Stability of cigar soliton

TL;DR

This paper investigates the global behavior of Ricci flow with finite width on R^2 using Kähler geometry, aiming to identify conditions under which the flow converges to the cigar soliton metric.

Contribution

It provides a sufficient condition for the Ricci flow's limiting metric to be the cigar soliton, extending understanding of flow convergence in Kähler geometry.

Findings

Identifies conditions for Ricci flow convergence to cigar soliton

Uses classification results to analyze limiting metrics

Connects finite width Ricci flow to known soliton solutions

Abstract

In this paper, we shall use the K\"ahler geometry formulation to study the global behavior of the Ricci flow on $R^2$. The geometric feature of our Ricci flow is that it has finite width. Our aim is to determine the limiting metric (which corresponds an eternal Ricci flow) obtained by L.F.Wu. We can use the classification result of Daskalopoulos-Sesum to give a sufficient condition such that the limiting metric of L.F.Wu is the metric of cigar soliton.