.
\end{lemma}
\begin{proof}
Sobolev maps are approximately differentiable almost everywhere.
Truncate the positive-rank sets where their nonzero singular values
lie in $[M^{-1},M]$, and choose $M$ so large that the lost derivative
integral is small. Rank-zero points have zero derivative energy and
need not be included. On each retained part, pass to compact subsets
with continuous jet data and uniform approximate differentiability,
losing an arbitrarily small further derivative integral.
These are $C^1$ Whitney jets after a further compact refinement. To
see the value condition, take two nearby retained points and compare
their approximate first-order expansions at a common good point in
the overlap of balls of radius comparable to their separation. The
uniform density losses can be made smaller than a fixed fraction of
the overlap, so such a point exists. Subtracting the two expansions,
using continuity of the jets, and then sending the differentiation
error to zero gives the Whitney remainder estimate. The $C^1$
extension theorem~\cite[Part~I, Section~4, Theorem~I]{Whitney1934} supplies the extensions. At full rank, continuity
of the derivative and the inverse function theorem give local
invertibility. Injectivity on the compact set, together with
compactness, makes the extension injective on a sufficiently small
neighborhood of the whole compact piece. Its derivative and inverse
derivative bounds can be kept within $C_M$.
For ranks one and two, cover the bounded rank-$k$ matrix range by
finitely many $\delta$-balls centered at rank-$k$ matrices with
comparable nonzero singular values. Refine the compact pieces
accordingly, losing an arbitrarily small integral to make the finite
pieces disjoint. Their number does not enter the local chart
bounds. Injectivity of $f$ makes their compact images disjoint as
well. Regularity of the integrable measure $(1+|Df|^p)\,dx$ gives
\eqref{la:excess-neighborhood}; continuity of $f^{-1}$ permits target
buffers whose preimages stay inside these source neighborhoods.
The initial losses can be allocated to give
\eqref{la:jet-tail}.
\end{proof}
Use Lemma~\ref{lt:source-mesh} with mesh sizes at most $lh$, deep
cubical regions around the $K_i$, and a common macro shift within
each such region. At rank $k=1,2$, choose orthonormal coordinates
$x=(y,z)$ with $z\in\ker D_i$ and $y\in\R^k$. At rank three use
Lemma~\ref{la:radial-blocks}. A failed macro block is left as ordinary
fine cells. Outside the retained macro blocks all cells are ordinary,
including the locally finite cells approaching $\partial\Omega$.
The anisotropic target grids of Lemma~\ref{lt:sharp-grid} are needed
only at rank one and at rank two when $p=2$. Their long directions
span $\operatorname{range}D_i$, their aspect parameter is $\eta>0$,
and a linear normalization $L_i$ can be chosen so that
\begin{equation}\label{la:normalization}
L_iD_i(y,z)=(y,0),\qquad
\|L_i\|+\|L_i^{-1}\|\le C_M.
\end{equation}
The range and its orthogonal complement remain orthogonal in this
normalization. Elsewhere the target grid can be isotropic.
Consider one rank-$k$ candidate macro cube $Q$ of side $h$. Let $B$
be its concentric product cube obtained by removing one fine layer
from each side, let $Y$ be its projection onto the $y$-coordinates,
and let $z_0$ be its central kernel coordinate. Put
\[
S=Y\times\{z_0\}.
\]
Construct a Lipschitz map
\begin{equation}\label{la:flat-collapse}
P_0(y,z)=(y,p_y(z)):Q\longrightarrow Q
\end{equation}
which is the identity on $\partial Q$ and collapses $B$ onto $S$.
For $y\in Y$, collapse the inner kernel cube to $z_0$ and extend
radially and linearly to the outer kernel boundary. Outside $Y$,
blend with the identity using a piecewise smooth $y$-cutoff. These
formulas may be chosen semialgebraic and piecewise $C^1$ on closed
pieces, with Lipschitz bound $C_l$. They preserve $y$ exactly, so
\begin{equation}\label{la:collapse-model-identity}
D_iDP_0=D_i.
\end{equation}
For $0<\lambda\le1$, put
\begin{equation}\label{la:positive-collapse}
P_\lambda=(1-\lambda)P_0+\lambda\Id.
\end{equation}
For each fixed $y$ the kernel radial slopes are positive. The map is
triangular in $(y,z)$, fixes the boundary, and is bi-Lipschitz on $Q$.
