Higher systolic inequalities for 3-dimensional contact manifolds

Abbondandolo, Alberto; Lange, Christian; Mazzucchelli, Marco

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

]

Abbondandolo, Alberto ¹ ; Lange, Christian ² ; Mazzucchelli, Marco 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 :

Abbondandolo, Alberto ¹ ; Lange, Christian ² ; Mazzucchelli, Marco ³

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Abbondandolo, Alberto; Lange, Christian; Mazzucchelli, Marco. 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. http://www.numdam.org/articles/10.5802/jep.195/

Bibliographie
Cité par

[AB19] Abbondandolo, Alberto; Benedetti, Gabriele On the local systolic optimality of Zoll contact forms, 2019 | arXiv

[ABHSa18] Abbondandolo, Alberto; Bramham, Barney; Hryniewicz, Umberto L.; Salomão, Pedro A. S. Sharp systolic inequalities for Reeb flows on the three-sphere, Invent. Math., Volume 211 (2018) no. 2, pp. 687-778 | DOI | MR | Zbl

[ABHSa19] Abbondandolo, Alberto; Bramham, Barney; Hryniewicz, Umberto L.; Salomão, Pedro A. S. Contact forms with large systolic ratio in dimension three, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), Volume 19 (2019) no. 4, pp. 1561-1582 | MR | Zbl

[AG21] Albach, Bernhard; Geiges, Hansjörg Surfaces of section for Seifert fibrations, Arnold Math. J., Volume 7 (2021) no. 4, pp. 573-597 | DOI | MR | Zbl

[APB14] Álvarez Paiva, J. C.; Balacheff, F. Contact geometry and isosystolic inequalities, Geom. Funct. Anal., Volume 24 (2014) no. 2, pp. 648-669 | DOI | MR | Zbl

[Ban78] Banyaga, Augustin Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique, Comment. Math. Helv., Volume 53 (1978) no. 2, pp. 174-227 | DOI | Zbl

[Ban97] Banyaga, Augustin The structure of classical diffeomorphism groups, Mathematics and its Appl., 400, Kluwer Academic Publishers Group, Dordrecht, 1997 | DOI

[Bes78] Besse, Arthur L. Manifolds all of whose geodesics are closed, Ergeb. Math. Grenzgeb., 93, Springer-Verlag, Berlin-New York, 1978 | DOI | MR

[BK21] Benedetti, Gabriele; Kang, Jungsoo A local contact systolic inequality in dimension three, J. Eur. Math. Soc. (JEMS), Volume 23 (2021) no. 3, pp. 721-764 | DOI | MR | Zbl

[Bot80] Bottkol, Matthew Bifurcation of periodic orbits on manifolds and Hamiltonian systems, J. Differential Equations, Volume 37 (1980) no. 1, pp. 12-22 | DOI | MR | Zbl

[BP94] Bialy, Misha; Polterovich, Leonid Geodesics of Hofer’s metric on the group of Hamiltonian diffeomorphisms, Duke Math. J., Volume 76 (1994) no. 1, pp. 273-292 | DOI | MR | Zbl

[Cal70] Calabi, Eugenio On the group of automorphisms of a symplectic manifold, Problems in analysis, Princeton University Press, Princeton, NJ, 1970, pp. 1-26 | MR | Zbl

[CGM20] Cristofaro-Gardiner, Daniel; Mazzucchelli, Marco The action spectrum characterizes closed contact 3-manifolds all of whose Reeb orbits are closed, Comment. Math. Helv., Volume 95 (2020) no. 3, pp. 461-481 | DOI | MR | Zbl

[CK94] Croke, Christopher B.; Kleiner, Bruce Conjugacy and rigidity for manifolds with a parallel vector field, J. Differential Geom., Volume 39 (1994) no. 3, pp. 659-680 http://projecteuclid.org/euclid.jdg/1214455076 | MR | Zbl

[Eps72] Epstein, D. B. A. Periodic flows on three-manifolds, Ann. of Math. (2), Volume 95 (1972), pp. 66-82 | DOI | MR | Zbl

[Gei22] Geiges, Hansjörg What does a vector field know about volume?, J. Fixed Point Theory Appl., Volume 24 (2022) no. 2, 23, 26 pages | DOI | MR

