Some remarks on the Kaehler geometry of LeBrun's Ricci flat metrics on C^2

TL;DR

This paper explores the geometric properties of LeBrun's Ricci flat metrics on C^2, showing they are generally not balanced unless flat, and analyzing the Englis expansion related to these metrics.

Contribution

It demonstrates that LeBrun's metrics are not balanced unless flat and establishes conditions for the existence and properties of the Englis expansion.

Findings

LeBrun's metrics are not balanced unless flat.

An Englis expansion always exists for these metrics.

The coefficient a_3 vanishes only for the flat metric.

Abstract

In this paper we investigate the balanced condition (in the sense of Donaldson) and the existence of an Englis expansion for the LeBrun's metrics on $C^2$. Our first result shows that a LeBrun's metric on $C^2$ is never balanced unless it is the flat metric. The second one shows that an Englis expansion of the Rawnsley's function associated to a LeBrun's metric always exists, while the coefficient $a_3$ of the expansion vanishes if and only if the LeBrun's metric is indeed the flat one.