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Isoperimetric inequalities vs. upper curvature bounds

Stephan Stadler, Stefan Wenger

TL;DR

The paper develops an analytic criterion linking isoperimetric data to upper curvature bounds in complete metric spaces: a space $X$ has curvature bounded above by $\kappa$ in the Alexandrov sense if its Dehn function is dominated by the model $\delta_{\kappa}$. The authors prove this via minimal discs constructed in ultralimits and a Plateau-problem framework that yields intrinsic isoperimetric control, even in non-locally-compact settings. They establish a stable, local version: if a sequence of spaces has Dehn-function bounds relative to $\delta_{\kappa}$, every ultralimit is locally CAT($\kappa$); they also derive consequences for asymptotic cones, showing CAT(0) behavior (or trees in the strict case) when the Dehn data is appropriately scaled. The results provide a robust bridge between isoperimetric inequalities and curvature in non-smooth and non-compact environments, with implications for geometric group theory and coarse geometry.

Abstract

The Dehn function of a metric space measures the area necessary in order to fill a closed curve of controlled length by a disc. As a main result, we prove that a length space has curvature bounded above by $κ$ in the sense of Alexandrov if and only if its Dehn function is bounded above by the Dehn function of the model surface of constant curvature $κ$. This extends work of Lytchak and the second author from locally compact spaces to the general case. A key ingredient in the proof is the construction of minimal discs with suitable properties in certain ultralimits. Our arguments also yield quantitative local and stable versions of our main result. The latter has implications on the geometry of asymptotic cones.

Isoperimetric inequalities vs. upper curvature bounds

TL;DR

The paper develops an analytic criterion linking isoperimetric data to upper curvature bounds in complete metric spaces: a space has curvature bounded above by in the Alexandrov sense if its Dehn function is dominated by the model . The authors prove this via minimal discs constructed in ultralimits and a Plateau-problem framework that yields intrinsic isoperimetric control, even in non-locally-compact settings. They establish a stable, local version: if a sequence of spaces has Dehn-function bounds relative to , every ultralimit is locally CAT(); they also derive consequences for asymptotic cones, showing CAT(0) behavior (or trees in the strict case) when the Dehn data is appropriately scaled. The results provide a robust bridge between isoperimetric inequalities and curvature in non-smooth and non-compact environments, with implications for geometric group theory and coarse geometry.

Abstract

The Dehn function of a metric space measures the area necessary in order to fill a closed curve of controlled length by a disc. As a main result, we prove that a length space has curvature bounded above by in the sense of Alexandrov if and only if its Dehn function is bounded above by the Dehn function of the model surface of constant curvature . This extends work of Lytchak and the second author from locally compact spaces to the general case. A key ingredient in the proof is the construction of minimal discs with suitable properties in certain ultralimits. Our arguments also yield quantitative local and stable versions of our main result. The latter has implications on the geometry of asymptotic cones.
Paper Structure (27 sections, 26 theorems, 117 equations, 1 figure)

This paper contains 27 sections, 26 theorems, 117 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main result
  3. Minimal surfaces in ultralimits of locally non-compact spaces
  4. Generalizations
  5. Motivation and strategy for Theorem \ref{['thm_minimal']}
  6. Further questions
  7. Organization
  8. Acknowledgements
  9. Basics on metric spaces
  10. Notation
  11. Intrinsic metric of a map
  12. Ultralimits
  13. CAT($\kappa$)
  14. Sobolev theory
  15. Basics
  16. ...and 12 more sections

Key Result

Theorem 1

Let $X$ be a complete length space and $r_0\in(0,2D_\kappa]$. Suppose that holds for all $r\in(0,r_0)$. Then every closed ball of radius at most $\frac{r_0}{4}$ in $X$ is convex and CAT($\kappa$). Moreover, if $r_0=2D_\kappa$, then $X$ is a CAT($\kappa$) space.

Figures (1)

  • Figure 1: A framed collar $U$ with frame $G$.

Theorems & Definitions (33)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 2.1
  • Lemma 2.2
  • Definition 3.1
  • Lemma 3.2
  • Definition 3.3
  • ...and 23 more