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

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

2011-03-31

arxiv.org/abs/1103.6158v1

Mathematics

Differential Geometry

19 pages

Scientific paper

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.

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