Extremal Kaehler metrics induced by finite or infinite dimensional complex space forms

TL;DR

This paper investigates extremal Kaehler metrics induced by complex space forms, proving that radial metrics in finite dimensions are space forms and extending results to infinite dimensions with constant scalar curvature, analyzing stability and curvature conditions.

Contribution

It establishes conditions under which extremal Kaehler metrics induced by complex space forms are themselves space forms, extending results to infinite-dimensional settings with stability assumptions.

Findings

Radial extremal metrics in finite dimensions are complex space forms.

Infinite-dimensional extremal metrics with constant scalar curvature are analyzed.

Stability conditions imply non-positive holomorphic sectional curvature.

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 Kaehler-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 non-positive holomorphic sectional curvature.