Summary of the Kähler Mass Paper

whenever this limit exists. Here Σk(r) ≡ ∂Bn r (0)/Γk denotes the Euclidean sphere of radius r in the k-th conical end of M , all suband superscripts refer to the standard Euclidean coordinates on Rn, ν is the Euclidean exterior unit normal to Σk(r), and dA is the Euclidean area measure. Bartnik [1] and Chruściel [2] discovered sharp conditions on the decay rate of Φ∗g − g0 ensuring that the limit in (1) exists and is independent of Φ. We provide an alternative proof of their result under the assumption that (M, g) is Kähler. In fact, in this case, we obtain an explicit formula for the mass in terms of the total scalar curvature, the Kähler class, and the first Chern class. Using this formula, we are then able to prove the Positive Mass Theorem in the Kähler case.