  In relativity, the energy of a moving particle depends on the observer, and
the rest mass is the minimal energy seen among all observers. The Wang-Yau
quasi-local mass for a surface in spacetime introduced in [7] and [8] is
defined by minimizing quasi-local energy associated with admissible isometric
embeddings of the surface into the Minkowski space. A critical point of the
quasi-local energy is an isometric embedding satisfying the Euler-Lagrange
equation. In this article, we prove results regarding both local and global
minimizing properties of critical points of the Wang-Yau quasi-local energy. In
particular, under a condition on the mean curvature vector we show a critical
point minimizes the quasi-local energy locally. The same condition also implies
that the critical point is globally minimizing among all axially symmetric
embedding provided the image of the associated isometric embedding lies in a
totally geodesic Euclidean 3-space.
