具有曲率下界的Kahler流形

黎曼几何中，具有Ricci曲率下界的流形研究具有悠久的历史。在这个课题中，申请人计划系统地研究具有Ricci曲率下界的Kahler流形。其关键在于研究复结构和曲率之间的关系。具体的问题包括：1. 丘成桐的单值化猜想 2. Kahler流形的Gromov-Hausdorff 极限研究 3. Ricci平坦Kahler流形（尤其是非紧）。可能用到的工具包括Gromov-Hausdorff收敛理论，Hormander L2理论，代数几何，偏微分方程等。..申请人在之前对以上问题已经有一些重要的结果。.1. 发现Kahler流形三环定理。.2. 证明了倪磊猜想。.3. 证明了丘成桐有限生成猜想。.4. 只加一个自然条件的情况下证明了丘成桐单值化猜想。.5. 合作者推广了田刚院士著名的Partial C0估计。

In Riemannian geometry，the study of manifolds with Ricci curvature lower bound has a long history. The applicant plans to study Kahler manifolds with Ricci curvature lower bound systematically. The key point is the relation between curvature and complex structure. The problems include: 1. Yau's uniformization conjecture 2. Gromov-Hausdorff limits of Kahler manifolds 3. Ricci flat Kahler manifolds (especially the noncompact case). Possible tools include Gromov-Hausdorff convergence theory, Hormander L2 theory, algebraic geometry, partial differential equations, etc...The applicant has made important contributions to the problems above. .1. Discovery of a three circle theorem on Kahler manifolds..2. Proof of Ni's conjecture on polynomial growth holomorphic functions..3. Proof of Yau's finite generation conjecture..4. Proof of Yau's uniformization conjecture, under a very natural assumption..5. Generalization of Prof. Tian's famous Partial C0 estimate（joint with collaborator).