On canonical radial Kaehler metrics

TL;DR

This paper characterizes when a radial Kaehler metric is Kaehler-Einstein, linking it to extremal metrics, generalized scalar curvatures, and Kaehler-Ricci solitons, providing a comprehensive classification.

Contribution

It establishes necessary and sufficient conditions for radial Kaehler metrics to be Kaehler-Einstein, connecting various curvature properties and soliton structures.

Findings

Kaehler-Einstein iff extremal and associated to a Kaehler-Ricci soliton

Two different generalized scalar curvatures are constant

Extremal (not cscK) with constant generalized scalar curvature

Abstract

We prove that a radial Kaehler metric g is Kaehler-Einstein if and only if one of the following conditions is satisfied: 1. g is extremal and it is associated to a Kaehler-Ricci soliton; 2. two different generalized scalar curvatures of g are constant; 3. g is extremal (not cscK) and one of its generalized scalar curvature is constant.