Perelman's Functionals on Cones: Construction of Type III Ricci Flows Coming Out of Cones

摘要

In this paper, we are interested in conical structures of manifolds with respect to the Ricci flow. In a first part, we study Perelman's lambda and nu functionals of cones and characterize their finiteness in terms of the lambda-functional of the link. As an application, we characterize manifolds with conical singularities on which a lambda-functional can be defined and get upper bounds on the nu-functional of asymptotically conical manifolds. We then present an adaptation of the proof of Perelman's pseudolocality theorem and prove that cones over some perturbations of the unit sphere can be smoothed out by type III immortal solutions of the Ricci flow.