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 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 <= 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 (Sigma) over tilde of Sigma in N, there exists a critical point of the Wang-Yau quasi-local energy of (Sigma) over tilde.