Critical points of Wang-Yau quasi-local energy

TL;DR

This paper proves a theorem about the critical points of Wang-Yau quasi-local energy for certain spacelike surfaces in spacetimes satisfying energy conditions, revealing conditions for local minima and critical points.

Contribution

It establishes conditions under which the Wang-Yau quasi-local energy has strict local minima and critical points for specific spacelike surfaces in general relativity.

Findings

Brown-York mass is a strict local minimum of Wang-Yau energy under given conditions.

Existence of critical points for perturbed surfaces in the spacetime.

Conditions relating mean curvature and isometric embedding in R^3.

Abstract

In this paper, we prove the following theorem regarding the Wang-Yau quasi-local energy of a spacelike two-surface in a spacetime: Let $\Sigma$ be a boundary component of some compact, time-symmetric, spacelike hypersurface $\Omega$ in a time-oriented spacetime $N$ satisfying the dominant energy condition. Suppose the induced metric on $\Sigma$ has positive Gaussian curvature and all boundary components of $\Omega$ have positive mean curvature. Suppose $H \le H_0$ where $H$ is the mean curvature of $\Sigma$ in $\Omega$ and $H_0$ is the mean curvature of $\Sigma$ when isometrically embedded in $R^3$. If $\Omega$ is not isometric to a domain in $R^3$, then 1. the Brown-York mass of $\Sigma$ in $\Omega$ is a strict local minimum of the Wang-Yau quasi-local energy of $\Sigma$, 2. on a small perturbation $\tilde{\Sigma}$ of $\Sigma$ in $N$, there exists a critical point of the Wang-Yau quasi-local energy of $\tilde{\Sigma}$.