3-manifolds with Yamabe invariant greater than that of $\Bbb{RP}\sp 3$

Abstract

We show that, for all nonnegative integers $k, l, m$ and $ n$, the Yamabe invariant of $$ \#(\mathbb{RP}^3)\# \ell(\mathbb{RP}^2 \#m(S^2 \times S^1)\#n(S^2 \tilde \times S^1)$$ is equal to the Yamabe invariant of $\mathbb{RP}^3$, provided $k + \ell \geq 1$. We then complete the classification (started by Bray and the second author) of all closed 3-manifolds with Yamabe invariant greater than that of $\mathbb{RP}^3$. More precisely, we show that such maniforlds are either $S^3$ or finite connected sums $ \# m (S^2 \times S^1) \# n (S^2 \tilde \times S^1)$, where $S^2 \tilde \times S^1$ is the nonorientable $S^2$-bundle over $S^1$.

A key ingredient is Aubin’s Lemma [3], 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 function for the conformal Laplacians on specific finite and normal infinite Riemannian coverings, will al- low us to construct a family of nice test functions on the finite coverings and thus prove the desired result.