Calculus of Variations and Geometric Measure Theory
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A. Mondino - A. Templeton-Browne

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

created by mondino on 20 Jul 2021

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Submitted Paper

Inserted: 20 jul 2021
Last Updated: 20 jul 2021

Year: 2021

ArXiv: 2107.08110 PDF

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 Ω⊂M satisfies that every point has a neighbourhood U⊂Ω such that the supremum of the Hawking mass of surfaces contained in U is non-positive, then Ω is locally isometric to Euclidean R3 (resp. locally isometric to the Hyperbolic 3-space H3). 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 R3 (resp. globally isometric to the Hyperbolic 3-space H3). 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 R3) 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


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