Budget the shell edges after $P_0$, and also its shell faces when
$p=2$. Include $S$, its tile edges, and all lower-dimensional
intersections that occur in these templates. In source and target
units divided by $h$, require that the sum of the $p$-power
tangential derivative errors from the model on these finitely many
pieces is at most $\tau^p$. On a $P_0$-image this means the error
from $D_iDP_0=D_i$; on $S$ it means the error from its $y$-columns.
The unprojected section is required to carry its Sobolev trace even
when $k=2,p<2$.
\begin{lemma}[Successful kernel blocks]\label{la:kernel-selection}
For fixed $M,l,\tau$, the total expected volume of candidate
rank-one or rank-two blocks failing these trace tests tends to zero
as first $\delta\to0$ and then $h\to0$, with sufficiently small
excess neighborhoods. In successful blocks the transferred
unparametrized shell data have converted cost
\begin{equation}\label{la:kernel-shell-budget}
O_M(l)\sum_Q|Q|+\text{an arbitrarily small error}.
\end{equation}
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.
\end{lemma}
\begin{proof}
Translation averaging on the fixed product templates, with block
weight $h^3$, bounds the expected sum of normalized trace errors by
\begin{equation}\label{la:trace-test-expectation}
C_l\sum_i\int_{N_{Ch}(K_i)}|Df-D_i|^p.
\end{equation}
The measured curve strata, and the measured surface strata when $p=2$,
are covered by Lemma~\ref{lt:trace-slicing}. For the additional
unprojected rank-two plane section when $p<2$, ordinary Sobolev slicing
and translation Fubini supply the $W^{1,p}$ trace used in these tests.
This step requires neither an image-area formula nor a two-dimensional
Lusin property. On $K_i$, the integrand is at most
$\delta^p$; off $K_i$ it is bounded by
$C_M(1+|Df|^p)$, whose excess integral is small. Markov's inequality
therefore makes the failed volume small relative to the fixed
threshold $\tau^p$.
There are $O(l^{-2})$ shell cells of side $lh$. In grid units, with
the longest side restored after the anisotropic rescaling,
\eqref{la:collapse-model-identity} prevents inflation of the model
term. The factors from $DP_0$ and from anisotropy multiply only
$Df-D_i$. Lemma~\ref{lt:curve-probability}, followed by H\"older
on each curve, gives the edge length-power budgets. At $p=2$,
Lemma~\ref{lt:budget-expectation} and the squared derivative trace
integrals give the face budgets. A bounded model derivative costs
$C_M(lh)^3$ per shell cell after conversion; multiplying by
$O(l^{-2})$ gives $C_Mlh^3$ per macro block. For fixed $l,\eta$, the
error contributions are made as small as desired by $\tau$ and the
budget-transfer tolerances. All relevant images lie deep inside
their prescribed target patches for small $h$, by the source and
target buffers. This proves \eqref{la:kernel-shell-budget}.
\end{proof}
\subsection{Parametrizing the retained line and plane sections}
For a successful rank-$k$ block we shall find a bi-Lipschitz
self-homeomorphism $\sigma_Y$ of $Y$, preserving its tile complex,
and a tangential map $h_Y$ into the target. If $V$ denotes the initial
map $H_0$ after all the required output motions, the relation is
\begin{equation}\label{la:section-parametrization}
h_Y(y)=V(\sigma_Y(y),z_0).
\end{equation}
The aim is
\begin{equation}\label{la:section-goal}
\sum_{\text{successful }Q}
h^{3-k}\int_Y |Dh_Y-D_i|_y^p
\quad\hbox{arbitrarily small}.
\end{equation}
Only the $y$-columns of $D_i$ occur in this formula. The factor
$h^{3-k}$ is the volume of the kernel fibers, up to fixed constants.
\paragraph{Rank one.}
On each interval tile, reparametrize the transferred curve at
constant speed in the normalization $L_i$. By
Lemma~\ref{lt:sharp-grid}, its expected length budget, divided by
the tile length, is at most
\begin{equation}\label{la:sharp-length-budget}
1+C_M\eta+o_{\tau}(1)+o_{\mathrm{transfer}}(1),
\end{equation}
where $l$ and $\eta$ have already been fixed. Its budget is also at
least $1-o(1)$: the section trace test gives almost correct endpoint
differences, and a sufficiently fine transfer preserves the values
to a still smaller error. Consequently the total positive excess,
weighted by tile length times $h^2$, has arbitrarily small expectation.
