Critical points of Wang-Yau quasi-local energy

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