Mathematical Sciences: Evolution Equations in Geometry

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The principal investigator will study a method for showing that all closed 3-manifolds with positive scalar curvature are obtained from spherical space forms by performing connected sums. An analogous problem for 4-manifolds with positive isotropic curvature will be analyzed. Other parts of the project include investigations into the four-dimensional Schoenflies conjecture, as well as studies of Kaehler and symplectic structures. A closed curve on a sphere, which does not have any intermediate intersections, separates the plane into two disks. Analogously, a properly smooth sphere lying in ordinary 3-space bounds one ball, and bounds a second ball if infinity is collapsed to a point. The principal investigator will study a method using heat flow to solve the analogous problem for three- dimensional smooth spheres lying in four-dimensional space.