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Ultralimits of Sobolev maps and stability of Dehn functions

Toni Ikonen, Stefan Wenger

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 $κ$ via an isoperimetric inequality for curves.

Ultralimits of Sobolev maps and stability of Dehn functions

Abstract

We show that the ultralimit of a bounded sequence of Lipschitz maps into pointed metric spaces extends naturally to -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 via an isoperimetric inequality for curves.
Paper Structure (31 sections, 23 theorems, 119 equations)

This paper contains 31 sections, 23 theorems, 119 equations.

Table of Contents

  1. Introduction and main results
  2. Background
  3. Ultralimits of Sobolev maps
  4. Stability of Dehn functions
  5. Applications
  6. Outlines of the proofs and additional results
  7. Final remark and structure of the paper
  8. Acknowledgements
  9. Preliminaries
  10. Curves and length distances
  11. Hausdorff measures
  12. Non-principal ultrafilters
  13. Ultralimits of spaces
  14. Ultralimits of mappings
  15. Injective metric spaces
  16. ...and 16 more sections

Key Result

Theorem 1.1

There is a unique map sending a $p$-bounded sequence $( u_m \colon \Omega \to X_m )$ to a Sobolev map $u_\omega\in W^{1,p}(\Omega, X_\omega )$ with the following properties:

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4: stadler-wenger-2025-isoperimetric-inequalities-vs-upper-curvature-bounds
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 2.1
  • proof
  • Definition 3.1
  • ...and 31 more