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

Jian Wang

doi:10.1016/j.crma.2019.02.001

Geometry/Differential geometry

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

Presented by: the Editorial Board

Jian Wang ¹

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

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

Abstract
Résumé

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 $S^{2} \times R$ .

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 $S^{2} \times R$ .

Received: 2018-06-28
Accepted: 2019-02-07
Published online: 2019-02-21

DOI: 10.1016/j.crma.2019.02.001

Author's affiliations:

Jian Wang ¹

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

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     title = {Simply connected open 3-manifolds with slow decay of positive scalar curvature},
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     pages = {284--290},
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     language = {en},
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Jian Wang. Simply connected open 3-manifolds with slow decay of positive scalar curvature. Comptes Rendus. Mathématique, Volume 357 (2019) no. 3, pp. 284-290. doi : 10.1016/j.crma.2019.02.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.02.001/

References
Cited by

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[19] R. Schoen; S.-T. Yau 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] J. Stallings Group Theory and Three-Dimensional Manifolds, Yale University Press, New Haven, 1971

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

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