Complex Hyperbolicity, Value Distribution and Non-Positive Curvature

本课题的研究重点是复双曲和非等维值分布理论，它们都研究了从复线到复流形的整个全纯映射的像的行为。一般推测，在负弯曲流形或各种一般类型中的这种全纯曲线具有很大的约束。在最近与Yum-Tong Siu合作的几篇论文中，研究者解决了Lang关于整条全纯曲线关于一个充足因子的值分布的猜想，并在二维空间中实质上解决了Kobayashi关于复射影平面上曲线补的双曲性的猜想；这方面的研究将继续进行。一个长期的目标是了解复双曲性与丢番图近似理论的关系。在第二部分，作者考虑了负曲率流形的几个方面。在过去的研究中，作者从几个方面研究了这类流形在不同附加条件下的均匀化问题。他建议在与他早期研究项目相关的几个方向上工作，包括研究这种流形的几何和分析方面及其与普遍覆盖、基本群和贝蒂数的关系。在几何学中，人们研究的模型一方面要优雅、相对简单地描述，另一方面要表现出丰富的几何结构。一个精心选择的负弯曲空间恰好是一个很好的研究对象。从物理学的角度来看，研究弯曲或非平坦的空间是很自然的，因为广义相对论的结果是，人们意识到宇宙本身的模型不是平坦的。负弯曲几何结构的研究领域非常丰富，因为来自不同数学分支的思想可以被应用于产生美丽的结果。本提案中提出的大多数项目都涉及对这些空间的澄清和分类。

The focus of this project is on the research of complex hyperbolicity and non-equidimensional value distribution theory, both of which study the behaviour of the image of an entire holomorphic map from the complex line to a complex manifold. It is generally conjectured that such entire holomorphic curves in a negatively curved manifold or a variety of general type are greatly constrained. In several recent joint papers with Yum-Tong Siu, the investigator has solved a conjecture of Lang about the value distribution of an entire holomorphic curve with respect to an ample divisor, and in dimension two essentially solved a conjecture of Kobayashi about hyperbolicity of the complement of a curve in the complex projective plane; research in this direction will be continued. A long term goal is to understand the relation of complex hyperbolicity to diophantine approximation theory. In the second part of the proposal, the author considers several aspects of manifolds with negative curvature. In the past the author has studied several aspects of uniformization of such manifolds with different conditions attached. He proposes to work on several directions related to his earlier research projects, including study of geometric and analytic aspects of such a manifold and their relations to its universal covering, fundamental group and Betti numbers. In geometry, people study models which on one hand should be elegant and relatively simple to discribe and on the other hand should display rich geometric structures. A well chosen negatively curved space happens to be a good candidate to study. The study of curved or non-flat spaces is very natural from a physics point of view because as a result of general relativity, people realize that the model of the universe itself is not flat. The area of the study of negatively curved geometric structures is very fertile in the sense that ideas from different branches of mathematics can be applied to produce beautiful results. Most of the projects pro posed in this proposal concern the clarification and classification of such spaces.