A Donaldson type functional on a holomorphic Finsler vector bundle

TL;DR

This paper introduces a Donaldson type functional for holomorphic Finsler vector bundles, generalizing classical Donaldson functionals, and characterizes Finsler-Einstein metrics as its unique critical points on a Kähler manifold.

Contribution

It develops a new Donaldson type functional in complex Finsler geometry and proves Finsler-Einstein metrics are its only critical points, solving a problem posed by Kobayashi.

Findings

The functional attains its minimum at Finsler-Einstein metrics.

Finsler-Einstein metrics are the unique critical points of the functional.

The work generalizes Donaldson's functional to complex Finsler geometry.

Abstract

In this paper, we solve a problem of Kobayashi posed in \cite{Ko4} by introducing a Donaldson type functional on the space $F^+(E)$ of strongly pseudo-convex complex Finsler metrics on $E$ -- a holomorphic vector bundle over a closed K\"ahler manifold $M$. This Donaldson type functional is a generalization in the complex Finsler geometry setting of the original Donaldson functional and has Finsler-Einstein metrics on $E$ as its only critical points, at which this functional attains the absolute minimum.