Harmonic mean curvature flow in Riemannian manifolds and Ricci flow on noncompact manifolds.

Abstract

This thesis treats two topics about geometric flows. One topic concerns the deformation of hypersurfaces in negatively curved Riemannian manifolds using fully nonlinear parabolic equations defined in terms of the normal curvatures. It is shown that in a simply-connected negatively curved manifold, any strictly convex compact initial hypersurface flowing by Harmonic Mean Curvature Flow produces a solution which converges to a single point in finite time, and becomes spherical as the limit point is approached. We give several examples of hypersurfaces #30;t(Mn) evolving in time with speed determined by functions of the normal curvatures in an (n + 1)-dimensional hyperbolic manifold; we emphasize the case of flow by harmonic mean curvature. The examples converge to a totally geodesic submanifold of any dimension from 1 to n, and include cases which exist for infinite time. The other topic concerns Ricci flow on noncompact manifolds. Following Deane Yang’s sketch, we give the first detailed proof of the short-time existence of the local Ricci flow introduced by him. Using the local Ricci flow, we prove the short-time existence of the Ricci flow on noncompact manifolds whose Ricci curvature has a global lower bound and sectional curvature has only a local average integral bound. The short-time existence of the Ricci flow on noncompact manifolds was studied by Wan-Xiong Shi in 1989, who required a uniform bound of curvature tensors. As a corollary of our main theorem, we prove short-time existence in this more general context.