The Entropy of Ricci Flows with Type-I Scalar Curvature Bounds

TL;DR

This paper investigates the entropy behavior of Ricci flows with Type-I scalar curvature bounds at singularities, extending previous results and characterizing the singular set via heat kernel analysis, especially in four dimensions.

Contribution

It generalizes the understanding of Ricci flow singularities by analyzing entropy convergence under weaker Type-I scalar curvature bounds and characterizes the singular set using heat kernel density functions.

Findings

Entropy of conjugate heat kernel converges to soliton entropy.

Singular Ricci soliton in dimension 4 has finitely many conical orbifold singularities.

Results extend previous work to weaker curvature assumptions.

Abstract

In this paper, we extend the theory of Ricci flows satisfying a Type-I scalar curvature condition at a finite-time singularity. In [Bam16], Bamler showed that a Type-I rescaling procedure will produce a singular shrinking gradient Ricci soliton with singularities of codimension 4. We prove that the entropy of a conjugate heat kernel based at the singular time converges to the soliton entropy of the singular soliton, and use this to characterize the singular set of the Ricci flow solution in terms of a heat kernel density function. This generalizes results previously only known with the stronger assumption of a Type-I curvature bound. We also show that in dimension 4, the singular Ricci soliton is smooth away from finitely many points, which are conical smooth orbifold singularities