Uniqueness of balanced metrics on complex vector bundles.

Let E → M be a holomorphic vector bundle over a compact Kähler manifold (M,ω). We prove that if E admits a ω-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 ω-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 ω-balanced Kähler maps into Grassmannians.