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

We prove rigidity results involving the Hawking mass for surfaces immersed in a 3-dimensional, complete Riemannian manifold $(M,g)$ with non-negative scalar curvature (respectively, with scalar curvature bounded below by $-6$ ). Roughly, the main result states that if an open subset $\Omega \subset M$ satisfies that every point has a neighbourhood $U\subset \Omega$ such that the supremum of the Hawking mass of surfaces contained in $U$ is non-positive, then $\Omega$ is locally isometric to Euclidean $\mathbb {R}^3$ (respectively, locally isometric to the Hyperbolic 3-space ${\mathbb {H}}^3$ ). Under mild asymptotic conditions on the manifold $(M,g)$ (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 $M$ has a neighbourhood $U$ such that the supremum of the Hawking mass of surfaces contained in $U$ is non-positive, then $(M,g)$ is globally isometric to Euclidean $\mathbb {R}^3$ (respectively, globally isometric to the Hyperbolic 3-space ${\mathbb {H}}^3$ ). Also, if the space is not flat (respectively, not of constant sectional curvature $-1$ ), 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 $\mathbb {R}^{3}$ ) 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