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

TL;DR

This paper proves a new Schoenflies extension theorem for certain bi-Lipschitz homeomorphisms with integrable derivatives, establishing sharp regularity conditions for extensions in Euclidean spaces.

Contribution

It introduces a novel extension theorem for locally bi-Lipschitz homeomorphisms with p-integrable second derivatives, extending the classical Schoenflies theorem to a broader regularity class.

Findings

Extension theorem holds for 1 < p < n with specified regularity

The theorem is essentially sharp, with limitations for p > n due to exotic spheres

Provides conditions for homeomorphic extensions preserving regularity

Abstract

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.