On the existence of isoperimetric regions in manifolds with nonnegative Ricci curvature and Euclidean volume growth

Abstract

In this paper we provide new existence results for isoperimetric sets of large volume in Riemannian manifolds with nonnegative Ricci curvature and Euclidean volume growth. We find sufficient conditions for their existence in terms of the geometry at infinity of the manifold. As a byproduct we show that isoperimetric sets of big volume always exist on manifolds with nonnegative sectional curvature and Euclidean volume growth. Our method combines an asymptotic mass decomposition result for minimizing sequences, a sharp isoperimetric inequality on nonsmooth spaces, and the concavity property of the isoperimetric profile. The latter is new in the generality of noncollapsed manifolds with Ricci curvature bounded below.

37 pages

References in corpus (5)

• Isoperimetric sets in spaces with lower bounds on the Ricci curvature
• Sobolev inequalities in manifolds with nonnegative curvature
• Some isoperimetric comparison theorems for convex bodies in Riemannian manifolds
• Sobolev inequalities on manifolds with nonnegative Bakry-Émery Ricci curvature
• Rectifiability of RCD(K,N) spaces via $\delta$-splitting maps

Cited by in corpus (7)

• Asymptotic isoperimetry on non collapsed spaces with lower Ricci bounds
• Isoperimetric sets in spaces with lower bounds on the Ricci curvature
• 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
• Non-negative Ricci curvature and Minimal graphs with linear growth
• Isoperimetric problem and structure at infinity on Alexandrov spaces with nonnegative curvature
• On splitting complete manifolds via infinity harmonic functions