Geometric Analysis on Complete Manifolds

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Abstract: "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.