$SO(2)\times SO(3)$-invariant Ricci solitons and ancient flows on $\mathbb{S}^4$

TL;DR

This paper proves that all $SO(2)\times SO(3)$-invariant Ricci solitons on the 4-sphere are likely round, by establishing curvature bounds and analyzing ancient Ricci flow solutions.

Contribution

It establishes uniform bounds for invariant Ricci solitons on $\mathbb{S}^4$, providing evidence that the only such solitons are the round sphere.

Findings

Bounded Riemann curvature and volume for invariant Ricci solitons

Injectivity radius bounded below by a universal constant

Identification of the 'pancake' ancient Ricci flow solution

Abstract

Consider the standard action of $SO(2)\times SO(3)$ on $\mathbb{R}^5=\mathbb{R}^2\oplus \mathbb{R}^3$. We establish the existence of a uniform constant $\mathcal{C}>0$ so that any $SO(2)\times SO(3)$-invariant Ricci soliton on $\mathbb{S}^4\subset \mathbb{R}^5$ with Einstein constant $1$ must have Riemann curvature and volume bounded by $\mathcal{C}$, and injectivity radius bounded below by $\frac{1}{\mathcal{C}}$. This observation, coupled with basic numerics, gives strong evidence to suggest that the only $SO(2)\times SO(3)$-invariant Ricci solitons on $\mathbb{S}^4$ are round. We also encounter the so-called `pancake' ancient solution of the Ricci flow.