In this note we introduce a Yang-Mills bar equation on complex vector bundles E provided with a Hermitian metric over compact Hermitian manifolds. According to the Koszul-Malgrange criterion any holomorphic structure on E can be seen as a solution to this equation. We show the existence of a non-trivial solution to this equation over compact Kähler manifolds as well as a short time existence of...

We start by constructing a Hilbert manifold T of orientation preserving diffeomorphisms of the circle (modulo the group of bi-holomorphic self-mappings of the disc). This space, which could be thought of as a completion of the universal Teichmüller space, is endowed with a right-invariant Kähler metric. Using results from the theory of quasiconformal mappings we construct an embedding of T into...

In the first part of this work, the Poisson equation on complete noncompact manifolds with nonnegative Ricci curvature is studied. Sufficient and necessary conditions for the existence of solutions with certain growth rates are obtained. Sharp estimates on the solutions are also derived. In the second part, these results are applied to the study of curvature decay on complete Kähler manifolds. ...

In the first part of my talk, we consider special metrics on holomorphic bundles. We will recall the classical Hitchin-Kobayashi correspondence (Donaldson-Uhlenbeck-Yau theory) of stability and HermitianEinstein metrics on holomorphic vector bundles; and some generalizations of the classical Hitchin-Kobayashi correspondence, specially, we will focus on non-compact case; furthermore, We’ll discu...

We study the existence of L holomorphic sections of invariant line bundles over Galois coverings of Zariski open sets in Moishezon manilolds. We show that the von Neuman dimension of the space of L holomorphic sections is bounded below under reasonable curvature conditions. We also give criteria for a a compact complex space with isolated singularities and some related strongly pseudoconcave ma...

The curvature KT (w) of a contraction T in the Cowen-Douglas class B1(D) is bounded above by the curvature KS∗(w) of the backward shift operator. However, in general, an operator satisfying the curvature inequality need not be contractive. In this note, we characterize a slightly smaller class of contractions using a stronger form of the curvature inequality. Along the way, we find conditions o...

In complex general relativity, Lorentzian space-time is replaced by a four-complex-dimensional complex-Riemannian manifold, with holomorphic connection and holomorphic curvature tensor. A multisymplectic analysis shows that the Hamiltonian constraint is replaced by a geometric structure linear in the holomorphic multimomenta, providing some boundary conditions are imposed on two-complex-dimensi...

The analysis of holomorphic sections of high powers L of holomorphic ample line bundles L → M over compact Kähler manifolds has been widely applied in complex geometry and mathematical physics. Any polarized Kähler metric g with respect to the ample line bundle L corresponds to the Ricci curvature of a hermitian metric h on L. Any orthonormal basis {SN 0 , ..., S dN} of H(M,L ) induces a holomo...

Applying a well known result for attracting fixed points of biholomorphisms [4, 6], we observe that one immediately obtains the following result: if (Mn, g) is a complete non-compact gradient Kähler-Ricci soliton which is either steady with positive Ricci curvature so that the scalar curvature attains its maximum at some point, or expanding with non-negative Ricci curvature, then M is biholomor...