Mathematical Sciences: Differential Geometry and Partial Differential Equations

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The principal investigator, Richard Schoen, will continue his research on the construction of Riemannian metrics satisfying curvature equations, and on differential equations involving mappings between Riemannian manifolds. He will study the Yamabe- type problem for non-compact manifolds. For compact manifolds he will investigate certain geometrically natural uniqueness questions. For harmonic maps, he will consider regularity questions for stationary maps, existence questions for higher critical points of the energy functional, and generic uniqueness questions for minima. He also intends to study a geometrically natural functional which is unchanged upon replacement of a map by its inverse. This should be a useful tool for studying situations involving one-one maps. The construction of metrics with certain curvature restrictions is an important aspect of differential geometry with ramifications in mathematical physics.