Non-collapsed finite time singularities of the Ricci flow on compact K\"ahler surfaces are of Type I

TL;DR

This paper proves that non-collapsed finite time singularities of Ricci flow on compact Kähler surfaces are always of Type I, and they are modeled on a specific shrinking Ricci soliton, clarifying the nature of such singularities.

Contribution

The paper establishes that all non-collapsed finite time singularities in this setting are of Type I, linking them to a known Ricci soliton model.

Findings

All non-collapsed finite time singularities are of Type I.

Such singularities are modeled on the Feldman-Ilmanen-Knopf shrinking Ricci soliton.

The result builds on previous work to classify singularity types.

Abstract

We show that any non-collapsed finite time singularity of the Ricci flow on a compact K\"ahler surface is of Type I. Combined with a previous result of the first author, Cifarelli, and Deruelle, it follows that any such singularity is modeled on the shrinking Ricci soliton of Feldman-Ilmanen-Knopf on the total space of the line bundle $\mathcal{O}_{\mathbb{P}^1}(-1)\to\mathbb{P}^{1}$.