The constant speeds are bounded by $C_M$ simultaneously, using
Lemma~\ref{lt:curve-probability}. To see that its grid-unit constants
do not deteriorate with $\eta$, split the normalized trace derivative
into the tangent model and the error. The model has size bounded by
$C_M$ in longest-side-restored grid units. The error has $p$-mean at
most $C_{\eta,l}\tau$, and is made small after the grid parameters
are fixed.
Here is the elementary sharpness calculation. On a unit normalized
interval, write $u$ for the constant-speed curve, $s=|u'|$, and
$d=u(1)-u(0)$. If $e$ is the desired unit direction, then
\begin{equation}\label{la:length-sharpness}
\int_0^1|u'-e|^2=s^2+1-2e\cdot d.
\end{equation}
Since $s\le C_M$, $|d-e|$ is small, and $s\ge1-o(1)$, the right side
is bounded by $C_M(s-1)_+$ plus an arbitrarily small error.
For $1\le p\le2$, H\"older converts this squared error to a
$p$-error. Rescaling the tiles and summing with the fiber weights
proves \eqref{la:section-goal}. In particular the argument works at
$p=1$; it does not appeal to strict convexity of the $L^1$ norm.
\paragraph{Rank two at the critical exponent.}
Now $k=p=2$. For every planar tile, use the area budget in the
normalization $L_i$. Lemma~\ref{lt:sharp-grid}, applied to its
oriented tangent $2$-vector and the squared derivative trace error,
gives expected normalized area at most
\begin{equation}\label{la:sharp-area-budget}
1+C_M\eta+o_\tau(1)+o_{\mathrm{transfer}}(1).
\end{equation}
There is also the lower bound $1-o(1)$ for every sufficiently fine
generic sample. Indeed the edge tests make the original boundary
trace uniformly close, after translation and normalization, to the
boundary of the identity square. The transfer preserves that
closeness. Orthogonal projection of the transferred disk to the
model plane has degree one at every point a small distance inside
the square. It therefore covers that inner square, and its area is
at least the inner-square area. Since the transferred area is no
larger than its budget plus a chosen small error, the same lower
bound holds for the budget. These tiles lie away from all exempt
star centers.
It follows that the weighted sum of the positive area excesses has
arbitrarily small expectation. To obtain the needed boundary
parametrization, subdivide each common edge into $J$ fixed intervals
and use constant speed on each interval, preserving its endpoints.
Choose $J$ large before the finer trace and transfer tolerances.
Lemma~\ref{lt:curve-probability} bounds the normalized speeds by
$C_M$; the error part is $C_{\eta,l,J}\tau$ and is made small later.
Every reparametrized point stays within its $J$-interval, so the
boundary values converge uniformly to the ideal translated square
as $J\to\infty$ and the finer errors tend to zero. The boundary
squared derivative integral is uniformly bounded.
Absorb these common edge changes by coning in each tile, then apply
Lemma~\ref{lt:disk}. Fix a desired derivative tolerance $\zeta>0$.
The sharp assertion of Lemma~\ref{lt:disk} supplies an area-excess tolerance
$\rho>0$ and a boundary-closeness tolerance, using the already fixed
speed bound. On tiles whose normalized area is at most $1+\rho$,
it gives squared derivative error at most $\zeta$ in tile units.
For the remaining tiles, the sum of their converted tile measures
is bounded by $\rho^{-1}$ times the positive excess. The crude disk
estimate bounds their energy by a constant times area plus the
bounded boundary energy. Thus their total converted energy is at
most
\[
C_M(1+\rho^{-1})\,
\bigl(\hbox{total converted positive excess}\bigr)
+o(1).
\]
Make the positive excess small after fixing $\rho$. This proves
\eqref{la:section-goal} also in the critical rank-two case. The
finite-piece and edge regularity needed by the disk estimate will
be checked below before the volume gluing.
\paragraph{Rank two below the critical exponent.}
Suppose $k=2$ and $1\le p<2$. The macro selection has already fixed
finitely many plane sections. On each $S$, first discard points
where the tangential derivative differs substantially from the
injective $y$-columns of $D_i$. On the retained set its two singular
values lie between bounds depending only on $M$. The trace test and
Chebyshev's inequality show that the discarded integral of
$1+|D_yf|^p$ 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
arbitrarily small additional integral. The jets extend to ambient
charts $b$ by adding a normal column with the ambient orientation of
$f$. Their derivative and inverse derivative bounds are $C_M$.
