Some remarks on Wang–Yau quasi-local mass

We review the definition of Wang–Yau quasi-local mass from the point of view of the gravitational Hamiltonian. This makes clear the relation between Wang–Yau definition and Brown–York or even global ADM definition. We make a brief comment on admissibility condition in the definition of the 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 graph blowing up at marginally outer trapped surfaces.