3-Manifolds with Yamabe invariant greater than that of $\RP^3$
Kazuo Akutagawa, André Neves
TL;DR
We classify all closed 3-manifolds with Yamabe invariant greater than that of $\mathbb{RP}^3$, showing they are either $S^3$ or a finite connected sum of copies of $S^2×S^1$ and the nonorientable $S^2$-bundle over $S^1$. The argument combines Aubin's lemma with inverse mean curvature flow and an analysis of Green's functions for the conformal Laplacian on finite and normal infinite coverings to construct effective test functions on coverings and compare Yamabe constants. The approach uses conformal deformations to Yamabe metrics, asymptotically flat manifolds via Green's functions, Hawking mass monotonicity under IMCF, and Kobayashi-type inequalities to bound $Y(M)$ by the model value $Y_2$, yielding $Y(M) \le Y_2$ and the desired classification. This completes the program and clarifies how scalar curvature invariants govern the topology of closed 3-manifolds, with implications for geometric analysis on 3-manifolds.
Abstract
We complete the classification (started by Bray and the second author) of all closed 3-manifolds with Yamabe invariant greater than that of $\RP^3$, by showing that such manifolds are either $S^3$ or finite connected sums $# m(S^2 \times S^1) # n(S^2 \tilde{\times} S^1)$ for $m + n \geq 1$, where $S^2 \tilde{\times} S^1$ is the nonorientable $S^2$-bundle over $S^1$. A key ingredient is Aubin's Lemma, which says that if the Yamabe constant is positive, then it is strictly less than the Yamabe constant of any of its non-trivial finite conformal coverings. This lemma, combined with inverse mean curvature flow and with analysis of the Green's functions for the conformal Laplacians on specific finite and normal infinite Riemannian coverings, will allow us to construct a family of nice test functions on the finite coverings and thus prove the desired result.