Strong field behavior of Wang-Yau quasi-local energy

We consider the strong field behavior of the Wang-Yau quasi-local energy. In particular, we examine the limit of the Wang-Yau quasi-local energy as the defining spacelike 2-sphere Σ approaches an apparent horizon from the exterior. We find that if the apparent horizon Σ cannot be isometrically embedded into R3, the Wang-Yau quasi-local energy diverges at the horizon. Further, in this situation, the optimal embedding equation does not admit any solution that is smooth up to the horizon.