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

TL;DR

The paper establishes rigidity results for surfaces in 3D manifolds with non-negative or bounded scalar curvature, linking Hawking mass behavior to local and global geometric characterizations, and provides bounds for the Bartnik mass.

Contribution

It introduces new rigidity theorems connecting Hawking mass conditions to the local and global geometry of manifolds, and proposes a novel notion of 'sup-Hawking mass' with desirable properties.

Findings

If Hawking mass supremum is non-positive locally, the manifold is locally isometric to Euclidean or hyperbolic space.

Under mild asymptotic conditions, local conditions imply global isometry to Euclidean or hyperbolic space.

Explicit positive lower bounds for Hawking and Bartnik masses are derived for non-flat manifolds.

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.