Geometry/Differential geometry
Simply connected open 3-manifolds with slow decay of positive scalar curvature
Comptes Rendus. Mathématique, Volume 357 (2019) no. 3, pp. 284-290.

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 R3. Furthermore, using this result, we prove that any open simply connected 3-manifold M with π2(M)=Z and a complete metric as above is diffeomorphic to S2×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 R3. De plus, en utilisant ce résultat, nous montrons que toute 3-variété ouverte simplement connexe M telle que π2(M)=Z, munie d'une métrique complète comme celle ci-dessus, est difféomorphe à S2×R.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2019.02.001
Wang, Jian 1

1 Université Grenoble Alpes, Institut Fourier, 100, rue des Maths, 38610 Gières, France
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Wang, Jian. 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. http://www.numdam.org/articles/10.1016/j.crma.2019.02.001/

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