Thus the leftover part of the section satisfies
\begin{equation}\label{la:section-leftover}
h^{-2}\int_{\text{leftover}}(1+|D_yf|^p)
\le C_M\tau^p+\text{an arbitrarily small error}.
\end{equation}
Choose disjoint compact neighborhoods on the 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.
Inside each section use a new macro grid of side $s$ and a fine grid
of side $ls$, with $s\to0$ only after the macro data have been fixed.
To respect the original tile seams, insert their fixed coordinate
levels, omit shifted $s$-levels within $s/4$ of a seam, and subdivide
each remaining gap into $N$ equal pieces. The resulting rectangles
have controlled aspect ratios. Near a seam, every nonfixed
subdivision level has a shift coefficient of magnitude at least
$1/N$, so its translation density is bounded for fixed $l$.
Only boundedly many such levels occur per seam in an $O(s)$ band.
Therefore their edge trace integrals, multiplied by $ls$, are bounded
in expectation by $C_l$ times the section derivative integral on
that shrinking band. The fixed seam lines have finite trace
integrals and their converted weights tend to zero. Both
contributions vanish as $s\to0$.
Apply Lemma~\ref{la:radial-blocks} with $m=2$ on the new compacts,
simultaneously with the preparation of $H_0$. All target radial and
vertex motions avoid the already-budgeted fixed graph because of the
clearance just established. On each successful micro block, make
the radial source change only as a parametrization of the section,
and keep its controlled core. Parametrize the remaining planar
edges at constant speed, except for the prescribed core edges, and
apply Lemma~\ref{lt:collapse} to the remaining two-dimensional cells.
It applies because $p<2$. Do the same on fallback micro cells.
These changes fit together to form a tile-preserving bi-Lipschitz
map $\sigma_Y$ of the section. They are not inserted as an
additional ambient precomposition.
On the compact jets in the micro cores, the map equals $b$, apart
from the bounded-derivative caps of Lemma~\ref{la:radial-blocks}; its
derivative is $D_yf$ there. The cap fraction and shell contributions
are $O_M(l)$ in section units. The derivative is bounded by $C_M$
on the remaining core portions. The shell spokes have image length
$O_M(\gamma s)$, so constant-speed parametrization gives their
required bounded energy. Ordinary outer and fallback edges use the
individual high-probability length bounds, followed by the planar
collapse estimate. The latter has the form
\[
\int_D|Dg|^p
\le C(ls)\int_{\partial D}|D_tg|^p+\text{a chosen error}.
\]
Translation averaging and \eqref{la:section-leftover} therefore give
\begin{equation}\label{la:micro-section-error}
h^{-2}\int_Y|Dh_Y-D_i|_y^p
\le C_M l+C_M\tau^p+o_s(1)
+\text{an arbitrarily small error}.
\end{equation}
Here failed micro blocks contribute vanishing cost by their failed
area and bounded compact jets; elsewhere the ordinary converted
trace cost is bounded by the leftover integral. The constants in
that latter bound depend on $M$, not on later micro-chart widths or
their number. The seam errors were handled separately above.
Multiplying by $h$ and summing proves \eqref{la:section-goal}.
On the original tile edges, the same construction gives an upper
$p$-energy bound by a fixed constant times the original trace
$p$-energy plus chosen errors. There are no micro output motions
on those edges.
\subsection{From section cores to volume cells}
We now keep all finite section data and output motions fixed. On a
rank-$k$ macro core use
\begin{equation}\label{la:kernel-core-map}
G_\lambda(y,z)
=V\bigl(P_\lambda(\sigma_Y(y),z)\bigr),\qquad(y,z)\in B.
\end{equation}
Since $P_0$ is flat on $B$, its argument here is
\[
(\sigma_Y(y),z_0+\lambda(z-z_0)).
\]
Lemma~\ref{la:composition-limit} gives
\begin{equation}\label{la:kernel-core-limit}
DG_\lambda\longrightarrow[Dh_Y,0]
\quad\hbox{in }L^p(B)\qquad(\lambda\downarrow0).
