Mass and rigidity in almost K\"ahler geometry

Abstract

We derive an explicit formula for the ADM mass of asymptotically locally Euclidean (ALE) almost K\"ahler manifolds. The formula expresses the mass in terms of the total Hermitian scalar curvature and topological data associated with the underlying almost complex structure, extending a result of Hein and LeBrun in the K\"ahler ALE case. Our approach is based on a spin$^\mathbb {C}$ adaptation of Witten's proof of the positive mass conjecture in the spin case and is therefore distinct from previous complex-geometric methods. In dimension $4$, we prove a positive mass theorem and a Penrose-type inequality for asymptotically Euclidean (AE) almost K\"ahler manifolds. We also study rigidity phenomena of almost K\"ahler ALE manifolds. We prove that an almost K\"ahler-Einstein ALE manifold with nonnegative scalar curvature and certain decay assumptions is necessarily K\"ahler-Einstein. In particular, any four dimensional Ricci-flat almost K\"ahler manifold with maximal volume growth and curvature in $L^2$ is K\"ahler, yielding new evidence towards the Bando--Kasue--Nakajima conjecture. We also discuss analogous rigidity results for asymptotically locally flat (ALF) manifolds.