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 $(M,g)$ with non-negative scalar curvature (resp. 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 ${\mathbb{R}}^3$ (resp. 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 \emph{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 ${\mathbb{R}}^3$ (resp. globally isometric to the Hyperbolic 3-space ${\mathbb{H}}^3$). Also, if the space is not flat (resp. 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.

39 pages. Final version, to appear in the Journal of the London Mathematical Society

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Cited by in corpus (1)

• Implications of some mass-capacity inequalities