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


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.


Hubert L. Bray

Department of Mathematics, Columbia University, New York, NY 10027, United States

André Neves

Instituto Superior Tećnico, Lisbon, Portugal and Department of Mathematics, Stanford Universit, Stanford, CA 94305, United States