Complex Differential Geometry and Rigidity

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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.