Compactness of immersions with local Lipschitz representation

TL;DR

This paper proves compactness results for immersions that can be locally represented as Lipschitz graphs, with specific bounds on Lipschitz constants and volume, in various codimensions.

Contribution

It establishes new compactness theorems for Lipschitz-graph immersions in codimension 1 and higher, with bounds on Lipschitz constants.

Findings

Compactness holds for fixed Lipschitz constant L and bounded volume in codimension 1.

In arbitrary codimension, compactness is valid for L ≤ 1/4.

Results extend the understanding of geometric limits of Lipschitz-graph immersions.

Abstract

We consider immersions admitting uniform representations as an L-Lipschitz graph. In codimension 1, we show compactness for such immersions for arbitrary fixed finite L and uniformly bounded volume. The same result is shown in arbitrary codimension for L less than or equal to 1/4.