In this paper we compute the σ-invariants (sometimes also called the smooth Yamabe invariants) of ℝℙ3 and ℝℙ2×S1 (which are equal) and show that the only prime 3-manifolds with larger σ-invariants are S3, S2×S1, and S2×~S1 (the nonorientable S2 bundle over S1). More generally, we show that any 3-manifold with σ-invariant greater than ℝℙ3 is either S3, a connect sum with an S2 bundle over S1, or has more than one nonorientable prime component. A corollary is the Poincaré conjecture for 3-manifolds with σ-invariant greater than ℝℙ3.
Surprisingly these results follow from the same inverse mean curvature flow techniques which were used by Huisken and Ilmanen in  to prove the Riemannian Penrose Inequality for a black hole in a spacetime. Richard Schoen made the observation  that since the constant curvature metric (which is extremal for the Yamabe problem) on ℝℙ3 is in the same conformal class as the Schwarzschild metric (which is extremal for the Penrose inequality) on ℝℙ3 minus a point, there might be a connection between the two problems. The authors found a strong connection via inverse mean curvature flow.