The principal investigator will continue his study of Riemannian metrics and spectral geometry. In particular, he will consider the following problems: find conditions for the solvability of the equation of prescribed scalar curvature on spheres, find bounds for the Sobolev constant of a Riemannian manifold in terms of its spectral data, and construct extremal metrics with respect to functionals arising from spectral considerations. The research supported by this award generally will attempt to relate the local geometric information on a manifold or surface with global information about the surface. The local information is given in terms of partial differential equations on the surface and the global information involves globally defined solutions to these partial differential equations. The theme of relating local to global information is a common one in differential geometry. One of the locally defined operators to be considered here is the Laplacian which is perhaps the most important operator for applications to engineering or physics.
