Geometric Flows On Compact Manifolds PDF Download

We study the Ricci flow and cross curvature flow of a class of warped product metrics which are suitable for studying these flows on manifolds which admit a circle action, possibly with fixed points. This includes studying geometric flows on manifolds with boundary and discussing the necessary boundary conditions one must impose to obtain a well-defined flow. We derive the induced flows on the orbit space and study mixed boundary value problems of systems corresponding to the Ricci and cross curvature flows which are suitable for our situation. This involves studying the Ricci flow and cross curvature flow of a metric which is degenerate on the boundary of a manifold. We prove short time existence of solutions for these systems with certain assumptions made about the form of the warping function and metric on the orbit space in terms of parallel geodesic coordinates. We then study the evolution of various geometric quantities of interest along with evolution equations involving the warping function. We also derive the evolution of the first and second fundamental forms under these flows which are useful in studying these and other boundary value problems. We calculate certain quantities related to the curvature and cross curvature at the boundary of the manifold which may be used to study long time existence of solutions with additional conditions made on the warping functions and metric on the orbit space.