Higher systolic inequalities for 3-dimensional contact manifolds

Alberto Abbondandolo; Christian Lange; Marco Mazzucchelli

doi:10.5802/jep.195

Higher systolic inequalities for 3-dimensional contact manifolds
[Inégalités systoliques d’ordre supérieur pour les variétés de contact de dimension

3

]

Alberto Abbondandolo ¹ ; Christian Lange ² ; Marco Mazzucchelli ³

¹ Ruhr Universität Bochum, Fakultät für Mathematik Gebäude IB 3/65, D-44801 Bochum, Germany
² Ludwig-Maximilians-Universität München, Mathematisches Institut Theresienstraße 39, D-80333 Munich, Germany
³ CNRS, UMPA, École Normale Supérieure de Lyon 46 allée d’Italie, 69364 Lyon, France

Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 807-851.

Résumé
Abstract

Une forme de contact est dite de type Besse si son flot de Reeb est périodique. On prouve que les formes de contact de type Besse sur les variétés connexes fermées de dimension $3$ sont les maximiseurs locaux de certains rapports systoliques d’ordre supérieur. Notre résultat étend des théorèmes antérieurs pour les formes de contact de type Zoll, c’est-à-dire les formes de contact dont le flot de Reeb définit une action libre du cercle.

A contact form is called Besse when the associated Reeb flow is periodic. We prove that Besse contact forms on closed connected $3$ -manifolds are the local maximizers of suitable higher systolic ratios. Our result extends earlier ones for Zoll contact forms, that is, contact forms whose Reeb flow defines a free circle action.

Reçu le : 2021-09-03
Accepté le : 2022-05-05
Publié le : 2022-05-16

DOI : 10.5802/jep.195

Classification : 53D10
Keywords: Systolic inequalities, Besse contact forms, Seifert fibrations, Calabi homomorphism
Mot clés : Inégalités systoliques, formes de contact de type Besse, fibrations de Seifert, homomorphisme de Calabi

Affiliations des auteurs :

Alberto Abbondandolo ¹ ; Christian Lange ² ; Marco Mazzucchelli ³

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Higher systolic inequalities for 3-dimensional~contact manifolds},
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Alberto Abbondandolo; Christian Lange; Marco Mazzucchelli. Higher systolic inequalities for 3-dimensional contact manifolds. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 807-851. doi : 10.5802/jep.195. https://jep.centre-mersenne.org/articles/10.5802/jep.195/

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