Mathematical Sciences: Isoperimetric Inequalities for Eigenvalue Ratios

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Work supported by this award concentrates on geometric questions related to partial differential operators defined on surface-like regions known as manifolds. One can visualize the general thrust of the work by considering a vibrating surface or membrane and asking for analytic quantities which capture the geometric properties of the surface. One such class of quantities consists of eigenvalues of the Laplace operator - a partial differential operator - defined on the surface with certain boundary conditions imposed on its boundary. It had long been conjectured that the ratio of the first two Dirichlet eigenvalues of the Laplacian was maximized for surfaces which were flat discs. This (the Payne-Polya-Weinberger conjecture) has recently been verified by the principal investigator. This project will seek to describe surfaces which maximize other eigenvalue ratios. At the same time, efforts will be made to extend results obtained in Euclidean space to manifolds. Since much of what is known is confined to two-dimensional surfaces, extensions to surfaces and manifolds of higher dimension will also be addressed.