Tangent Flows of K\"ahler Metric Flows

TL;DR

This paper advances the understanding of singularities and tangent flows in noncollapsed Ricci flows within the K"ahler setting, revealing new structural properties and symmetries of the metric flows.

Contribution

It establishes the equality of certain singular strata and demonstrates the existence of isometric actions on tangent flows, using novel parabolic regularizations.

Findings

Singular strata satisfy $ ext{S}^{2j}= ext{S}^{2j+1}$.

Tangent flows admit nontrivial isometric one-parameter actions.

Parabolic regularizations of heat kernel potentials are developed for analysis.

Abstract

We improve the description of $\mathbb{F}$-limits of noncollapsed Ricci flows in the K\"ahler setting. In particular, the singular strata $\mathcal{S}^k$ of such metric flows satisfy $\mathcal{S}^{2j}=\mathcal{S}^{2j+1}$. We also prove an analogous result for quantitative strata, and show that any tangent flow admits a nontrivial one-parameter action by isometries, which is locally free on the cone link in the static case. The main results are established using parabolic regularizations of conjugate heat kernel potential functions based at almost-selfsimilar points, which may be of independent interest.