Compact Proof of the Positivity of Quasi-Local Masses for a class of Initial Data

Abstract: We prove a purely quasi-local positivity theorem for the Wang--Yau mass for a class of initial data whose Jang deformation, after a boundary-preserving conformal reduction to zero scalar curvature, lies in a sufficiently small transverse--traceless (TT) perturbative neighborhood of a strictly convex Euclidean fill-in.We explicitly construct a nontrivial class of physical initial data whose admissible Jang reductions realize this TT-generated sector. The argument reduces the Wang--Yau energy to the Brown--York mass of the resulting scalar-flat compact metric, together with nonnegative bulk terms determined by the Jang deformation, and establishes strict positivity by computing the second variation of the Brown--York functional at the Euclidean metric in transverse--traceless directions. The proof is entirely confined to the compact fill-in and uses neither an asymptotically flat extension nor the positive mass theorem. This gives a partial answer to a question of R. Schoen concerning a genuinely quasi-local proof of positivity for quasi-local mass.