"Boundary Behavior of Compact Manifolds With Scalar Curvature Lower Bou" by Aghil Alaee, Pei Ken Hung et al.
 

Mathematics

Boundary Behavior of Compact Manifolds With Scalar Curvature Lower Bounds and Static Quasi-Local Mass of Tori

Document Type

Article

Abstract

A classic result of Shi and Tam states that a 2-sphere of positive Gauss and mean curvature bounding a compact 3-manifold with nonnegative scalar curvature must have total mean curvature not greater than that of the isometric embedding into Euclidean 3-space, with equality only for domains in this reference manifold. We generalize this result to 2-tori of Gauss curvature greater than −1, which bound a compact 3-manifold having scalar curvature not less than −6 and at least one other boundary component satisfying a ‘trapping condition’. The conclusion is that the total weighted mean curvature is not greater than that of an isometric embedding into the Kottler manifold, with equality only for domains in this space. Examples are given to show that the assumption of a secondary boundary component cannot be removed. The result gives a positive mass theorem for the static Brown-York mass of tori, in analogy to the Shi-Tam positivity of the standard Brown-York mass, and represents the first such quasi-local mass positivity result for nonspherical surfaces. Furthermore, we prove a Penrose-type inequality in this setting.

Publication Title

Proceedings of the American Mathematical Society

Publication Date

5-2025

Volume

153

Issue

5

First Page

2153

Last Page

2167

ISSN

0002-9939

DOI

10.1090/proc/17223

Keywords

differential geometry, general relativity and quantum cosmology (gr-qc)

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