Some remarks on the Kähler geometry of the Taub-NUT metrics

In this article, we investigate the balanced condition and the existence of an Engliš expansion for the Taub-NUT metrics on ℂ^2. Our first result shows that a Taub-NUT metric on ℂ^2 is never balanced unless it is the flat metric. The second one shows that an Engliš expansion of the Rawnsley's function associated to a Taub-NUT metric always exists, while the coefficient a_3 of the expansion vanishes if and only if the Taub-NUT metric is indeed the flat one.