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

Kazuo Akutagawa Tokyo University of Science André Neves Princeton University

Differential Geometry mathscidoc:1609.10009

Journal of Differential Geometry, 75, (3), 359-386, 2007
We show that, for all nonnegative integers k, Г, m and n, theYamabe invariant of #k(RP3)#Г(RP2 】 S1)#m(S2 】 S1)#n(S2. 】S1) is equal to the Yamabe invariant of RP3, provided k + Г ∶ 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 RP3. More precisely, we show that such manifolds are either S3 or finite connected sums #m(S2 】 S1)#n(S2. 】S1),where S2. 】S1 is the nonorientable S2-bundle over S1. A key ingredient is Aubin’s Lemma [3], which says that if the Yamabe constant is positive, then it is strictly less than the Yam- abe 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 allow us to construct a family of nice test functions on the finite coverings and thus prove the desired result.
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@inproceedings{kazuo20073-manifolds,
  title={3-manifolds with Yamabe invariant greater than that of $\ Bbb {RP}\ sp 3$},
  author={Kazuo Akutagawa, and André Neves},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160908193451526765666},
  booktitle={Journal of Differential Geometry},
  volume={75},
  number={3},
  pages={359-386},
  year={2007},
}
Kazuo Akutagawa, and André Neves. 3-manifolds with Yamabe invariant greater than that of $\ Bbb {RP}\ sp 3$. 2007. Vol. 75. In Journal of Differential Geometry. pp.359-386. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160908193451526765666.
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