Some rigidity results for the Hawking mass and a lower bound for the Bartnik capacity

Abstract

We prove rigidity results involving the Hawking mass for surfaces immersed in a 3-dimensional, complete Riemannian manifold ( M , g ) $(M,g)$ with non-negative scalar curvature (respectively, with scalar curvature bounded below by − 6 $-6$ ). Roughly, the main result states that if an open subset Ω ⊂ M $\Omega \subset M$ satisfies that every point has a neighbourhood U ⊂ Ω $U\subset \Omega$ such that the supremum of the Hawking mass of surfaces contained in U $U$ is non-positive, then Ω $\Omega$ is locally isometric to Euclidean R 3 $\mathbb {R}^3$ (respectively, locally isometric to the Hyperbolic 3-space H 3 ${\mathbb {H}}^3$ ). Under mild asymptotic conditions on the manifold ( M , g ) $(M,g)$ (which encompass as special cases the standard ‘asymptotically flat’ or, respectively, ‘asymptotically hyperbolic’ assumptions) the previous quasi-local rigidity statement implies a global rigidity: if every point in M $M$ has a neighbourhood U $U$ such that the supremum of the Hawking mass of surfaces contained in U $U$ is non-positive, then ( M , g ) $(M,g)$ is globally isometric to Euclidean R 3 $\mathbb {R}^3$ (respectively, globally isometric to the Hyperbolic 3-space H 3 ${\mathbb {H}}^3$ ). Also, if the space is not flat (respectively, not of constant sectional curvature − 1 $-1$ ), the methods give a small yet explicit and strictly positive lower bound on the Hawking mass of suitable spherical surfaces. We infer a small yet explicit and strictly positive lower bound on the Bartnik mass of open sets (non-locally isometric to Euclidean R 3 $\mathbb {R}^{3}$ ) in terms of curvature tensors. Inspired by these results, in the appendix we propose a notion of ‘sup-Hawking mass’ which satisfies some natural properties of a quasi-local mass.