Optimal integral pinching results

Bour, Vincent; Carron, Gilles

doi:10.24033/asens.2238

Optimal integral pinching results
[Résultats de pincement intégral optimaux]

Bour, Vincent ; Carron, Gilles

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 1, pp. 41-70.

Résumé
Abstract

La formule de Bochner-Weitzenböck implique qu'une variété riemannienne compacte dont l'opérateur de courbure est strictement positif a tous ses nombres de Betti triviaux. Nous obtenons un tel résultat d'annulation sous des hypothèses de pincement intégral sur la courbure. Nos résultats sont optimaux et nous analysons les cas d'égalités. Il s'agit d'une extension à la dimension supérieure d'un résultat de M. Gursky.

In this article, we generalize the classical Bochner-Weitzenböck theorem for manifolds satisfying an integral pinching on the curvature. We obtain the vanishing of Betti numbers under integral pinching assumptions on the curvature, and characterize the equality case. In particular, we reprove and extend to higher degrees and higher dimensions a number of integral pinching results obtained by M. Gursky for four-dimensional closed manifolds.

Publié le : 2015-09-24

MR Zbl

DOI : 10.24033/asens.2238

Classification : 53C21; 58E35, 53C24.
Keywords: Yamabe invariant, curvature pinching, Betti number, harmonic forms.
Mot clés : Invariant de Yamabe, pincement de la courbure, nombre de Betti, formes harmoniques.

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     author = {Bour, Vincent and Carron, Gilles},
     title = {Optimal integral pinching results},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {41--70},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 48},
     number = {1},
     year = {2015},
     doi = {10.24033/asens.2238},
     mrnumber = {3335838},
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}

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Bour, Vincent; Carron, Gilles. Optimal integral pinching results. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 1, pp. 41-70. doi : 10.24033/asens.2238. http://www.numdam.org/articles/10.24033/asens.2238/

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