Classification of prime 3-manifolds with $\sigma$-invariant greater than $\Bbb{RP}^3$

Abstract

In this paper we compute the $\sigma$-invariants (sometimes also called the smooth Yamabe invariants) of $\mathbb{RP}^3$ and $\mathbb{RP}^2 \times S^1$ (which are equal) and show that the only prime $3$-manifolds with larger $\sigma$-invariants are $S^3$, $S^2 \times S^1$, and $S^2 \tilde\times S^1$ (the nonorientable $S^2$ bundle over $S^1$). More generally, we show that any $3$-manifold with $\sigma$-invariant greater than $\mathbb{RP}^3$ is either $S^3$, a connect sum with an $S^2$ bundle over $S^1$, or has more than one nonorientable prime component. A corollary is the Poincaré conjecture for $3$-manifolds with $\sigma$-invariant greater than $\mathbb{RP}^3$.

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 $\mathbb{RP}^3$ is in the same conformal class as the Schwarzschild metric (which is extremal for the Penrose inequality) on $\mathbb{RP}^3$ minus a point, there might be a connection between the two problems. The authors found a strong connection via inverse mean curvature flow.