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A local isoperimetric inequality for balls with nonpositive curvature

Mohammad Ghomi, John Ioannis Stavroulakis

TL;DR

This work proves a sharp local isoperimetric inequality for balls under metrics with nonpositive curvature, showing that small $\\mathcal{C}^2$-perturbations of the Euclidean ball cannot decrease the isoperimetric ratio $I$ and that equality forces a Euclidean ball. The authors reduce to a star-shaped domain $\\Omega_{\\overline{g}} \\subset \\\mathbb{R}^n$ in normal coordinates, described by a radial boundary function $f_{\\overline{g}}$, and apply Rauch comparison together with a Jacobian analysis to compare $I(\\Omega_{\\overline{g}})$ with $I(\\Omega_{\\delta^k})$. They extend the local result to curvature upper bounds $k \\le 0$ and connect the conclusion to the Cartan–Hadamard conjecture, providing a rigidity statement for perturbations that preserve the isoperimetric ratio. The methods illuminate how local metric perturbations affect global isoperimetric properties in nonpositively curved spaces and yield a concrete local verification of the conjecture.

Abstract

We show that small perturbations of the metric of a ball in Euclidean n-space to metrics with nonpositive curvature do not reduce the isoperimetric ratio. Furthermore, the isoperimetric ratio is preserved only if the perturbation corresponds to a homothety of the ball. These results establish a sharp local version of the Cartan-Hadamard conjecture.

A local isoperimetric inequality for balls with nonpositive curvature

TL;DR

This work proves a sharp local isoperimetric inequality for balls under metrics with nonpositive curvature, showing that small -perturbations of the Euclidean ball cannot decrease the isoperimetric ratio and that equality forces a Euclidean ball. The authors reduce to a star-shaped domain in normal coordinates, described by a radial boundary function , and apply Rauch comparison together with a Jacobian analysis to compare with . They extend the local result to curvature upper bounds and connect the conclusion to the Cartan–Hadamard conjecture, providing a rigidity statement for perturbations that preserve the isoperimetric ratio. The methods illuminate how local metric perturbations affect global isoperimetric properties in nonpositively curved spaces and yield a concrete local verification of the conjecture.

Abstract

We show that small perturbations of the metric of a ball in Euclidean n-space to metrics with nonpositive curvature do not reduce the isoperimetric ratio. Furthermore, the isoperimetric ratio is preserved only if the perturbation corresponds to a homothety of the ball. These results establish a sharp local version of the Cartan-Hadamard conjecture.
Paper Structure (3 sections, 6 theorems, 20 equations)

This paper contains 3 sections, 6 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the Main Result

Key Result

Theorem 1.1

There exists $\varepsilon>0$ such that for all metrics $g\in \mathcal{M} _{0}(B^n)$ with $|g-\delta|_{\mathcal{C}^2(B^n)}\leqslant\varepsilon$, $\mathop{\mathrm{I}}\nolimits(B^n_g)\geqslant \mathop{\mathrm{I}}\nolimits(B^n_\delta).$ Furthermore, $\mathop{\mathrm{I}}\nolimits(B^n_g)= \mathop{\mathrm

Theorems & Definitions (11)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 1 more