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.
