Critical points of Wang-Yau quasi-local energy

الأصل · EN

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 ≤ H₀ where H is the mean curvature of Σ in Ω and H₀ is the mean curvature of Σ when isometrically embedded in R³. If Ω is not isometric to a domain in R³, 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 Σ of Σ in N, there exists a critical point of the Wang-Yau quasi-local energy of Σ.