\end{equation}
The outer map $V$ has finitely many closed-piece $C^1$ formulas on
the compact configurations: this is true for $H_0$ and for each
radial, angular, and cap map. The inner maps 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 one or two.
The core boundary values preserve the images of its individual facets.
Indeed, set
\[
w_\lambda=V\circ P_\lambda,\qquad
T_B(y,z)=(\sigma_Y(y),z).
\]
Then $G_\lambda=w_\lambda\circ T_B$ on $B$. Since $\sigma_Y$ preserves
the tile complex, $T_B$ preserves each fine facet and edge of
$\partial B$. In particular,
\[
G_\lambda(F)=w_\lambda(F)
\quad\hbox{for every such facet }F.
\]
Thus these boundary values can be prescribed on the adjacent shell
cells without changing their base face images. The product formula
need not be extended through the shell; its base map there is
$w_\lambda=V\circ P_\lambda$.
Apply Lemma~\ref{la:composition-limit} separately on its edges and
faces, with limiting parameter map $P_0$. Their lengths and areas
converge to the transferred budgets, or to smaller upper bounds
when rank is lost.
We record the boundary scaling of the prescribed core data. Facets
normal to a kernel coordinate have limiting energy bounded by
\begin{equation}\label{la:normal-kernel-face}
C h^{2-k}\int_Y|Dh_Y|^p.
\end{equation}
For $k=2$, facets normal to a $y$-coordinate use the original tile-edge
parametrizations on $\partial Y$; their total energy per macro block
is $O_M(h^2)$. For $k=1$ their limiting energy is zero. The total
limiting edge energy on the subdivided $\partial B$ is
\begin{equation}\label{la:core-edge-energy}
O_M(h/l).
\end{equation}
For the critical rank-two case this follows from the uniform edge
speed bounds. Below the critical exponent use instead the
individual original tile-edge energy estimates just proved; the
trace tests bound their sum. Edges replicated in kernel directions
have the same tangential data, and kernel-axis edges have zero
limiting energy. Multiplying \eqref{la:normal-kernel-face} by $lh$
and \eqref{la:core-edge-energy} by $(lh)^2$ gives
$O_M(l)$ times the macro-volume contributions, using
\eqref{la:section-goal} for the first expression. Small positive
$\lambda$ preserves these estimates with arbitrarily small error.
\begin{lemma}[Completion of uncontrolled cells]\label{la:cell-completion}
Let $D$ be an uncontrolled source three-cell of scale $H$ in the
construction. Its base embedding $v:D\to\R^3$ is bi-Lipschitz and is
given by finitely many $C^1$ formulas extending to neighborhoods of
their closed pieces. Write $g$ for the prescribed or subsequently
chosen boundary data. On every prescribed core face $F$,
assume that $g|_F=v|_F\circ s_F$, where $s_F:F\to F$ is a
bi-Lipschitz self-homeomorphism preserving each edge. These self-maps
agree on shared edges. When $p=2$, their edge restrictions have
derivatives with essential limits almost everywhere.
Then the cell can be reparametrized bi-Lipschitzly, preserving its
base image and all prescribed boundary data, with derivative cost
at most a constant times
\begin{equation}\label{la:cell-cost}
\begin{split}
H\sum_{F\text{ prescribed}}\int_F|D_tg|^p
&+H^2\sum_{e\subset\partial D}\int_e|D_tg|^p\\
&+\mathbf1_{\{p=2\}}\,
H\sum_{F\text{ uncontrolled}}\operatorname{Area}(v|_F)
\end{split}
\end{equation}
plus any prescribed positive error. An unprescribed edge of length
comparable to $H$ may be given energy at most
$C\operatorname{length}(v(e))^pH^{1-p}$.
The constants depend only on $p$ and the fixed shape bounds, and
adjacent cells can be completed with identical shared-face data.
\end{lemma}
\begin{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~\ref{lt:collapse} in dimension two gives
face energy bounded by $CH$ times its boundary $p$-energy plus a
chosen error. If $p=2$, use the crude estimate of
Lemma~\ref{lt:disk}; after scaling it bounds the face energy by a
constant times its base image area plus $H$ times its 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
almost-everywhere 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 prescribed kernel edges, the source self-map is the restriction
of $T_B$. In the rank-one case it is the interval arclength change;
in the critical rank-two case it is the arclength change on each fixed
$J$-subinterval 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, even though the
resulting curves need not have constant speed for the base map
$w_\lambda$. On radial-core facets the prescribed map already equals
$v$, so $s_F=\Id$. Thus all prescribed edges needed for a critical
face satisfy the stated condition. Coning the edge changes preserves the
compatibility required by Lemma~\ref{lt:disk}.
