Local Hölder continuity of the isoperimetric profile in complete noncompact Riemannian manifolds with bounded geometry

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 $\left(1-\frac{1}{n}\right)$-Hölder 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.

17 pages. This is an improvement of the result about the continuity of the isoperimetric profile function contained in arXiv:1404.3245. The arguments used here are a slight modification of the ones already used in arXiv:1404.3245. The paper is already accepted in Geometriae Dedicata

References in corpus (1)

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

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• The isoperimetric problem via direct method in noncompact metric measure spaces with lower Ricci bounds
• On clusters and the multi-isoperimetric profile in Riemannian manifolds with bounded geometry
• On the Hamilton's isoperimetric ratio in complete Riemannian manifolds of finite volume
• Some rigidity results for the Hawking mass and a lower bound for the Bartnik capacity