  We review Wang-Yau quasi-local definitions along the line of gravitational
Hamiltonian. This makes clear the connection and difference between Wang-Yau
definition and Brown-York or even global ADM definition. We make a brief
comment on admissibility condition in Wang-Yau quasi-lcoal mass. We extend the
positivity proof for Wang-Yau quasi-local energy to allow possible presence of
strictly stable apparent horizons through establishing solvability of Dirac
equation in certain 3-manifolds that possess cylindrical ends, as in the case
of Jang's graph blowing up at marginally outer trapped surfaces.
