Rigidity of compact manifolds and positivity of quasi-local mass

Abstract

In this paper, we obtain some rigidity theorems on compact manifolds with nonempty boundary. The results may be related to the positivity of some quasi-local mass of Brown–York type. The main argument is to use monotonicity of quantities similar to the Brown–York quasi-local mass in a foliation of quasi-spherical metrics. Together with a hyperbolic version of positivity of a mass quantity, we obtain our main results.

On the volume functional of compact manifolds with boundary with constant scalar curvature

“…Let Σ be any given compact strictly convex hypersurface in R 3 . If (Ω, g) is a compact Riemannian 3-manifold with nonnegative scalar curvature whose boundary is isometric to Σ and has positive mean curvature H, then it was proved in [19] [20] that (43) does hold if Σ is a geodesic sphere in H 3 and (Ω, g) has scalar curvature no less than −6, which is consistent with Proposition 3.1. It remains an interesting question to know whether (43) is true for a compact 3-manifold (Ω, g) whose boundary is isometric to a geodesic sphere in Σ 3 + and whose scalar curvature is greater than or equal to +6.…”

“…As a direct application of Theorem 3.1 and Corollary 2.2, we have Before we proceed to discuss properties of general critical metrics of V (·), we want to relate the result in Proposition 3.1 to the results in [19] and [20]. Let Σ be any given compact strictly convex hypersurface in R 3 .…”

Quasi-Local Energy-Momentum and Angular Momentum in General Relativity

“…This result is generalized in [459] by weakening the energy condition, in which case lower estimates of the Brown-York energy can still be given. For some rigidity theorems connected with this positivity result, see [461]; and for their generalization for higher dimensional spin manifolds, see [329]. …”

A Minkowski Inequality for Hypersurfaces in the Anti‐de Sitter‐Schwarzschild Manifold

“…The time component of the embedding can be chosen to be cosh r, which is the same as the static potential here. In fact, the same weighting factor was considered in [23], where another quasi-local mass with the hyperbolic space as reference was studied. We remark that the total mass for asymptotically hyperbolic manifolds has been considered by many authors; see, e.g., [1,6,7,20,26,27].…”