Communications in Analysis and Geometry

Volume 21 (2013)

Number 2

Hawking mass and local rigidity of minimal two-spheres in three-manifolds

Pages: 409 – 433



Davi Máximo (Department of Mathematics, The University of Texas, Austin, Tx., U.S.A.)

Ivaldo Nunes (Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, Brazil)


We study rigidity of minimal two-spheres $\Sigma$ that locally maximize the Hawking mass on a Riemannian three-manifold with a positive lower bound on its scalar curvature. After assu\-ming strict stability of $\Sigma$, we prove that a neighborhood of it in $M$ is isometric to one of the deSitter–Schwarzschild metrics on $(-\epsilon,\epsilon)\times \Sigma$. We also show that if $\Sigma$ is a critical point for the Hawking mass on the deSitter–Schwarzschild manifold $\mathbb{R} \times \mathbb{S}^2$ and can be written as a graph over a slice $\Sigma_r=\{r\}\times\mathbb{S}^2$, then $\Sigma$ itself must be a slice, and moreover that slices are indeed local maxima amongst competitors that are graphs with small $C^2$-norm.

Published 9 April 2013