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

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 assuming 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 $\mathbb{R}\times\Sph^2$ 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.