Mass Inequality and Stability of the Positive Mass Theorem For K\"ahler Manifolds

Abstract

We prove an integral inequality and two stability results for the ADM mass on AE K\"ahler manifolds of all complex dimensions. The inequality bounds the ADM mass from below by an integral of the scalar curvature and the Hessian of certain holomorphic coordinate functions arising from the complex coordinates at infinity. Using this, we first prove a stability result for any sequence of AE K\"ahler manifolds with ADM mass converging to zero. We conclude that, for any such sequence, there exist subsets of each with vanishing boundaries in the limit such that the complements converge to Euclidean space in the pointed Gromov-Hausdorff sense. This gives the first stability result of the Positive Mass Theorem for K\"ahler manifolds, or more generally, of manifolds without strong curvature or volume conditions or for a very explicit family of metrics in real dimensions greater than three. If we furthermore impose a uniform lower bound on the Ricci curvature, the second stability theorem shows the same result without taking the complement of a sequence of vanishing sets. Finally, we find three new families of AE K\"ahler manifolds with vanishing mass in the limit for which the stability results apply.