SO(2)×SO(3)$SO(2)\times SO(3)$‐invariant Ricci solitons and ancient flows on S4$\mathbb {S}^4$

Abstract

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