Local H\"older continuity of the isoperimetric profile in complete noncompact Riemannian manifolds with bounded geometry

TL;DR

This paper proves the local Hölder continuity of the isoperimetric profile in complete noncompact Riemannian manifolds with bounded geometry, establishing continuity and equivalence of profile formulations, and discusses upper semicontinuity under general metrics.

Contribution

It establishes the local Hölder continuity of the isoperimetric profile in manifolds with bounded geometry and proves the equivalence of different profile formulations.

Findings

Isoperimetric profile is locally Hölder continuous with exponent (1 - 1/n).

Weak and strong formulations of the isoperimetric profile are equivalent.

Upper semicontinuity of the isoperimetric profile holds under any metric.

Abstract

For a complete noncompact connected Riemannian manifold with bounded geometry $M^n$, we prove that the isoperimetric profile function $I_{M^n}$ is a locally $\left(1-\frac{1}{n}\right)$-H\"older continuous function and so in particular it is continuous. Here for bounded geometry we mean that $M$ have $Ricci$ curvature bounded below and volume of balls of radius $1$, uniformly bounded below with respect to its centers. We prove also the equivalence of the weak and strong formulation of the isoperimetric profile function in complete Riemannian manifolds which is based on a lemma having its own interest about the approximation of finite perimeter sets with finite volume by open bounded with smooth boundary ones of the same volume. Finally the upper semicontinuity of the isoperimetric profile for every metric (not necessarily complete) is shown.