Mass, Kaehler Manifolds, and Symplectic Geometry

TL;DR

This paper demonstrates that techniques from 4-dimensional symplectic geometry can be used to establish a mass formula for asymptotically Euclidean Kaehler manifolds with weaker fall-off conditions, simplifying previous proofs.

Contribution

It provides a new proof of the mass formula for ALE Kaehler manifolds in dimension four using symplectic geometry, requiring only Chrusciel fall-off conditions.

Findings

Mass formula for ALE Kaehler manifolds in dimension four

New proof of Penrose-type inequality under weaker conditions

Application of symplectic geometry techniques to geometric analysis

Abstract

In the author's previous joint work with Hans-Joachim Hein, a mass formula for asymptotically locally Euclidean (ALE) Kaehler 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 Chrusciel fall-off conditions that sufficed in higher dimensions. The present article, however, shows that techniques of $4$-dimensional symplectic geometry can be used to obtain all the major results of the previous paper, assuming only Chrusciel-type fall-off. In particular, the present article presents a new a proof of our Penrose-type inequality for the mass of an asymptotically Euclidean Kaehler manifold that only requires Chrusciel metric fall-off.