Differential Geometry in the Large

AbstractAward：DMS-0503735首席研究员：Peter Li PI将继续与他的合著者王家平共同研究某些类完备流形的刚性和有限性。 受维滕和丘德威定理的启发，PI和Wang证明了n维完备流形的刚性和有限性结果，这些完备流形的Ricci曲率下有界且Laplacian的谱下有界。 其中一个主要项目是考虑一个比上述更宽松的假设。 而不是假设该谱有一个正的下界，李和王将证明流形上的加权庞加莱不等式是有效的。 更一般的假设将扩大类的流形考虑，甚至包括欧几里德空间的维数至少为3。其他与庞加莱不等式有关的项目也将被考虑。特别是它与椭圆情形下的Agmon距离、线性抛物Schroedinger方程情形下的Li-Yau的工作以及非线性抛物（Ricci流）情形下的Perelman的L-E的关系，Witten-Yau的工作得出了宇宙的某种物理模型的结论，证明了虫洞是不存在的。 该项目的一个更广泛的目的是在几何环境中寻求他们工作的实质性概括。 特别是，这个研究项目将使人们进一步理解无限（无限）几何对象。从这个意义上说，一个可能的长期影响是理解宇宙的其他物理模型。 对几何结构的理解也将对材料科学产生更广泛的影响。这条调查线将产生直接影响到理论的偏微分方程的行为，许多物理模型和生物模型。 它还涉及到许多工程问题，如液晶、传热和成像。

AbstractAward: DMS-0503735Principal Investigator: Peter LiThe PI will continue the joint investigation with his coauthor,Jiaping Wang, on rigidity and finiteness properties of certain classesof complete manifolds. Motivated by the theorem of Witten and Yau,the PI and Wang proved some rigidity and finiteness results forn-dimensional complete manifolds with Ricci curvature bounded frombelow and has the spectrum of the Laplacian bounded from below by apositive constant. One of the main projects is to consider a morerelaxed hypothesis than the above mentioned. Instead of assuming thatthe spectrum has a positive lower bound, Li and Wang will considermanifolds on which a weighted Poincare inequality is valid. The moregeneral hypothesis will enlarge the class of manifolds underconsideration to include even Euclidean space of dimension at least3. Other projects related to the wieghted Poincare inequality willalso be considered. In particular, its relationship to Agmon'sdistance in the elliptic setting, the work of Li-Yau in the linearparabolic Schroedinger equation setting, and the Perelman's L-lengthin the non-linear parabolic (Ricci flow) setting.The work of Witten-Yau drew conclusion on a certain physical model ofthe universe and showed that there is no worm-hole. A broader purposeof the project is to seek a substantial generalization of their workin a geometric setting. In particular, this research project willgive further understanding of unbounded (infinite) geometric objects.In this sense, a possible long-term effect is the understanding ofother physical models of the universe. The understanding of geometricstructure will also have broader impact on material science. This lineof investigation will yield direct implications to the theory ofpartial differential equations governing the behavior of many physicalmodels and biological models. It is also related to many engineeringproblems, such as, liquid crystals, heat transfer, and imaging.