The assembled boundary map $g$ gives the face-preserving bi-Lipschitz
self-homeomorphism $v^{-1}\circ g$ of $\partial D$. Apply
Lemma~\ref{lt:collapse} to this boundary self-map in dimension three,
where $p\le2<3$.
It contributes a factor $H$ times the true boundary-face energy.
Substituting the preceding face estimates gives
\eqref{la:cell-cost}. This last collapse requires the radial
compatibility of the base parametrization and a fixed bi-Lipschitz
boundary map; it does not require stronger smoothness of the entire
boundary reparametrization. Allocate the intermediate errors so
their converted sum is below the prescribed cell error.
\end{proof}
\subsection{Global choices, summability, and strong convergence}
\begin{proof}[Proof of Theorem~\ref{la:approximation}]
We first construct locally bi-Lipschitz approximants; smoothing is
performed only after their global derivative costs have been
estimated. If necessary, compose with a reflection to fix an
orientation convention, and undo it at the end.
\paragraph{Order of the fixed data.}
Let $\varepsilon_0>0$ be a tail tolerance, to be sent to zero.
Choose the compact rank truncation and its bound $M$ in
Lemma~\ref{la:jet-selection}. Choose $l$ so small that all the
$O_M(l)$ contributions in volume and section shells and in cap
regions are below a prescribed auxiliary tolerance. Recall that
$\gamma=l/K_0$.
With $M,l$ fixed, choose the sharp derivative tolerances for the
line and plane tiles. The speed bounds entering those tolerances
are fixed in terms of $M$. When $k=p=2$, choose the 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 the factors
depending on $l,\eta,J$. Next choose $\delta$ in
\eqref{la:rank-model} small enough, refine the compact pieces, and
choose their nested excess neighborhoods so that
\eqref{la:excess-neighborhood} is small after multiplication by every
now fixed inflation factor. These bounds do not depend on the
widths of the neighborhoods or on the number of their compact
pieces. Only then choose $h$ small and make the macro success
selections. The planar micro configurations, when present, are
chosen afterward, using \eqref{la:section-leftover} and then small
$s$. Their ambient chart constants are controlled by $M$.
Fix the bounded radial area weights and compact clearance margins
before choosing the target budget mesh. Choose that mesh using
Proposition~\ref{lt:budget-transfer} and Lemmas~\ref{lt:budget-expectation},
\ref{lt:curve-probability}, and~\ref{lt:sharp-grid}. This determines
$H_0$, with the required exact affine germs. The small radial
parameters, angular attenuation parameters, and vertex contractions
are then 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~\ref{la:cell-completion}, with summable positive
error prescriptions. Lemma~\ref{pre:local-tolerances} permits the
locally finite fine choices and these summable prescriptions.
\paragraph{Controlling ordinary and failed cells.}
We give the accounting that makes this order of choices sufficient.
For ordinary cells meeting no compact jet set and no enlarged special
neighborhood, source sampling bounds the converted trace integrals in
expectation by
\begin{equation}\label{la:ordinary-tail}
C_p\int_{\Omega\setminus\bigcup_iK_i}|Df|^p.
\end{equation}
On the simultaneous individual curve-bound event, H\"older's inequality
bounds the converted edge costs by these source trace integrals. When
$p=2$, the conditional expectation over the later target mesh bounds
the converted face-area budgets by the corresponding source traces.
Substituting in \eqref{la:cell-cost} gives precisely the codimension
conversion factors $H$ for faces and $H^2$ for edges. The constants in
\eqref{la:ordinary-tail} are ordinary mesh and budget constants,
independent of $M,l,\eta$.
In a failed macro block, the compact-jet integrands are bounded by a
constant depending on $M$. For the ordinary fine grid, the total
converted size of edges and faces in that block is $O(h^3)$, with
fixed factors from any special target normalization allowed.
The converted source trace integrals over its compact-jet portions
are therefore bounded deterministically by $C h^3$. After the source
mesh and failed blocks have been selected, the target area-budget
expectation applies to these fixed traces. On the individual
curve-bound event, H\"older's inequality gives the same bound for
converted edge costs. Thus the small failed-block volume controls
their total without requiring a source trace law conditional on
failure. On off-compact portions, discard the failure indicator
before applying the unconditional source trace bounds.
