Differential Geometry by way of Partial Differential Equations

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ABSTRACT DMS - 0202508.The first proposed project is to understand the topology of complete manifolds whose Ricci curvature is bounded from below by a negative multiple of the bottom of the spectrum of the Laplace operator. In this case, the bottom of the spectrum is assumed to be positive. The second project of the Principal Investigator is to investigate the size of the space of harmonic functions with polynomial growth on complete manifolds with non-negative sectional curvature. In particular, the primary goal is to prove that the dimension of the space of harmonic functions of polynomial growth with degree at most d is bounded by the dimension of an analogous space in Euclidean space of the same dimension. The third project is to understand the geometry of stable minimal hypersurfaces of Euclidean space. More generally, he would like to study minimal hypersurfaces with have finite index. The PI also proposes to support a postdoc, Dr. Xiaofeng Sun, in his research projects. Dr. Sun proposes to study the regularity theory of harmonic maps into a metric space. These proposed projects have the unified theme of using analytical techniques in studying the underlining geometric spaces. In many instances, topological and geometrical conclusions can be drawn by detail analysis of solutions of some appropriate partial differential equations defined on the space.