The balanced metrics and cscK metrics on certain holomorphic ball bundles

Abstract

In this paper, we use the well-known Calabi ansatz, further generalized by Hwang-Singer, to study the existence of balanced metrics and constant scalar curvature Kähler (cscK for short) metrics on certain holomorphic ball bundles M which are locally expressed as M={(z,u,w)∈Ω×Cd1×Cd2:eλ1ϕ(z)‖u‖2+eλ2ϕ(z)‖w‖2<1}. Let g be the Kähler metric on M associated with the Kähler forms locally expressed as ω=−1∂∂‾(ϕ(z)+F(eλ1ϕ(z)‖u‖2+eλ2ϕ(z)‖w‖2)). Firstly, we obtain sufficient and necessary conditions for g to be cscK metrics. Secondly, using this result, we obtain necessary and sufficient conditions for mg to be balanced metrics for all sufficiently large positive integer numbers m. Finally, we obtain complete cscK metrics and balanced metrics on the ball bundles over simply connected Riemann surfaces. The main contribution of this paper is the explicit construction of complete, non-compact cscK metrics and balanced metrics.