Differential Geometry in the Large

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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.