Radial Projectively Induced Canonical K\"ahler Metrics: Rigidity and Classification

Abstract

We study radial K\"ahler metrics on domains of $\mathbb{C}^n$, $n\geq 2$, admitting a K\"ahler immersion into a finite- or infinite-dimensional complex projective space. We classify those with constant non-negative scalar curvature: up to a linear change of coordinates, they are positive integer multiples of the Fubini-Study metric, the flat metric, or, in complex dimension two, generalized Burns-Simanca metrics. We also prove that every radial projectively induced K\"ahler-Einstein metric has constant holomorphic sectional curvature and is therefore a Fubini-Study, flat, or complex hyperbolic metric. Finally, we show that a radial infinitely projectively induced extremal K\"ahler metric has unbounded maximal radial domain if and only if it is scalar-flat.