Dehn functions and H\"older extensions in asymptotic cones

TL;DR

This paper introduces a new perspective on the Dehn function, proving its invariance under quasi-isometries, establishing H"older extension properties, and analyzing its behavior in asymptotic cones, especially for quadratic Dehn functions.

Contribution

It presents a new definition of the Dehn function, extends quasi-isometry invariance, and explores H"older extensions and asymptotic cone properties for spaces with quadratic Dehn functions.

Findings

Dehn function invariance under quasi-isometries for broad classes of spaces

H"older extension properties in spaces with quadratic Dehn functions

Quadratic Dehn function behavior preserved in ultralimits and asymptotic cones

Abstract

The Dehn function measures the area of minimal discs that fill closed curves in a space; it is an important invariant in analysis, geometry, and geometric group theory. There are several equivalent ways to define the Dehn function, varying according to the type of disc used. In this paper, we introduce a new definition of the Dehn function and use it to prove several theorems. First, we generalize the quasi-isometry invariance of the Dehn function to a broad class of spaces. Second, we prove H\"older extension properties for spaces with quadratic Dehn function and their asymptotic cones. Finally, we show that ultralimits and asymptotic cones of spaces with quadratic Dehn function also have quadratic Dehn function. The proofs of our results rely on recent existence and regularity results for area-minimizing Sobolev mappings in metric spaces.