Ultralimits of Sobolev maps and stability of Dehn functions

TL;DR

This paper develops a framework for ultralimits of Sobolev maps into metric spaces, demonstrating their stability and applying this to prove the stability of Dehn functions, with implications for curvature bounds and isoperimetric inequalities.

Contribution

It introduces a natural extension of ultralimits to Sobolev maps and proves the stability of Dehn functions under ultraconvergence of length spaces.

Findings

Ultralimits of Sobolev maps preserve key properties.

Dehn functions are stable under ultraconvergence.

Simplified proof of curvature bounds via isoperimetric inequalities.

Abstract

We show that the ultralimit of a bounded sequence of Lipschitz maps into pointed metric spaces extends naturally to $p$-bounded sequences of Sobolev maps and that this ultralimit for Sobolev maps enjoys desirable properties. We use this to prove the stability of Dehn functions under ultraconvergence of pointed length spaces, thus resolving a problem posed by several researchers in the field. As an application, we obtain a simpler proof of a recent result of Stadler--Wenger, previously proved in the locally compact case by Lytchak--Wenger, characterizing spaces of curvature bounded above by $\kappa$ via an isoperimetric inequality for curves.