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 ≤ 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 $${\mathbb R^3}$$ . If Ω is not isometric to a domain in $${\mathbb R^3}$$ , 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 $${\tilde{\Sigma}}$$ of Σ in N, there exists a critical point of the Wang–Yau quasi-local energy of $${\tilde{\Sigma}}$$
.