On homogeneous complex manifolds with negative definite canonical Hermitian form

Complex Monge-ampère Equations on Hermitian Manifolds

We study complex Monge-Ampère equations in Hermitian manifolds, both for the Dirichlet problem and in the case of compact manifolds without boundary. Our main results extend classical theorems of Yau [43] and Aubin [1] in the Kähler case, and those of Caffarelli, Kohn, Nirenberg and Spruck [9] for the Dirichlet problem in C n . As an application we study the problem of finding geodesics in the ...

A Note on Homogeneous Complex Contact Manifolds

Homogeneous two-manifolds with an invariant two-form

We study a 2-dimensional manifold that is homogeneous acted on by a 3-dimensional Lie group G, and that has a 2-form invariant under G. We show that such a manifold can be realized as a surface in the affine 3-space and list such realizations. Mathematics Subject Classification(2000). 53C30, 53C42, 53C45

Vanishing theorems on Hermitian manifolds

We prove the vanishing of the Dolbeault cohomology groups on Hermitian manifolds with ddc-harmonic Kähler form and positive (1, 1)-part of the Ricci form of the Bismut connection. This implies the vanishing of the Dolbeault cohomology groups on complex surfaces which admit a conformal class of Hermitian metrics, such that the Ricci tensor of the canonical Weyl structure is positive. As a coroll...

Strictly Hermitian Positive Definite Functions

Let H be any complex inner product space with inner product < ·, · >. We say that f : | C → | C is Hermitian positive definite on H if the matrix ( f(< z,z >) )n r,s=1 (∗) is Hermitian positive definite for all choice of z, . . . ,z in H, all n. It is strictly Hermitian positive definite if the matrix (∗) is also non-singular for any choice of distinct z, . . . ,z in H. In this article we prove...