Rotation invariant singular K\"ahler metrics with constant scalar curvature on $\mathbb{C}^n$

TL;DR

This paper classifies rotation invariant constant scalar curvature Kähler metrics on complex Euclidean spaces minus the origin, providing explicit solutions in lower dimensions and proving non-existence results for negative scalar curvature in certain cases.

Contribution

The authors reduce the scalar curvature equation to a 2nd order ODE system and explicitly classify all such metrics with zero or positive scalar curvature, also establishing non-existence of negative cases in specific dimensions.

Findings

Complete classification of zero or positive csck metrics in lower dimensions

Explicit solutions obtained through solving ODEs

Non-existence of negative csck metrics for n=2,3

Abstract

The scalar curvature equation for rotation invariant K\"ahler metrics on $\mathbb{C}^n \backslash \{0\}$ is reduced to a system of ODEs of order 2. By solving the ODEs, we obtain complete lists of rotation invariant zero or positive csck on $\mathbb{C}^n \backslash \{0\}$ in lower dimensions. We also prove that there does not exist negative csck on $\mathbb{C}^n \backslash \{0\}$ for $n=2,3$.