A Donaldson type functional on a holomorphic Finsler vector bundle

Abstract

Abstract. In this paper, we solve a problem of Kobayashi posed in [15] 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 Kähler 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.

Geodesic-Einstein metrics and nonlinear stabilities

“…Moreover, by using X. Chen's geodesic approximation lemma (cf. [5], Lemma 7; also [13], Lemma 2.3), we get the following theorem:…”

“…Moreover, by using X. Chen's geodesic approximation lemma (cf. [5], Lemma 7; also [13], Lemma 2.3), we get the following theorem: Theorem 0.1. The functional L(•, ψ) attains its absolute minimum at the geodesic-Einstein metrics on L.…”

“…By the Kobayashi correspondence (cf. [16], [12], [13]), a Finsler metric G on E induces a natural admissible metric on O P (E) (1). In this case, we can prove that the induced metric on O P (E) (1) is geodesic-Einstein if and only if G is Finsler-Einstein.…”

“…For any Finsler metric G on E, let φ be the corresponding Hermitian metric on the line bundle O P (E) (1), which is also an admissible metric on O P (E) (1) (cf. [16], [12], [13]). With respect to a holomorphic trivialization of E → M , by the standard procedure, one gets a local homogeneous holomorphic coordinate system {[ζ]|ζ = (ζ 1 , · · · , ζ r ) = 0} on the fibres of P (E).…”

“…Also recall that a Finsler-Einstein vector bundle E → M is semistable (cf. [13]). Here a natural question is whether a Finsler-Einstein vector bundle admits a Hermitian-Einstein metric.…”