Uniqueness of balanced metrics on holomorphic vector bundles

Let E→ME→M be a holomorphic vector bundle over a compact Kähler manifold (M,ω)(M,ω). We prove that if EE admits a ωω-balanced metric (in X. Wang’s terminology (Wang, 2005 [3])) then it is unique. This result together with Biliotti and Ghigi (2008) [14] implies the existence and uniqueness of ωω-balanced metrics of certain direct sums of irreducible homogeneous vector bundles over rational homogeneous varieties. We finally apply our result to show the rigidity of ωω-balanced Kähler maps into Grassmannians.