On the balanced condition for the Eguchi-Hanson metric

TL;DR

This paper proves that the scaled Eguchi-Hanson metric on the blow-up of complex two-space at the origin is never balanced for any positive integer scale.

Contribution

It establishes a new non-balancedness result for the Eguchi-Hanson metric under all positive integer scalings.

Findings

$mg_{EH}$ is not balanced for any positive integer $m$

The result applies to the Eguchi-Hanson metric on the blow-up of $\\mathbb{C}^2$

Provides insight into the geometric properties of the Eguchi-Hanson metric

Abstract

Let $g_{EH}$ be the Eguchi-Hanson metric on the blow-up of $\mathbb{C}^2$ at the origin. In this paper we show that $mg_{EH}$ is not balanced for any positive integer $m$.