Perelman's functionals on cones and Construction of type III Ricci flows coming out of cones

TL;DR

This paper investigates the behavior of Perelman's functionals on conical manifolds, characterizes their finiteness, and constructs type III Ricci flows originating from cones, advancing understanding of singularity smoothing.

Contribution

It analyzes Perelman's functionals on cones, characterizes their finiteness, and constructs type III Ricci flows from conical structures, extending Ricci flow theory.

Findings

Finiteness of Perelman's functionals on cones depends on the link's $\lambda$-functional.

Manifolds with conical singularities can have well-defined $\lambda$-functionals.

Cones over perturbed spheres can be smoothed by type III Ricci flows.

Abstract

In this paper, we are interested in conical structures of manifolds with respect to the Ricci flow and, in particular, we study them from the point of view of Perelman's functionals. 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 on the Ricci flow.