## Journal of Differential Geometry

- J. Differential Geom.
- Volume 75, Number 3 (2007), 359-386.

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

K. Akutagawa and A. Neves

#### 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.

#### Article information

**Source**

J. Differential Geom., Volume 75, Number 3 (2007), 359-386.

**Dates**

First available in Project Euclid: 30 March 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.jdg/1175266277

**Digital Object Identifier**

doi:10.4310/jdg/1175266277

**Mathematical Reviews number (MathSciNet)**

MR2301449

**Zentralblatt MATH identifier**

1119.53027

#### Citation

Akutagawa, K.; Neves, A. 3-manifolds with Yamabe invariant greater than that of $\Bbb{RP}\sp 3$. J. Differential Geom. 75 (2007), no. 3, 359--386. doi:10.4310/jdg/1175266277. https://projecteuclid.org/euclid.jdg/1175266277