"Geometric Analysis on Complete Manifolds" by Peter Li
The goal of this project is to obtain further understanding of the underlying geometric and topological structure of an unbounded (infinite) geometric object (manifold). The techniques being utilized will involve understanding solutions of some partial differential equations on the manifold in terms of the curvature and other geometric invariants. On the other hand, knowledge of the solutions of these partial differential equations will yield additional information on the geometric and topological structure of the manifold. This general theme is the essence of this proposal and the development of various techniques involved in this project will be very useful in future studies of complete, noncompact manifolds. 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.
In a broader point of view, the proposed project will yield further understand of a geometric object. The shape of and structures of geometric objects are important related to many scientific fields. Examples of these are the structure of black holes and worm holes in physics, the structure of DNA given by the double helix in biology, the structure of molecules in chemistry, and behavior of liquid crystal in engineering. This Principal Investigator will also contribute to the national needs of producing more mathematicians by involving graduate students and postdoctoral scholars in his research. Their involvement may be in the form of direct collaborations, advising, and learning new material through seminars. This process will be an important component in training the new generation of mathematicians.
Together with coauthor Jiaping Wang, the PI proved that if a complete, n-dimensional manifold of finite volume has Ricci curvature bounded from below by -(n-1) and if 0 is the only eigenvalue in the spectrum below (n-1)^2/4, then the manifold must have finitely many ends (cusps). In particular, the number of cusps can be estimated by the ratio of the volume of the manifold and the volume of any fixed unit ball. The PI also completed a book on 'Geometry Analysis' (280 pages) for beginning researcher in the subject. The book is published in the 'Cambridge Studies in Advanced Mathematics' series by Cambridge University Press. Under the support of this grant, a female graduate student, My-An Tran, has written a research article jointly with SY Li entitled 'Infimum of the spectrum of Laplace-Beltrami operator on a bounded pseudoconvex domain with a Kaehler metric of Bergman type.' The PI is also supervising two Ph.D. students, Lihan Wang and Fei He, who are currently working on their dissertations. Both students have already published 1 paper each while being supported by the grant. The paper by Lihan Wang is entitled "Eigenvalues of the weighted p-Laplacian" that has been accepted for publication by the Proceedings of AMS. The paper by Fei He is entitled "Remarks on the extension of the Ricci flow" that has been accepted for publication by the Journal of Geometric Analysis.
