Spectrum, Geometry and Topology of Complete Manifolds

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The proposal concerns some aspects on the interactions among the spectrum, topology and geometry of Riemannian manifolds. One of them is the behavior of eigenfunctions, both on compact and non-compact manifolds. It aims to gain a better understanding of the structure of spectrum on complete manifolds. The study will also lead to sharp estimates of the higher eigenvalues. Another aspect is on the size of the bottom spectrum of complete manifolds. The issues to be addressed include finding an optimal upper bound in terms of the curvature lower bounds and uncovering the effect on the geometry and topology when such an upper bound is attained. Particular emphasis is on complete resolution of some of the remaining open problems from the recent studies in this direction. The principal investigator also proposes to develop a corresponding theory for the Hodge Laplacian on complete manifolds.The spectrum or the vibrating frequencies of an geometric object is closely related to its shape and structure. The proposal is to study some aspects of this relation via new mathematical formulations and approaches. Of particular interest here is to determine the size and structure of the spectrum via curvature, a quantity that can be measured locally and describes how curved the geometric object is. Successful completion of the project will provide deeper insight into this important relation. It will also have interesting applications in both physics theory and other sciences.