Communications in Analysis and Geometry

Volume 28 (2020)

Number 7

The isoperimetric problem of a complete Riemannian manifold with a finite number of $C^0$‑asymptotically Schwarzschild ends

Pages: 1577 – 1601



Abraham Enrique Muñoz Flores (Instituto de Matemática e Estatística, UERJ-Universidade do Estado do Rio de Janeiro, Brazil)

Stefano Nardulli (Centro de Matemática Cognição e Computação, UFABC-Universidade Federal do ABC, SP, Brazil)


We show existence and we give a geometric characterization of isoperimetric regions for large volumes, in $C^2$-locally asymptotically Euclidean Riemannian manifolds with a finite number of $C^0$-asymptotically Schwarzschild ends. This work extends previous results contained in [EM13b], [EM13a], and [BE13]. Moreover strengthening a little bit the speed of convergence to the Schwarzschild metric we obtain existence of isoperimetric regions for all volumes for a class of manifolds that we named $C^0$-strongly asymptotic Schwarzschild, extending results of [BE13]. Such results are of interest in the field of mathematical general relativity.

The first-named author was partially supported by Capes.

Received 24 July 2016

Accepted 22 April 2018

Published 7 December 2020