The two senior investigators will study the relationship between Lp bounds on curvature and the geometry and topology of a Riemannian manifold. They will work towards a proof that a new topological invariant, if zero, implies an F-structure in the sense of Cheeger-Gromov. The relationship between the isoperimetric constant and the curvature of a manifold will be investigated. The investigators will generalize the Heintze- Karcher estimate on the volume of a geodesic neighborhood. A lower bound on the shortest closed geodesic would be very useful. A computer will be used to compute some nontrivial examples of convergent Riemannian manifolds. The standard sphere has positive Gaussian curvature everywhere. Even when deformed, the average curvature remains positive. Such is not the case for the two-holed torus. Regardless of how one of these is deformed, the average curvature must be negative. In higher dimensions the picture is much more complicated. Other sorts of curvature, such as Ricci curvature, play the central roles. A central problem in differential geometry, which the two investigators will address, is how the averages of these curvatures relates to topological properties such as numbers of holes.
