On the extension of bi-Lipschitz mappings

Lipschitz null-homotopy of mappings $S^3 \rightarrow S^2$

This work focuses on important step in quantitative topology: given homotopic mappings from $S^m$ to $S^n$ of Lipschitz constant $L$, build the (asymptotically) simplest homotopy between them (meaning having the least Lipschitz constant). The present paper resolves this problem for the first case where Hopf invariant plays a role: $m = 3$, $n = 2$, constructing a homotopy with Lipschitz constant $O(L)$

Uniform Lipschitz extension in bounded curvature

We prove a uniform extension result for contracting maps defined on subsets of Hadamard manifolds subject to curvature bounds.

粘合微分同胚、双 Lipschitz 映射与同胚

Gluing diffeomorphisms, bi-Lipschitz mappings and homeomorphisms

本文阐释 Palais 关于光滑球上微分同胚整体扩张的初等证明，并将该扩张结果推广到同胚和双 Lipschitz 同胚，且指出推广依赖稳定同胚定理与环域定理。

Lipschitz homotopy and density of Lipschitz mappings in Sobolev spaces

We construct a smooth compact n-dimensional manifold Y with one point singularity such that all its Lipschitz homotopy groups are trivial, but Lipschitz mappings Lip(S^n,Y) are not dense in the Sobolev space W^{1,n}(S^n,Y). On the other hand we show that if a metric space Y is Lipschitz (n-1)-connected, then Lipschitz mappings Lip(X,Y) are dense in N^{1,p}(X,Y) whenever the Nagata dimension of X is bounded by n and the space X supports the p-Poincare inequality.

An implicit function theorem for Lipschitz mappings into metric spaces

We prove a version of the implicit function theorem for Lipschitz mappings $f:\mathbb{R}^{n+m}\supset A \to X$ into arbitrary metric spaces. As long as the pull-back of the Hausdorff content $\mathcal{H}_{\infty}^n$ by $f$ has positive upper $n$-density on a set of positive Lebesgue measure, then, there is a local diffeomorphism $G$ in $\mathbb{R}^{n+m}$ and a Lipschitz map $π:X\to \mathbb{R}^n$ such that $π\circ f\circ G^{-1}$, when restricted to a certain subset of $A$ of positive measure, is a the orthogonal projection of $\mathbb{R}^{n+m}$ onto the first $n$-coordinates. This may be seen a…

A Schoenflies extension theorem for a class of locally bi-Lipschitz homeomorphisms

In this paper we prove a new version of the Schoenflies extension theorem for collared domains in Euclidean n-space: for 1 < p < n, locally bi-Lipschitz homeomorphisms between collared domains with locally p-integrable, second-order weak derivatives admit homeomorphic extensions of the same regularity. Moreover, the theorem is essentially sharp. The existence of exotic 7-spheres shows that such extension theorems cannot hold for p > n = 7.