  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.
