# Rotationally symmetric Ricci flow on $\mathbb{R}^{n+1}$

## TL;DR

This paper investigates the behavior of rotationally symmetric Ricci flows on Euclidean space, proving singularity types and convergence to solitons under various initial conditions, and confirming a conjecture by Chow and Tian.

## Contribution

It provides new results on singularity formation and long-term behavior of Ricci flow with rotational symmetry, including proofs of conjectures and classification of solutions.

## Key findings

- Flow develops Type-II singularity and converges to Bryant soliton for cylindrical asymptotics.
- Flow is immortal if curvature decays at infinity.
- Flow encounters Type-I singularity with sufficiently pinched necks.

## Abstract

We study the Ricci flow on $\mathbb{R}^{n+1}$, with $n\geq 2$, starting at some complete bounded curvature rotationally symmetric metric $g_{0}$. We first focus on the case where $(\mathbb{R}^{n+1},g_{0})$ does not contain minimal hyperspheres; we prove that if $g_{0}$ is asymptotic to a cylinder then the solution develops a Type-II singularity and converges to the Bryant soliton, while if the curvature of $g_{0}$ decays at infinity, then the solution is immortal. As a corollary, we prove a conjecture by Chow and Tian about Perelman's standard solutions. We then consider a class of asymptotically flat initial data $(\mathbb{R}^{n+1},g_{0})$ containing a neck and we prove that if the neck is sufficiently pinched, in a precise way, the Ricci flow encounters a Type-I singularity.