Characterization of invariant complex Finsler metrics on the complex Grassmann manifold

Let P := U ( p + q ) /U ( p ) x U ( q ) be the complex Grassmann manifold and F : T 1 ,0 P -> [0 , + infinity) be an arbitrary U ( p + q ) -invariant strongly pseudoconvex complex Finsler metric. We prove that F is necessary a K & auml;hler-Berwald metric which is not necessary Hermitian quadratic. We also prove that F is Hermitian quadratic if and only if F is a constant multiple of the canonical U ( p + q ) -invariant K & auml;hler metric on P . In particular on the complex projective space CP n = U ( n + 1) /U ( n ) x U (1), there exists no U ( n + 1) -invariant strongly pseudoconvex complex Finsler metric other than a constant multiple of the Fubini-Study metric. These invariant metrics are of particular interesting since they are the most important examples of strongly pseudoconvex complex Finsler metrics on P which are elliptic metrics in the sense that they enjoy very similar holomorphic sectional curvature and bisectional curvature properties as that of the U ( p + q ) -invariant K & auml;hler metrics on P , nevertheless, these invariant metrics are not necessary Hermitian quadratic, hence provide nontrivial explicit examples for complex Finsler geometry in the compact cases. (c) 2024 Elsevier B.V. All rights reserved.