SO(2)xSO(3)$SO(2)\times SO(3)$-invariant Ricci solitons and ancient flows on S4$\mathbb {S}∧4$

Consider the standard action of |SO(2) x SO(3) on R-5 = R-2 circle plus R-3. We establish the existence of a uniform constant C > 0 so that any SO(2) x SO(3)-invariant Ricci soliton on S-4 subset of R-5 with Einstein constant 1 must have Riemann curvature and volume bounded by C, and injectivity radius bounded below by 1/c. This observation, coupled with basic numerics, gives strong evidence to suggest that the only SO(2) x SO(3)-invariant Ricci solitons on S-4 are round. We also prove existence of the socalled 'pancake' ancient solution of the Ricci flow.