Singularity Analysis of the Ricci Flow and Monotonicity Properties of Entropy Functionals

Jun-Fang Li, McGill University

Abstract. We study the profiles of singularity models for the Ricci flow, especially the classification of steady states of Ricci flow equations. We are interested in singularity analysis with Ricci flow solutions. Various integral quantities which have monotonic properties along a Ricci flow play a crucial role here. We will introduce several functionals of entropy type and study their first variation formulas. Furthermore, the first variation vanishes if and only if the manifold is a steady state of a Ricci flow solution. As applications, we improve previous work on the evolution of eigenvalues along the Ricci flow.