关于负曲率或非正曲率完备凯勒流形结构的研究

The most important objective of geometry is to study the structure of space while curvature is crucial for the objective. Comparing to the study of positively or nonnegatively curved complete Kähler manifolds, the study of negatively or nonpositively curved complete Kähler manifolds is still not enough. The structure of such kinds of space is not quite clear currently. Our previous works reveal that there is a close relation between negatively curved complete Kähler manifolds and the CR geometry of its boundary, and that geometric heat flows are useful for the study of Wu-Zheng’s splitting conjecture. In this project, we will further study the restriction of negatively curved complete Kähler metrics on complex structures, Wu-Zheng's splitting conjecture and explore the structure of negatively or nonpositively curved complete Kähler manifolds on the foundation of our previous works. We will mainly study the boundaries of negatively curved complete Kähler manifolds from the viewpoint of CR geometry, the existence of proper holomorphic embedding from bounded domains that admit a complete Kähler metric with sectional curvature not greater than a negative constant to unit balls, the complete points of Kähler metrics with sectional curvature not greater than a negative constant on the bidisc or bounded pseudo-convex domains and Wu-Zheng's splitting conjecture by geometric heat flows.

了解空间的结构是几何学最重要的任务，而曲率在空间结构的研究中具有决定性的作用。相对于正曲率或非负曲率完备凯勒流形结构的研究，对负曲率或非正曲率完备凯勒流形结构的研究还很不充分。学者们目前对于这类空间的结构还不太了解。前期研究表明，负曲率完备凯勒流形的结构与其边界的CR几何有密切的联系，而几何热流对于研究非正曲率完备凯勒流形的伍鸿熙和郑方阳教授的分解猜想颇为有效。本项目拟在前期研究的基础上，进一步研究负曲率完备凯勒度量对复结构的约束作用和伍鸿熙和郑方阳教授的分解猜想，探索负曲率或非正曲率完备凯勒流形的结构。我们将主要从CR几何的角度研究负曲率完备凯勒流形的边界，研究容许负常数截面曲率上界的完备凯勒度量的有界区域到复单位球中的常态全纯嵌入的存在性、双圆盘上甚至有界伪凸区域上具有负常数截面曲率上界的凯勒度量的完备点以及通过几何热流进一步解决伍鸿熙和郑方阳教授的分解猜想。

在项目执行期间，我们主要研究了如下内容：（i）带边流形上的Steklov特征值；（ii）数学广义相对论中的拟局域质量及等距嵌入；（iii）非零曲率下界下Sharp的Li-Yau型估计；（iv）一些其他问题及与几何相关的偏微分方程问题。在Stekelov特征值问题上，我们发展了微分形式的Steklov算子理论，将Hersch-Payne-Schiffer之前的经典工作推广到高维流形上。在拟局域质量问题上，我们揭示了此前未曾引起关注的CNT拟局域能量与Wang-Yau拟局域能量之间的联系。在Sharp的Li-Yau型估计上，我们引入Li-Yau乘数集，试图把Li-Yau估计归结为乘数集的比较。受到Bakry等工作的启发，我们还研究非线性形式的Li-Yau估计，获得3维双曲空间的Sharp的Li-Yau型估计。受到研究过程中各种自然启发，我们还研究图上的Steklov特征值及旋转不变调和微分同胚等若干与相关问题。