Continuity of the isoperimetric profile of a complete Riemannian manifold under sectional curvature conditions

TL;DR

This paper proves that the isoperimetric profile of certain complete Riemannian manifolds, including Hadamard and positively curved manifolds, is continuous and non-decreasing, under specific curvature conditions.

Contribution

It establishes the continuity and monotonicity of the isoperimetric profile for manifolds with a convex Lipschitz exhaustion function, extending known results to broader classes.

Findings

Isoperimetric profile is continuous and non-decreasing

Results apply to Hadamard and positively curved manifolds

Provides new insights into geometric analysis of Riemannian manifolds

Abstract

Let $M$ be a complete Riemannian manifold possessing a strictly convex Lipschitz continuous exhaustion function. We show that the isoperimetric profile of $M$ is a continuous and non-decreasing function. Particular cases are Hadamard manifolds and complete non-compact manifolds with strictly positive sectional curvatures.