Uniqueness problem of meromorphic mappings of a complete Kähler manifold into a projective space

In this article, we prove a new generalization of uniqueness theorems for meromorphic mappings of a complete Kähler manifold M into ℙn(ℂ) sharing hyperplanes in general position with a general condition on the intersections of the inverse images of these hyperplanes.

On uniqueness of meromorphic mappings from complete Kähler manifold into ℙn(ℂ)

In this paper, we prove a uniqueness theorem for meromorphic mappings of a complete Kähler manifold [Formula: see text] into [Formula: see text] sharing hyperplanes in general position under a general condition that the codimension of the intersection of inverse images of any [Formula: see text] hyperplanes is at least two.

A cup product lemma for continuous plurisubharmonic functions

A version of Gromov’s cup product lemma in which one factor is the (1, 0)-part of the differential of a continuous plurisubharmonic function is obtained. As an application, it is shown that a connected noncompact complete Kähler manifold that has exactly one end and admits a continuous plurisubharmonic function that is strictly plurisubharmonic along some germ of a [Formula: see text]-dimensional complex analytic set at some point has the Bochner–Hartogs property; that is, the first compactly supported cohomology with values in the structure sheaf vanishes.

A Liouville Property of Holomorphic Maps

We prove a Liouville property of holomorphic maps from a complete Kähler manifold with nonnegative holomorphic bisectional curvature to a complete simply connected Kähler manifold with a certain assumption on the sectional curvature.

L2 CASTELNUOVO–DE FRANCHIS, THE CUP PRODUCT LEMMA, AND FILTERED ENDS OF KÄHLER MANIFOLDS

Simple approaches to the proofs of the L2 Castelnuovo–de Franchis theorem and the cup product lemma which give new versions are developed. For example, assume that ω1 and ω2 are two linearly independent closed holomorphic 1-forms on a bounded geometry connected complete Kähler manifold X with ω2 in L2. According to a version of the L2 Castelnuovo–de Franchis theorem obtained in this paper, if ω1 ∧ ω2 ≡ 0, then there exists a surjective proper holomorphic mapping of X onto a Riemann surface for which ω1 and ω2 are pull-backs. Previous versions required both forms to be in L2.

GLOBAL INHOMOGENEOUS SCHRÖDINGER FLOW

In this paper, we consider the global existence of one-dimensional nonautonomous inhomogeneous Schrödinger flow. By exploiting geometric symmetries, we prove that, given a smooth initial map, the Cauchy problem of the one-dimensional nonautonomous inhomogeneous Schrödinger flow from S1 into a complete Kähler manifold with constant holomorphic sectional curvature admits a unique global smooth solution. As a corollary, we establish the global existence for the Cauchy problem of the inhomogeneous Heisenberg spin system.