On open manifolds admitting no complete metric with positive scalar curvature
Shi, Yuguang1; Wang, Jian2, 3; Wu, Runzhang1; Zhu, Jintian4
1Key Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing, 100871 (P. R. China)
2Department of Mathematics, Stony Brook University, 100 Nicolls Road, Stony Brook, NY 11794 (USA)
3Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 55 Zhongguancun East Road, Beijing 100190 (P. R. China)
4Institute for Theoretical Sciences, Westlake University, 600 Dunyu Road, 310030, Hangzhou, Zhejiang (P. R. China)
Annales de l'Institut Fourier, Online first, 35 p.

In this paper, we investigate the topological obstruction problem for positive scalar curvature and uniformly positive scalar curvature on open manifolds. We present a definition for open Schoen–Yau–Schick manifolds and prove that there is no complete metric with positive scalar curvature on these manifolds. Similarly, we define weak Schoen–Yau–Shick manifolds by analogy, which are expected to admit no complete metrics with uniformly positive scalar curvature.

Dans cet article, nous étudions le problème d’obstruction topologique lié à la courbure scalaire positive et à la courbure scalaire uniformément positive sur les variétés ouvertes. Nous proposons une définition des variétés ouvertes de type Schoen–Yau–Schick et démontrons qu’il n’existe aucune métrique complète possédant une courbure scalaire positive sur ces variétés. De manière analogue, nous définissons les variétés de type Schoen–Yau–Schick au sens faible, qui n’admettent aucune métrique complète avec une courbure scalaire uniformément positive.

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3762
Classification: 53C21
Keywords: Positive scalar curvature, open manifold, topological obstruction.
Mots-clés : Courbure scalaire positive, variétés ouvertes, obstruction topologique.

Shi, Yuguang  1 ; Wang, Jian  2 , 3 ; Wu, Runzhang  1 ; Zhu, Jintian  4

1 Key Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing, 100871 (P. R. China)
2 Department of Mathematics, Stony Brook University, 100 Nicolls Road, Stony Brook, NY 11794 (USA)
3 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 55 Zhongguancun East Road, Beijing 100190 (P. R. China)
4 Institute for Theoretical Sciences, Westlake University, 600 Dunyu Road, 310030, Hangzhou, Zhejiang (P. R. China) 
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Shi, Yuguang; Wang, Jian; Wu, Runzhang; Zhu, Jintian. On open manifolds admitting no complete metric with positive scalar curvature. Annales de l'Institut Fourier, Online first, 35 p.

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