A local isoperimetric inequality for balls with nonpositive curvature

TL;DR

This paper proves that small metric perturbations in nonpositively curved balls do not decrease the isoperimetric ratio, with equality only under homothety, thus establishing a local version of the Cartan-Hadamard conjecture.

Contribution

It provides a sharp local isoperimetric inequality for balls with nonpositive curvature, extending the Cartan-Hadamard conjecture to a local setting.

Findings

Small perturbations do not reduce the isoperimetric ratio.

The isoperimetric ratio is preserved only under homothety.

Results establish a local version of the Cartan-Hadamard 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.