Complex Differential Geometry and Rigidity

提案标题：复杂微分几何和刚性提案编号：DMS - 0203647 本文拟在微分几何中研究以下三个问题：1）具有退化Ricci张量的非正曲率黎曼流形的几何; 2）具有非正对分曲率的紧致Kaehler流形的结构; 3）复双曲空间的度量刚性/非刚性。对于题目1），提出者在真实的解析情形下得到了一些结果，并提出在此背景下研究光滑情形。 对于2），他研究了具有非正对分曲率的紧致Kaehler流形类。在一些额外的非退化假设下，他能够得到这种流形的结构结果，这可以被视为经典Howard-Smyth-Wu分裂定理的对偶版本。他想研究的一般情况下，以及其他一些相关问题，关于这种类型的流形。对于题目3），他提出研究复超的两个度量刚性问题！在二维或更高维度的弯曲空间中。一个是四分之一箍缩度量的非等变度量刚性问题，另一个是某些特殊类型的CAT（0）奇异度量的度量刚性问题。 拟议的研究属于微分几何领域，这是数学的一个分支，旨在理解给定空间的整体结构和曲率（衡量弯曲）之间的相互作用。在本研究中，空间的主要模型是黎曼流形和Kaehler流形。这一领域不仅对数学的许多分支很重要，而且对其他科学和工程如物理学也很重要。一个著名的例子是在爱因斯坦的广义相对论中使用黎曼几何。另一个例子是，现代控制理论几乎只使用微分几何中开发的工具。在微分几何领域中，自70年代初Mostow和Margulis发现非正曲空间和与之相关的刚性现象以来，非正曲空间的研究一直处于中心阶段。它在80、90年代引起了数学界的广泛关注。提议者和其他人试图研究Kaehler流形的这些主题，Kaehler流形是具有特殊类型结构的黎曼流形。弦理论的最新发展表明宇宙是通常的（四维）时空和一个紧致的六维截面的十维乘积空间，它应该是一种特殊的Kaehler流形（所谓的Calabi-Yau空间）。除了数学之外的原因和现有的/潜在的应用，Kaehler几何在纯数学和应用数学的许多领域中变得越来越重要。

Proposal title: Complex differential geometry and rigidityProposal number: DMS - 0203647 AbstractThe proposer plans to do research in the following three topics in differentialgeometry: 1) the geometry of nonpositively curved Riemannian manifolds with degenerate Ricci tensor; 2) the structure of compact Kaehler manifold with nonpositive bisectional curvature; 3) metric rigidity/non-rigidity for complex hyperbolic spaces. For topic 1), the proposer has obtained some results in the real analytic cases, and he proposes to investigate the smooth cases in this context. For 2), he has studied the class of compact Kaehler manifolds with nonpositive bisectional curvature. Under some additional non-degenerateness assumption, he was able to obtain a structural result for such manifolds, which can be viewed as the dual version of classical Howard-Smyth-Wu splitting theorem. He would like to study the general case, as well as some other related questions regarding this type of manifolds. For topic 3), he proposes to study two metric rigidity questions about the complex hyper!bolic space, in dimension two or higher. One is a non-equivariant metric rigidity question for quarter-pinched metrics, the other is for some special type of CAT(0) singular metrics. The proposed research lies in the field of differential geometry, which is a branch of mathematics aimed at understanding the interplay between the global structure andthe curvature (which measures the bending) of a given space. The major models of spaces in this study are Riemannian manifolds and Kaehler manifolds. This field is of importancenot only to many branches of mathematics, but also to other sciences and engineering such as physics. A well known example is the use of Riemannian geometry in Einstein's theory of the general relativity. Another example is that the modern control theory uses almost exclusively tools developed in differential geometry. Within the field of differential geometry, the study of nonpositively curved spaces and the rigidity phenomenon often associated with such spaces has been on the center stage since early 70's, after the discovery of Mostow and Margulis. It has drawn much attention in 80's and 90's from the mathematical world. The proposer and others are trying to investigate these topics for Kaehler manifolds, which are Riemannian manifolds with a special type of structure. The recent development in string theory suggests that the universe is the ten dimensional product space of the usual (four dimensional) space time and a compact six dimensionalcross section which should be a special kind of Kaehler manifold (the so-called Calabi-Yau spaces). Besides reasons and existing/potential application outside mathematics, Kaehler geometry has become more of more important within many fields of pure and applied mathematics.