Stability of Positive Mass Theorem for Static Quasi-Local Energy of Compact (Locally) Hyperbolic Graphical Manifolds

TL;DR

The paper analyzes the stability of the rigidity statement in the positive mass theorem for static quasi-local energy in hyperbolic settings. It derives lower bounds for the static Brown-York energy $m_{BY}^{S}$, establishes a volume-growth/volume-bound framework via a height function, and proves a flat-distance convergence of compact (locally) hyperbolic graphical manifolds to a reference product domain as the energy tends to zero. The main result shows that if $m_{BY}^{S}(\Sigma_i) \to 0$, then the manifolds converge to $\{0\}\times(\overline{U}\backslash U_o)$ in the sense of currents and their volumes converge, providing a rigorous stability statement. The approach combines hyperbolic Minkowski-type inequalities, isoperimetric bounds, and a four-region flat-distance decomposition to relate quasi-local energy to geometric closeness in the Federer–Fleming sense.

Abstract

In this paper, we consider compact graphical manifolds with boundary over (locally) hyperbolic static space. We prove the stability of the positive mass theorem with respect to the Federer--Fleming flat distance for the static quasi-local Brown-York energy of the outer boundary of compact (locally) hyperbolic graphical manifolds.

Theorems & Definitions (18)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Remark 2.3
• Lemma 3.1
• proof