Annals of Global Analysis and Geometry

, Volume 56, Issue 1, pp 97–112 | Cite as

Mass, Kähler manifolds, and symplectic geometry

  • Claude LeBrunEmail author


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.


Mass Asymptotically locally Euclidean Kähler Scalar curvature Symplectic Pseudo-holomorphic curve Penrose inequality 



This paper was largely written during a stay at the École Normale Supérieure in Paris, while the author was on sabbatical leave as a Simons Fellow. It is a particular pleasure to thank Olivier Biquard for his gracious hospitality in Paris, as well as for many stimulating and useful conversations. He would also like to thank the faculty of the ENS for offering such a warm welcome into their idyllic research environment.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsStony Brook UniversityStony BrookUSA

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