The principal investigator will analyze conformal metrics on compact Riemannian manifolds which either have or do not have boundaries. The existence of positive solutions to semilinear elliptic equations with non-linear boundary conditions will also be investigated. Additionally, topological obstructions to the existence of metrics with positive curvature on compact manifolds with boundary will be studied in detail. The sphere is the only two-dimensional compact surface which can have positive curvature everywhere; that is, at each point there are two distinct directions in which the surface bends "down." Points on a saddle have negative curvature for the saddle bends down in one direction, but up in other directions. Tori, or surfaces of donuts, cannot have positive curvature everywhere, regardless of how they are deformed. The principal investigator will study the analogous question for higher- dimensional compact hypersurfaces.
