  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
$\tilde{\Sigma}$.
