Kähler-Finsler rigidity of the Kobayashi metric and the Kähler base of weakly Kähler-Finsler metrics

In this paper, we establish a rigidity theorem: for every balanced, smooth, bounded pseudoconvex domain, its Kobayashi metric is a smooth, strongly pseudoconvex, Kähler-Finsler metric restricted on a neighborhood of the origin if and only if the domain is a Hermitian ellipsoid, biholomorphic to the unit ball. Thus for any balanced, smooth, bounded, strictly convex, and strongly pseudoconvex domain that is not biholomorphic to the unit ball, its Kobayashi metric is a non-Kähler-Finsler, weakly Kähler-Finsler metric. We also prove that the base manifold of every weak Kähler-Finsler metric is Kähler.