Simply connected open 3-manifolds with slow decay of positive scalar curvature

Wang, Jian

doi:10.1016/j.crma.2019.02.001

Geometry/Differential geometry

Simply connected open 3-manifolds with slow decay of positive scalar curvature
[3-Variétés simplement connexes ouvertes dont la courbure scalaire est positive avec une décroissance lente à l'infini]

Présenté par : the Editorial Board

Wang, Jian ¹

¹ Université Grenoble Alpes, Institut Fourier, 100, rue des Maths, 38610 Gières, France

Comptes Rendus. Mathématique, Tome 357 (2019) no. 3, pp. 284-290.

Résumé
Abstract

Le but de cet article est d'étudier la structure topologique de 3-variétés simplement connexes ouvertes dont la courbure scalaire présente une décroissance lente à l'infini. En particulier, nous montrons que la variété de Whitehead n'admet pas de métrique complète dont la courbure scalaire décroît lentement, et qu'en fait toute 3-variété contractible complète avec une telle métrique est difféomorphe à $R^{3}$ . De plus, en utilisant ce résultat, nous montrons que toute 3-variété ouverte simplement connexe M telle que $π_{2} (M) = Z$ , munie d'une métrique complète comme celle ci-dessus, est difféomorphe à $S^{2} \times R$ .

The goal of this paper is to investigate the topological structure of open simply connected 3-manifolds whose scalar curvature has a slow decay at infinity. In particular, we show that the Whitehead manifold does not admit a complete metric whose scalar curvature decays slowly, and in fact that any contractible complete 3-manifolds with such a metric is diffeomorphic to $R^{3}$ . Furthermore, using this result, we prove that any open simply connected 3-manifold M with $π_{2} (M) = Z$ and a complete metric as above is diffeomorphic to $S^{2} \times R$ .

Reçu le : 2018-06-28
Accepté le : 2019-02-07
Publié le : 2019-02-21

DOI : 10.1016/j.crma.2019.02.001

Affiliations des auteurs :

Wang, Jian ¹

¹ Université Grenoble Alpes, Institut Fourier, 100, rue des Maths, 38610 Gières, France

@article{CRMATH_2019__357_3_284_0,
     author = {Wang, Jian},
     title = {Simply connected open 3-manifolds with slow decay of positive scalar curvature},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {284--290},
     publisher = {Elsevier},
     volume = {357},
     number = {3},
     year = {2019},
     doi = {10.1016/j.crma.2019.02.001},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2019.02.001/}
}

TY  - JOUR
AU  - Wang, Jian
TI  - Simply connected open 3-manifolds with slow decay of positive scalar curvature
JO  - Comptes Rendus. Mathématique
PY  - 2019
SP  - 284
EP  - 290
VL  - 357
IS  - 3
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2019.02.001/
DO  - 10.1016/j.crma.2019.02.001
LA  - en
ID  - CRMATH_2019__357_3_284_0
ER  -

%0 Journal Article
%A Wang, Jian
%T Simply connected open 3-manifolds with slow decay of positive scalar curvature
%J Comptes Rendus. Mathématique
%D 2019
%P 284-290
%V 357
%N 3
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2019.02.001/
%R 10.1016/j.crma.2019.02.001
%G en
%F CRMATH_2019__357_3_284_0

Wang, Jian. Simply connected open 3-manifolds with slow decay of positive scalar curvature. Comptes Rendus. Mathématique, Tome 357 (2019) no. 3, pp. 284-290. doi : 10.1016/j.crma.2019.02.001. http://www.numdam.org/articles/10.1016/j.crma.2019.02.001/

Bibliographie
Cité par

[1] Atiyah, M.; Singer, I. The index of elliptic operators on compact manifolds, Bull. Amer. Math. Soc., Volume 69 (1962), pp. 422-433

[2] Batude, J.L. Singularité générique des applications différentiables de la 2-sphère dans une 3-variété différentiable, Ann. Inst. Fourier (Grenoble), Volume 21 (1971) no. 3, pp. 151-172

[3] Chang, S.; Weinberger, S.; Yu, G. Taming 3-manifolds using scalar curvature, Geom. Dedic., Volume 148 (2010), pp. 3-14

[4] Douglas, J. Solution of the problem of Plateau, Trans. Amer. Math. Soc., Volume 32 (1932), pp. 731-756

[5] Gromov, M.; Lawson, H. The positive scalar curvature and the Dirac operator on the complete Riemannian manifolds, Publ. Math. IHÉS, Volume 58 (1984), pp. 83-196

[6] Gulliver, R.D. Regularity of minimizing surfaces of prescribed mean curvature, Ann. of Math. (2), Volume 97 (1973), pp. 275-305

[7] Liu, G. 3-Manifolds with nonnegative Ricci curvature, Invent. Math., Volume 193 (2013) no. 2, pp. 367-375

[8] Morrey, C.B. The problem of Plateau in a Riemannian manifold, Ann. of Math. (2), Volume 49 (1948), pp. 807-851

[9] Morrey, C.B. Multiple Integrals in the Calculus of Variations, Springer-Verlag, Berlin, 1966

[10] Osserman, R. A proof of the regularity everywhere of the classical solution to Plateau's problem, Ann. of Math. (2), Volume 91 (1970), pp. 550-569

[11] Papakyriakopoulos, C. On Dehn's lemma and asphericity of knots, Ann. of Math. (2), Volume 66 (1957) no. 1, pp. 1-26

[12] Perelman, G. The entropy formula for Ricci flow and its geometric applications | arXiv

[13] Perelman, G. Ricci flow with surgery on the three manifolds | arXiv

[14] Perelman, G. Finite extinction time for the solution to Ricci flow on certain three-manifolds | arXiv

[15] Rado, T. On Plateau's problem, Ann. of Math. (2), Volume 31 (1930), pp. 457-469

[16] Rosenberg, H. Constant mean curvature surface in homogenously regular 3-manifold, Bull. Austral. Math. Soc., Volume 74 (2006), pp. 227-238

[17] Schoen, R.; Yau, S.-T. On the proof of the positive mass conjecture in general relativity, Commun. Math. Phys., Volume 65 (1979), pp. 45-76

[18] Schoen, R.; Yau, S.-T. Proof of the positive mass theorem. II, Commun. Math. Phys., Volume 79 (1981), pp. 231-260

[19] Schoen, R.; Yau, S.-T. Complete three-dimensional manifold with positive Ricci curvature and scalar curvature, Seminar on Differential Geometry, Annals of Mathematics Studies, vol. 102, Princeton University Press, Princeton, NJ, USA, 1982, pp. 209-228

[20] Stallings, J. Group Theory and Three-Dimensional Manifolds, Yale University Press, New Haven, 1971

[21] Whitehead, J.H.C. A certain open manifold whose group is unity, Quart. J. Math. Oxford, Volume 6 (1935), pp. 268-279

Cité par Sources :