Hermitian-Einstein metrics for vector bundles on complete Kähler manifolds

Abstract

Abstract. In this paper, we prove the existence of Hermitian-Einstein metrics for holomorphic vector bundles on a class of complete Kähler manifolds which include Hermitian symmetric spaces of noncompact type without Euclidean factor, strictly pseudoconvex domains with Bergman metrics and the universal cover of Gromov hyperbolic manifolds etc. We also solve the Dirichlet problem at infinity for the Hermitian-Einstein equations on holomorphic vector bundles over strictly pseudoconvex domains.

Hermitian Yang–Mills–Higgs Metrics on Complete Kähler Manifolds

“…In the second part of this paper, we study the existence of Hermitian Yang-Mills-Higgs metrics for a holomorphic vector bundle over a class of complete noncompact Kähler manifolds. We would like to point out that Ni and Ren [16,17] had discussed the existence of a Hermitian Einstein metric on a complete Kähler manifold, and we would adapt the techniques used by them. In section 4, we prove a long-time existence of the Hermitian Yang-Mills-Higgs heat equation on any complete Kähler manifold, under some assumptions on the initial metric and the holomorphic section φ.…”

Kobayashi–Hitchin correspondence for analytically stable bundles

“…There are many interesting studies of the Hermitian-Einstein metrics on vector bundles over Kähler manifolds with infinite volume. For example, see [4,34,36,38,39] and a more recent work [7]. We have already mentioned the pioneering work of Donaldson [9].…”

“…It has been known that the solvability of the Poisson equation on Y is useful for the construction of Hermitian-Einstein metrics of holomorphic vector bundles (E, ∂ E ) on Y . The idea has been efficiently used in [1,19,36,38,39]. In those cases, a Hermitian-Einstein metric h on (E, ∂ E ) is obtained if (E, ∂ E ) has a Hermitian metric h 0 such that ΛF (h 0 ) satisfies a decay condition, even if we do not assume (E, ∂ E , h 0 ) is analytically stable.…”

Stability and Hermitian–einstein Metrics for Vector Bundles on Framed Manifolds

“…We also want to point out the following result by Ni and Ren ( [27], see also [26]): In the situation of a complete Kähler manifold with a positive lower bound on the spectrum of the Laplace-Beltrami operator, they show that every Hermitian metric which is "asymptotically Hermitian-Einstein" (i. e. with its contracted curvature tensor satisfying a certain integrability condition) can be deformed into a Hermitian-Einstein metric. It turns out, however, that manifolds with metrics of Poincaré-type growth always have finite volume (see [15], proof of Lemma 1, p. 402) and thus clearly violate the above spectral bound condition.…”