Classification of Prime 3-Manifolds with Yamabe Invariant Greater than${\Bbb RP}^{3}

In this paper we compute the σ-invariants (sometimes also called the smooth Yamabe invariants) of${\Bbb RP}^{3}$and${\Bbb RP}^{2}\times S^{1}$(which are equal) and show that the only prime 3-manifolds with larger σ-invariants are S3, S2× S1, and$S^{2}\tilde{\times }S^{1}$(the nonorientable S2bundle over S1). More generally, we show that any 3-manifold with σ-invariant greater than${\Bbb RP}^{3}$is either S3, a connect sum with an S2bundle over S1, or has more than one nonorientable prime component. A corollary is the Poincaré conjecture for 3-manifolds with σ-invariant greater than${\Bbb RP}^{3}$. Surprisingly these results follow from the same inverse mean curvature flow techniques which were used by Huisken and Ilmanen in [7] to prove the Riemannian Penrose Inequality for a black hole in a spacetime. Richard Schoen made the observation [18] that since the constant curvature metric (which is extremal for the Yamabe problem) on${\Bbb RP}^{3}$is in the same conformal class as the Schwarzschild metric (which is extremal for the Penrose inequality) on${\Bbb RP}^{3}$minus a point, there might be a connection between the two problems. The authors found a strong connection via inverse mean curvature flow.