3-Manifolds with Yamabe invariant greater than that of $\RP^3$

TL;DR

This paper classifies all closed 3-manifolds with Yamabe invariant exceeding that of real projective space, showing they are either the 3-sphere or specific connected sums involving $S^2 imes S^1$ and its nonorientable bundle.

Contribution

It completes the classification of 3-manifolds with Yamabe invariant above a threshold, extending previous work by incorporating Aubin's Lemma and advanced geometric analysis techniques.

Findings

Manifolds are either $S^3$ or connected sums of $S^2 imes S^1$ and its nonorientable bundle.

Uses Aubin's Lemma to compare Yamabe constants across finite coverings.

Employs inverse mean curvature flow and Green's function analysis for conformal Laplacians.

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.