Gravitational Instantons, Weyl Curvature, and Conformally Kaehler Geometry

TL;DR

This paper studies Ricci-flat deformations of certain 4-manifolds, showing they remain Hermitian and often belong to previously classified families under mild conditions.

Contribution

It proves that Ricci-flat deformations of toric Hermitian ALF 4-manifolds are Hermitian and often fall into known classifications, extending previous classifications to deformations.

Findings

Deformations must be Hermitian under suitable conditions.

Deformations carry a non-trivial Killing vector field.

Deformed metrics often belong to previously classified families.

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.