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

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.

20 pages, 4 figures