On Homothetic Balanced Metrics

In this paper we study the set of balanced metrics (in Donaldson's terminology) on a compact complex manifold M which are homothetic to a given balanced one. This question is related to various properties of the Tian-Yau-Zelditch approximation theorem for Kahler metrics. We prove that this set is finite when $M$ admits a non-positive Kahler-Einstein metric, in the case of non-homogenous toric Kaehler-Einstein manifolds of dimension $\leq 4$ and in the case of Arezzo-Pacard constant scalar curvature metrics.