A momentum construction for circle-invariant Kahler metrics
  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. 
Andrew D. Hwang, Michael A. Singer