Whenever a nonuniversal constant can occur, its whole source trace
lies in one of the chosen enlarged source neighborhoods for small
enough meshes. Indeed its image meets a buffered anisotropic or
radial-motion zone, whose inverse image is compactly inside that
neighborhood; the small cell diameter and the fine initial-map
errors preserve this inclusion. Thus even an edge length-power
estimate can pay its finite factor on the \emph{whole} edge's
H\"older integral, then split this integral into the bounded
compact-jet part and the small excess part. By
\eqref{la:excess-neighborhood}, all these excess contributions are
as small as prescribed. No later large constant multiplies the
unprotected tail in \eqref{la:ordinary-tail}.
For full-rank radial blocks, Lemma~\ref{la:radial-blocks} gives the
small wedge budget and the contracted spoke lengths. On compact
points of a failed full-rank block, $F=\Id$, so their image points
cannot belong to another block's inset angular support. Outer
ordinary traces of retained blocks have the $O_M(l)$ converted cost
already computed. Any off-compact surface portion affected by an
angular map is estimated by the uncapped envelope, integrating in
$u_j$ before the later clearance-dependent caps. Its conditional
mean is bounded for the fixed trace template. This gives the same
small excess estimate. Nonincident edges avoid the angular
supports and hence never pay a power of that angular envelope.
Target-mesh suprema of the fixed bounded weights are allowed an
arbitrarily small enlargement error by a still finer target mesh.
The kernel shells are controlled by
Lemma~\ref{la:kernel-selection},
\eqref{la:normal-kernel-face}, and
\eqref{la:core-edge-energy}. Their prescribed face and edge
contributions in \eqref{la:cell-cost} are $O_M(l)$ in bulk units.
The radial volume shells have the same bound, since their prescribed
core faces have bounded derivatives and their radial edges have
length $O_M(\gamma h)$. All estimates are relative to the original
straight product or frustum cell sizes; no shape bound is imposed
on the intermediary image cells.
\paragraph{Simultaneous choices.}
Each estimate just used bounds a sum of nonnegative errors. In the
sharp line and plane cases, this sum is the positive excess, not the
signed total excess alone; the lower bounds in the section argument
are what allow that replacement. Choose the expectations smaller
than the desired tolerances with enough slack, and use Markov's
inequality for the finitely many aggregate objectives. The source
and micro configurations are fixed first, using the trace,
failed-volume, and angular-weight bounds. The target sample is
chosen next, meeting the expectation objectives together with the
arbitrarily high-probability individual curve bounds. Countably
many ordinary curves are handled with summable failure allowances
and local mesh upper bounds. Generic slicing and intersection
conditions have full measure on their templates; the star conditions
have already been accounted for by the small failed volume.
In particular, the sharp disk tolerances are selected using fixed
speed bounds. Make the positive excess small only after their area
threshold $\rho$ is chosen. The converted measure of bad sharp
tiles and their crude energy are then small by the explicit
$\rho^{-1}$ estimate above. This avoids any assertion that the
lattice intersection count itself converges pointwise on a
wrinkled trace. These choices yield deterministic configurations
with all the stated bounds simultaneously.
\paragraph{The resulting homeomorphism.}
Let $V$ be $H_0$ after the ambient radial-angular expansions and the
vertex contractions. Their supports are disjoint within each
family. Distinct compact configurations have separated target
neighborhoods; the much later micro-section supports were chosen
smaller still, and avoid all fixed graph data. In the rank-three
volume blocks precompose by the radial source squeezes. In kernel
blocks use $P_\lambda$ as the shell base map and
\eqref{la:kernel-core-map} on the core. Section micro squeezes enter
only through $\sigma_Y$. Complete all remaining edges, shared faces,
and volume cells as in Lemma~\ref{la:cell-completion}.
Each change is a self-parametrization relative to the same base
image, with matching boundary values. The meshes are locally finite,
and each interior compact meets only finitely many pieces. Finite
local straight cell geometry and the finite local bi-Lipschitz
constants give a locally bi-Lipschitz map across the interfaces.
It is a homeomorphism of $\Omega$ onto the same target as $V$.
