Mass in almost Kähler geometry

In this talk, I present an explicit formula for the ADM mass of asymptotically locally Euclidean (ALE) almost Kähler 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 Kähler ALE case. The proof is based on a spin-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, I show that one can prove a positive mass theorem and a Penrose-type inequality for asymptotically Euclidean (AE) almost Kähler manifolds using this formula.