Mass, Kähler manifolds, and symplectic geometry

In the author’s previous joint work with Hein (Commun Math Phys 347:183–221, 2016), a mass formula for asymptotically locally Euclidean Kähler manifolds was proved, assuming only relatively weak fall-off conditions on the metric. However, the case of real dimension 4 presented technical difficulties that led us to require fall-off conditions in this special dimension that are stronger than the Chruściel fall-off conditions that sufficed in higher dimensions. Nevertheless, the present article shows that techniques of four-dimensional symplectic geometry can be used to obtain all the major results of Hein-LeBrun (2016), assuming only Chruściel-type fall-off. In particular, the present article presents a new proof of our Penrose-type inequality for the mass of an asymptotically Euclidean Kähler manifold that only requires this very weak metric fall-off.