Communications in Analysis and Geometry

Volume 24 (2016)

Number 1

Existence of isoperimetric regions in non-compact Riemannian manifolds under Ricci or scalar curvature conditions

Pages: 115 – 138

DOI: https://dx.doi.org/10.4310/CAG.2016.v24.n1.a5

Authors

Andrea Mondino (Department of Mathematics, ETH, Zürich, Switzerland)

Stefano Nardulli (Departamento de Matemática, Instituto de Matemática, Cidade Universitária, Rio de Janeiro, Brasil)

Abstract

We prove existence of isoperimetric regions for every volume in non-compact Riemannian n-manifolds $(M,g), n \geq 2$, having Ricci curvature $\mathrm{Ric}_g \geq (n-1) k_0 g$ and being $C^0$-locally asymptotic to the simply connected space form of constant sectional curvature $k_0 \leq 0$; moreover in case $k_0 = 0$ we show that the isoperimetric regions are indecomposable. Our results apply to a large class of physically and geometrically relevant examples: Eguchi–Hanson metric and more generally ALE gravitational instantons, asymptotically hyperbolic Einstein manifolds, Bryant type solitons, etc. Finally, under assumptions on the scalar curvature, we prove existence of isoperimetric regions of small volume.

Keywords

isoperimetric problem, Ricci curvature, ALE gravitational instantons, asymptotically hyperbolic Einstein manifolds

2010 Mathematics Subject Classification

49Q10, 53A10, 53C42, 83C99

Published 6 June 2016