The Library
Some rigidity results for the hawking mass and a lower bound for the Bartnik capacity
Tools
Mondino, Andrea and Templeton‐Browne, Aidan (2022) Some rigidity results for the hawking mass and a lower bound for the Bartnik capacity. Journal of the London Mathematical Society, 106 (3). pp. 1844-1896. doi:10.1112/jlms.12612 ISSN 0024-6107.
Research output not available from this repository.
Request-a-Copy directly from author or use local Library Get it For Me service.
Official URL: https://doi.org/10.1112/jlms.12612
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.
Item Type: | Journal Article | ||||||||
---|---|---|---|---|---|---|---|---|---|
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
SWORD Depositor: | Library Publications Router | ||||||||
Journal or Publication Title: | Journal of the London Mathematical Society | ||||||||
Publisher: | Cambridge University Press | ||||||||
ISSN: | 0024-6107 | ||||||||
Official Date: | October 2022 | ||||||||
Dates: |
|
||||||||
Volume: | 106 | ||||||||
Number: | 3 | ||||||||
Page Range: | pp. 1844-1896 | ||||||||
DOI: | 10.1112/jlms.12612 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Open Access (Creative Commons) |
Request changes or add full text files to a record
Repository staff actions (login required)
View Item |