The initial $H_0$ is onto $\Omega'$, and every output change is a
self-homeomorphism supported inside $\Omega'$. Consequently the
assembled map $H$ is onto exactly $\Omega'$.
\paragraph{Derivative error.}
Let $\mathcal C$ be the union of the controlled volume cores. The
preceding bounds imply
\begin{equation}\label{la:outside-core-energy}
\int_{\Omega\setminus\mathcal C}|DH|^p
\le C_p\varepsilon_0+o(1),
\end{equation}
where the auxiliary errors, including $O_M(l)$, can be made as small
as prescribed in their stated order. The original derivative has
the same small bound on this complement: outside the compacts use
\eqref{la:jet-tail}, while on the compacts the shell fraction and
failed volume are small and the jets are bounded.
On a full-rank core, $H=b_i$ except on the small bounded-derivative
caps. On its compact jet set, $Db_i=Df$; on the remaining portion the
chart derivative is bounded and the excess integral is small.
Thus the derivative difference there is small. On a kernel core,
\eqref{la:kernel-core-limit} and \eqref{la:section-goal} give closeness
to $D_i$ with the correct fiber factor $h^{3-k}$. Equation
\eqref{la:rank-model} compares $D_i$ with $Df$ on $K_i$, and the
excess-neighborhood bound treats the remaining points. Choose the
finitely many $\lambda$'s sufficiently small after the other data.
Combining these estimates with \eqref{la:outside-core-energy} yields
\begin{equation}\label{la:derivative-final}
\int_\Omega|DH-Df|^p\le C_p\varepsilon_0+o(1).
\end{equation}
All uncontrolled-cell errors were chosen summably. The displayed
bounds therefore also establish $\int_\Omega|DH|^p<\infty$, rather
than only local Sobolev regularity.
\paragraph{Value convergence.}
The source self-parametrizations move points only within cells or
blocks whose maximum diameter tends to zero. The initial budget map
can have arbitrarily small value error, and the target motions are
supported in boxes of arbitrarily small diameter in physical
coordinates. On every interior compact, continuity of $f$ then
gives value convergence. Since the domains are bounded and all
images lie in the fixed bounded target, dominated convergence gives
$L^p$ convergence on $\Omega$. Choose the meshes and value
prescriptions so its error and \eqref{la:derivative-final} sum to less
than the prescribed $\eps$; this proves \eqref{la:goal}.
\end{proof}
==== END Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2//sections/04-low-assembly.tex ====
==== BEGIN Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2//sections/10-conclusion.tex ====
\section{Completion on the fixed target}\label{sec:completion}
\begin{proof}[Proof of Theorem~\ref{main:theorem}]
Fix $1\le p\le2$ and $\varepsilon>0$. Apply Theorem~\ref{la:approximation} with sufficiently small error, obtaining a locally bi-Lipschitz homeomorphism $H:\Omega\to\Lambda$ in $W^{1,p}$. Apply Theorem~\ref{sm:smoothing} to $H$ with a smaller error. The triangle inequality gives a smooth diffeomorphism $g:\Omega\to\Lambda$ such that
\[
\int_\Omega (|g-f|^p+|Dg-Df|^p)\,dx<\varepsilon.
\]
Both approximation theorems give global $W^{1,p}$ membership and retain
the target for each approximant. In particular, the proper-degree
argument in Lemma~\ref{sm:proper-degree}, used during smoothing,
establishes bijectivity before any limit is taken; the nonsingular
smooth differential then supplies the smooth inverse. Apply the
construction with $\varepsilon=2^{-j}$ to obtain
\eqref{main:convergence}.
\end{proof}
==== END Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2//sections/10-conclusion.tex ====
==== BEGIN Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2//references.bib ====
@incollection{Ball2001,
author = {Ball, John M.},
title = {Singularities and computation of minimizers for variational problems},
booktitle = {Foundations of Computational Mathematics (Oxford, 1999)},
editor = {DeVore, Ronald A. and Iserles, Arieh and S{\"u}li, Endre},
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@misc{OpenAIHigh2026,
author = {{OpenAI}},
title = {{Strong diffeomorphic approximation in three dimensions for $p>2$}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2-September-24-2026/main.pdf}{OAI:Strong-diffeomorphic-approximation-in-three-dimensions-for-p-gt-2-September-24-2026}},
year = {2026}
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==== END Strong-diffeomorphic-approximation-in-three-dimensions-for-1-le-p-le-2//references.bib ====