Principal Investigator: Peter Li
The PI will continue the joint investigation with his coauthor, Jiaping Wang, on rigidity and finiteness properties of certain classes of complete manifolds. Motivated by the theorem of Witten and Yau, the PI and Wang proved some rigidity and finiteness results for n-dimensional complete manifolds with Ricci curvature bounded from below and has the spectrum of the Laplacian bounded from below by a positive constant. One of the main projects is to consider a more relaxed hypothesis than the above mentioned. Instead of assuming that the spectrum has a positive lower bound, Li and Wang will consider manifolds on which a weighted Poincare inequality is valid. The more general hypothesis will enlarge the class of manifolds under consideration to include even Euclidean space of dimension at least 3. Other projects related to the wieghted Poincare inequality will also be considered. In particular, its relationship to Agmon's distance in the elliptic setting, the work of Li-Yau in the linear parabolic Schroedinger equation setting, and the Perelman's L-length in the non-linear parabolic (Ricci flow) setting.
The work of Witten-Yau drew conclusion on a certain physical model of the universe and showed that there is no worm-hole. A broader purpose of the project is to seek a substantial generalization of their work in a geometric setting. In particular, this research project will give further understanding of unbounded (infinite) geometric objects. In this sense, a possible long-term effect is the understanding of other physical models of the universe. The understanding of geometric structure will also have broader impact on material science. This line of investigation will yield direct implications to the theory of partial differential equations governing the behavior of many physical models and biological models. It is also related to many engineering problems, such as, liquid crystals, heat transfer, and imaging.
