Complex Differential Geometry and Rigidity

论文题目：复微分几何与刚性命题数：dms-0203647摘要作者计划在微分几何中做以下三个方面的研究：1)具有退化Ricci张量的非正弯曲黎曼流形的几何；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。对于主题1)，作者已经在实际分析案例中得到了一些结果，并建议在此背景下研究光滑案例。2)，他研究了一类具有非正对分曲率的紧致Kaehler流形。在一些附加的非退化假设下，他得到了这类流形的一个结构结果，这可以看作是经典的Howard-Smyth-Wu分裂定理的对偶版本。他想研究一般情况，以及与这种类型的流形有关的一些其他问题。对于主题3)，他建议研究二维或更高维的复超曲型空间的两个度量刚性问题。一个是关于四分之一夹紧度量的非等变度量刚性问题，另一个是关于某些特殊类型的CAT(0)奇异度量的度量刚性问题。拟议的研究位于微分几何领域，这是一个数学分支，旨在了解给定空间的整体结构和曲率(衡量弯曲程度)之间的相互作用。本文研究的空间模型主要有黎曼流形和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。一个众所周知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，现代控制理论几乎完全使用了微分几何中发展起来的工具。在微分几何领域，自70年代初S发现Mostow和Marguis以来，对非正曲线空间及其经常伴随的刚性现象的研究一直处于中心舞台上。它在80年代的S和90年代的S引起了数学界的极大关注。这位提出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)时空的十维乘积空间，是一个紧凑的六维截面，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，凯勒几何在纯数学和应用数学的许多领域中变得更加重要。

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.