Communications in Analysis and Geometry

Volume 25 (2017)

Number 1

Hawking mass and local rigidity of minimal surfaces in three-manifolds

Pages: 1 – 23

DOI: https://dx.doi.org/10.4310/CAG.2017.v25.n1.a1

Authors

A. Barros (Departamento de Matemática, Universidade Federal do Ceará, Fortaleza, CE, Brazil)

R. Batista (Departamento de Matemática, Universidade Federal do Piauí, PI, Brazil)

T. Cruz (Instituto de Matemática, Universidade Federal de Alagoas, Maceió, AL, Brazil)

Abstract

The aim of this paper is to generalize some recent local rigidity results for three-dimensional Riemannian manifolds $(M^3, g)$ with a bound on the scalar curvature. More precisely, we study rigidity of strictly stable minimal surfaces $\Sigma \subset M$ which locally maximize the Hawking mass on a Riemannian three-manifold $M$ whose scalar curvature is bounded from below by a negative constant. Moreover, we conclude that the metric of $M$ near $\Sigma$ must split as $g_a = dr^2 + u_a (r)^2 g_{\widetilde{\Sigma}}$ which is one the Kottler-Schwarzschild metric, where $g_{\widetilde{\Sigma}}$ is a metric of constant gaussian curvature.

Keywords

scalar curvature functional, Hawking mass, rigidity of three-manifolds

2010 Mathematics Subject Classification

Primary 53C21, 53C42. Secondary 58J60.

Received 15 November 2013

Published 9 June 2017