A momentum construction for circle-invariant Kahler metrics

TL;DR

This paper refines the conditions under which Calabi's ansatz can produce complete, constant scalar curvature Kähler metrics, including Einstein-Kähler metrics, on disk bundles over Kähler manifolds, revealing new families of such metrics.

Contribution

It provides weaker curvature conditions that guarantee the existence of new families of complete constant scalar curvature Kähler metrics using Calabi's ansatz.

Findings

Curvature conditions ensure existence of complete constant scalar curvature Kähler metrics.

These conditions are weaker than previous criteria, enabling new metric families.

Necessary conditions are established for the ansatz to produce multiple such metrics.

Abstract

An ansatz of Calabi allows construction of Kahler metrics in an Hermitian disk bundle over a Kahler manifold. We attempt to give a definitive treatment of this ansatz, with the following results: We give curvature conditions on the disk bundle that guarantee existence of families of complete Kahler metrics of constant scalar curvature, including some Einstein-Kahler metrics. These curvature conditions are somewhat weaker than those used by prior authors, and are shown to give rise to large families of metrics that seem to be new. Further, these curvature hypotheses are necessary, in the sense that if the ansatz gives two distinct metrics of constant scalar curvature in a disk bundle, then the curvature hypotheses are satisfied.