Critical points of Wang-Yau quasi-local energy

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 $Σ$ be a boundary component of some compact, time-symmetric, spacelike hypersurface $Ω$ in a time-oriented spacetime $N$ satisfying the dominant energy condition. Suppose the induced metric on $Σ$ has positive Gaussian curvature and all boundary components of $Ω$ have positive mean curvature. Suppose $H \le H_0$ where $H$ is the mean curvature of $Σ$ in $Ω$ and $H_0$ is the mean curvature of $Σ$ when isometrically embedded in $R^3$. If $Ω$ is not isometric to a domain in $R^3$, then 1. the Brown-York mass of $Σ$ in $Ω$ is a strict local minimum of the Wang-Yau quasi-local energy of $Σ$, 2. on a small perturbation $\tildeΣ$ of $Σ$ in $N$, there exists a critical point of the Wang-Yau quasi-local energy of $\tildeΣ$.