In this article, we study the set of balanced metrics given in Donaldson’s terminology (J. Diff. Geometry 59:479–522, 2001) 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 Kähler metrics. We prove that this set is finite when M admits a non-positive Kähler–Einstein metric, in the case of non-homogenous toric Kähler-Einstein manifolds of dimension ≤ 4 and in the case of the constant scalar curvature metrics found in Arezzo and Pacard (Acta. Math. 196(2):179–228, 2006; Ann. Math. 170(2):685–738, 2009).
On homothetic balanced metrics / C. Arezzo; A. Loi; F. Zuddas. - In: ANNALS OF GLOBAL ANALYSIS AND GEOMETRY. - ISSN 0232-704X. - 41(2012), pp. 473-491. [10.1007/s10455-011-9295-8]