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Paweł Goldstein

Publications and source records attributed to Paweł Goldstein.

13 recordsLinked to original sources
arXiv2026

Differentiability of Lipschitz mappings into metric spaces: area and co-area formulas

We give a self-contained exposition of metric differentiability for Lipschitz mappings from sets in Euclidean spaces into arbitrary metric spaces, together with the corresponding area and co-area formulas. The novelty is a new metric implicit function theorem for mappings into metric spaces, which is then used to give a direct and geometric proof of the co-area formula for Lipschitz mappings into metric spaces.

math.AP↗
arXiv2025

Gluing diffeomorphisms, bi-Lipschitz mappings and homeomorphisms

Cerf and Palais independently proved a remarkable result about extending diffeomorphisms defined on smooth balls in a manifold to global diffeomorphisms of the manifold onto itself. We explain Palais' argument and show how to extend it to the class of homeomorphisms and bi-Lipschitz homeomorphisms. While Palais' argument is surprising, it is elementary and short. However, its extension to bi-Lipschitz homeomorphisms and homeomorphisms requires deep results: the stable homeomorphism and the annulus theorems.

math.GT↗
arXiv2023

Constructing diffeomorphisms and homeomorphisms with prescribed derivative

We prove that for any measurable mapping $T$ into the space of matrices with positive determinant, there is a diffeomorphism whose derivative equals $T$ outside a set of measure less than $\varepsilon$. We use this fact to prove that for any measurable mapping $T$ into the space of matrices with non-zero determinant (with no sign restriction), there is an almost everywhere approximately differentiable homeomorphism whose derivative equals $T$ almost everywhere.

math.CA↗
arXiv2023

Smooth approximation of mappings with rank of the derivative at most $1$

It was conjectured that if $f\in C^1(\mathbb{R}^n,\mathbb{R}^n)$ satisfies $\operatorname{rank} Df\leq m<n$ everywhere in $\mathbb{R}^n$, then $f$ can be uniformly approximated by $C^\infty$-mappings $g$ satisfying $\operatorname{rank} Dg\leq m$ everywhere. While in general, there are counterexamples to this conjecture, we prove that the answer is in the positive when $m=1$. More precisely, if $m=1$, our result yields an almost-uniform approximation of locally Lipschitz mappings $f:Ω\to\mathbb{R}^n$, satisfying $\operatorname{rank} Df\leq 1$ a.e., by $C^\infty$-mappings $g$ with $\operatorname{rank} Dg\leq 1$, provided $Ω\subset\mathbb{R}^n$ is simply connected. The construction of the approximation employs techniques of analysis on metric spaces, including the theory of metric trees ($\mathbb{R}$-trees).

math.MG↗
arXiv2019

Jacobians of $W^{1,p}$ homeomorphisms, case $p=[n/2]$

We investigate a known problem whether a Sobolev homeomorphism between domains in $\mathbb{R}^n$ can change sign of the Jacobian. The only case that remains open is when $f\in W^{1,[n/2]}$, $n\geq 4$. We prove that if $n\geq 4$, and a sense-preserving homeomorphism $f$ satisfies $f\in W^{1,[n/2]}$, $f^{-1}\in W^{1,n-[n/2]-1}$ and either $f$ is Hölder continuous on almost all spheres of dimension $[n/2]$, or $f^{-1}$ is Hölder continuous on almost all spheres of dimensions $n-[n/2]-1$, then the Jacobian of $f$ is non-negative, $J_f\geq 0$, almost everywhere. This result is a consequence of a more general result proved in the paper. Here $[x]$ stands for the greatest integer less than or equal to $x$.

math.CA↗
arXiv2018

$C^1$ mappings in $\mathbb{R}^5$ with derivative of rank at most $3$ cannot be uniformly approximated by $C^2$ mappings with derivative of rank at most 3

We find a counterexample to a conjecture of Gałęski by constructing for some positive integers $m<n$ a mapping $f\in C^1(\mathbb{R}^n,\mathbb{R}^n)$ satisfying $\mathrm{rank}\, Df\leq m$ that, even locally, cannot be uniformly approximated by $C^2$ mappings $f_\varepsilon$ satisfying the same rank constraint $\mathrm{rank}\, Df_\varepsilon\leq m$.

math.CA↗
arXiv2018

Topologically nontrivial counterexamples to Sard's theorem

We prove the following dichotomy: if $n=2,3$ and $f\in C^1(\mathbb{S}^{n+1},\mathbb{S}^n)$ is not homotopic to a constant map, then there is an open set $Ω\subset\mathbb{S}^{n+1}$ such that $\mathrm{rank}\, df=n$ on $Ω$ and $f(Ω)$ is dense in $\mathbb{S}^n$, while for any $n\geq 4$, there is a map $f\in C^1(\mathbb{S}^{n+1},\mathbb{S}^n)$ that is not homotopic to a constant map and such that $\mathrm{rank}\, df<n$ everywhere. The result in the case $n\geq 4$ answers a question of Larry Guth.

math.CA↗
arXiv2018

Topological obstructions to continuity of Orlicz-Sobolev mappings of finite distortion

In the paper we investigate continuity of Orlicz-Sobolev mappings $W^{1,P}(M,N)$ of finite distortion between smooth Riemannian $n$-manifolds, $n\geq 2$, under the assumption that the Young function $P$ satisfies the so called divergence condition $\int_1^\infty P(t)/t^{n+1}\, dt=\infty$. We prove that if the manifolds are oriented, $N$ is compact, and the universal cover of $N$ is not a rational homology sphere, then such mappings are continuous. That includes mappings with $Df\in L^n$ and, more generally, mappings with $Df\in L^n\log^{-1}L$. On the other hand, if the space $W^{1,P}$ is larger than $W^{1,n}$ (for example if $Df\in L^n\log^{-1}L$), and the universal cover of $N$ is homeomorphic to $\mathbb{S}^n$, $n\neq 4$, or is diffeomorphic to $\mathbb{S}^n$, $n=4$, then we construct an example of a mapping in $W^{1,P}(M,N)$ that has finite distortion and is discontinuous. This demonstrates a new global-to-local phenomenon: both finite distortion and continuity are local properties, but a seemingly local fact that finite distortion implies continuity is a consequence of a global topological property of the target manifold $N$.

math.CA↗
arXiv2017

Finite distortion Sobolev mappings between manifolds are continuous

We prove that if $M$ and $N$ are Riemannian, oriented $n$-dimensional manifolds without boundary and additionally $N$ is compact, then Sobolev mappings $W^{1,n}(M,N)$ of finite distortion are continuous. In particular, $W^{1,n}(M,N)$ mappings with almost everywhere positive Jacobian are continuous. This result has been known since 1976 in the case of mappings $W^{1,n}(Ω,\mathbb{R}^n)$, where $Ω\subset\mathbb{R}^n$ is an open set. The case of mappings between manifolds is much more difficult.

math.CA↗
arXiv2017

Modulus of continuity of orientation preserving approximately differentiable homeomorphisms with a.e. negative Jacobian

We construct an a.e. approximately differentiable homeomorphism of a unit $n$-dimensional cube onto itself which is orientation preserving, has the Lusin property (N) and has the Jacobian determinant negative a.e. Moreover, the homeomorphism together with its inverse satisfy a rather general sub-Lipschitz condition, in particular it can be bi-Hölder continuous with an arbitrary exponent less than $1$.

math.CA↗