Characterization of invariant complex Finsler metrics on the complex Grassmann manifold

Abstract

Let $P:=U(p+q)/U\left(p\right)\times U\left(q\right)$ be the complex Grassmann manifold and $F:T^{1,0}P\to [0,+\infty )$ 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 ${\mathrm{CP}}^n=U(n+1)/U\left(n\right)\times U\left(1\right)$, 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.