Characterization of invariant complex Finsler metrics on the complex Grassmann manifold

Abstract

Let P:=U(p+q)/U(p)×U(q) be the complex Grassmann manifold and F:T1,0P→[0,+∞) be an arbitrary U(p+q)-invariant strongly pseudoconvex complex Finsler metric. We prove that F is necessary a Kähler-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 Kähler metric on P. In particular on the complex projective space CPn=U(n+1)/U(n)×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 Kähler 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.