Extremal Kähler Metrics Induced by Finite or Infinite-Dimensional Complex Space Forms

Abstract

In this paper, we address the problem of studying those complex manifolds M equipped with extremal metrics g induced by finite or infinite-dimensional complex space forms. We prove that when g is assumed to be radial and the ambient space is finite-dimensional, then (M, g) is itself a complex space form. We extend this result to the infinite-dimensional setting by imposing the strongest assumption that the metric g has constant scalar curvature and is well behaved (see Definition 1 in the Introduction). Finally, we analyze the radial Kahler–Einstein metrics induced by infinite-dimensional elliptic complex space forms and we show that if such a metric is assumed to satisfy a stability condition then it is forced to have constant nonpositive holomorphic sectional curvature.