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

We study rigidity of minimal two-spheres $\mathrm{\Sigma }$ 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 $\mathrm{\Sigma }$, we prove that a neighborhood of it in $M$ is isometric to one of the deSitter-Schwarzschild metrics on $(-ϵ,ϵ)\times \mathrm{\Sigma }$. We also show that if $\mathrm{\Sigma }$ is a critical point for the Hawking mass on the deSitter-Schwarzschild manifold $\mathbb{R}\times {\text{Sph}}^2$ and can be written as a graph over a slice ${\mathrm{\Sigma }}_r=\{r\}\times {\mathbb{S}}^2$, then $\mathrm{\Sigma }$ itself must be a slice, and moreover that slices are indeed local maxima amongst competitors that are graphs with small $C^2$-norm.