Complex Hyperbolicity, Value Distribution and Non-Positive Curvature

该项目的重点是复双曲性和非等维值分布理论的研究，这两个理论都研究了从复直线到复流形的整个全纯映射的像的行为。 一般认为，负曲流形或各种一般类型中的全纯曲线都受到极大的约束。 在最近的几个联合文件与Yum-Tong萧，调查解决了一个猜想郎的价值分布的整个全纯曲线就充分因子，并在第二次基本上解决了猜想小林的双曲补曲线在复射影平面;研究在这个方向将继续下去。 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。 在建议的第二部分，作者考虑了负曲率流形的几个方面。 作者在过去研究了这类流形在不同条件下的一致化的几个方面。 他建议工作的几个方向与他早期的研究项目，包括研究几何和分析方面的这样一个流形及其关系，其普遍覆盖，基本组和贝蒂数。 在几何学中，人们研究的模型一方面应该是优美的，相对简单的描述，另一方面应该显示丰富的几何结构。 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非常自然的，因为作为广义相对论的结果，人们认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是非常肥沃的，因为不同数学分支的思想可以应用于产生美丽的结果。 本提案中提出的大多数项目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.