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Mass in Kähler Geometry

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Abstract

We prove a simple, explicit formula for the mass of any asymptotically locally Euclidean (ALE) Kähler manifold, assuming only the sort of weak fall-off conditions required for the mass to actually be well-defined. For ALE scalar-flat Kähler manifolds, the mass turns out to be a topological invariant, depending only on the underlying smooth manifold, the first Chern class of the complex structure, and the Kähler class of the metric. When the metric is actually AE (asymptotically Euclidean), our formula not only implies a positive mass theorem for Kähler metrics, but also yields a Penrose-type inequality for the mass.

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Authors and Affiliations

  1. Department of Mathematics, University of Maryland, College Park, MD, 20742-4015, USA

    Hans-Joachim Hein

  2. Department of Mathematics, State University of New York, Stony Brook, NY, 11794-3651, USA

    Claude LeBrun

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  1. Hans-Joachim Hein
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  2. Claude LeBrun
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Correspondence to Claude LeBrun.

Additional information

Communicated by P. T. Chruściel

H.-J. Hein was research funded in part by NSF grant DMS-1514709.

C. LeBrun was research funded in part by NSF grant DMS-1510094.

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Hein, HJ., LeBrun, C. Mass in Kähler Geometry. Commun. Math. Phys. 347, 183–221 (2016). https://doi.org/10.1007/s00220-016-2661-4

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