Gravitational Instantons, Weyl Curvature, and Conformally Kaehler Geometry

Author:

Olivier Biquard, Paul Gauduchon, Claude LeBrun

Keyword:

Mathematics, Differential Geometry, Differential Geometry (math.DG)

journal:

date:

2023-10-21 16:00:00

Abstract

In a previous paper, the first two authors classified complete Ricci-flat ALF Riemannian 4-manifolds that are toric and Hermitian, but non-Kaehler. In this article, we consider general Ricci-flat deformations of such spaces, assuming only suitable fall-off conditions. Quite generally, we are able to show that such a deformation must be Hermitian, and must carry a non-trivial Killing vector field with fixed asymptotics. With mild additional hypotheses, we are then able to show that the new Ricci-flat metric must in fact belong to the family of previously classified metrics.

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