[GG82] Gromoll, Detlef; Grove, Karsten On metrics on $S^{2}$ all of whose geodesics are closed, Invent. Math., Volume 65 (1981/82) no. 1, pp. 175-177 | DOI | MR | Zbl

[Gin87] Ginzburg, V. L. New generalizations of Poincaré’s geometric theorem, Funktsional. Anal. i Prilozhen., Volume 21 (1987) no. 2, pp. 16-22 | MR

[Gir20] Giroux, Emmanuel Ideal Liouville domains, a cool gadget, J. Symplectic Geom., Volume 18 (2020) no. 3, pp. 769-790 | DOI | MR | Zbl

[GL18] Geiges, Hansjörg; Lange, Christian Seifert fibrations of lens spaces, Abh. Math. Sem. Univ. Hamburg, Volume 88 (2018) no. 1, pp. 1-22 Correction: Ibid. 91 (2021), no. 1, p. 145–150 | DOI | MR | Zbl

[JN83] Jankins, Mark; Neumann, Walter D. Lectures on Seifert manifolds, Brandeis Lect. Notes, 2, Brandeis University, Waltham, MA, 1983

[Kat73] Katok, A. B. Ergodic perturbations of degenerate integrable Hamiltonian systems, Izv. Akad. Nauk SSSR Ser. Mat., Volume 37 (1973), pp. 539-576 | MR | Zbl

[KL21] Kegel, Marc; Lange, Christian A Boothby-Wang theorem for Besse contact manifolds, Arnold Math. J., Volume 7 (2021) no. 2, pp. 225-241 | DOI | MR | Zbl

[Lan20] Lange, Christian On metrics on 2-orbifolds all of whose geodesics are closed, J. reine angew. Math., Volume 758 (2020), pp. 67-94 | DOI | MR | Zbl

[LM95] Lalonde, François; McDuff, Dusa Hofer’s $L^{\infty}$ -geometry: energy and stability of Hamiltonian flows. I, II, Invent. Math., Volume 122 (1995) no. 1, p. 1-33, 35–69 Erratum: Ibid. 123 (1996), no. 3, p. 613 | DOI | MR | Zbl

[LM04] Lisca, Paolo; Matić, Gordana Transverse contact structures on Seifert 3-manifolds, Algebraic Geom. Topol., Volume 4 (2004), pp. 1125-1144 | DOI | MR | Zbl

[LS21] Lange, Christian; Soethe, Tobias Sharp systolic inequalities for rotationally symmetric $2$ -orbifolds, 2021 | arXiv

[MR20] Mazzucchelli, Marco; Radeschi, Marco On the structure of Besse convex contact spheres, 2020 | arXiv

[MS17] McDuff, Dusa; Salamon, Dietmar Introduction to symplectic topology, Oxford Graduate Texts in Math., Oxford University Press, Oxford, 2017 | DOI

[Orl72] Orlik, Peter Seifert manifolds, Lect. Notes in Math., 291, Springer-Verlag, Berlin-New York, 1972 | DOI

[RW17] Radeschi, Marco; Wilking, Burkhard On the Berger conjecture for manifolds all of whose geodesics are closed, Invent. Math., Volume 210 (2017) no. 3, pp. 911-962 | DOI | MR | Zbl

[Sağ21] Sağlam, Murat Contact forms with large systolic ratio in arbitrary dimensions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), Volume 22 (2021) no. 3, pp. 1265-1308 | MR | Zbl

[Sul78] Sullivan, Dennis A foliation of geodesics is characterized by having no ‘tangent homologies’, J. Pure Appl. Algebra, Volume 13 (1978) no. 1, pp. 101-104 | DOI | MR | Zbl

[Tau07] Taubes, Clifford Henry The Seiberg-Witten equations and the Weinstein conjecture, Geom. Topol., Volume 11 (2007), pp. 2117-2202 | DOI | MR | Zbl

[Wad75] Wadsley, A. W. Geodesic foliations by circles, J. Differential Geom., Volume 10 (1975) no. 4, pp. 541-549 http://projecteuclid.org/euclid.jdg/1214433160 | MR | Zbl

[Wei73] Weinstein, Alan Normal modes for nonlinear Hamiltonian systems, Invent. Math., Volume 20 (1973), pp. 47-57 | DOI | MR | Zbl

[Zil83] Ziller, Wolfgang Geometry of the Katok examples, Ergodic Theory Dynam. Systems, Volume 3 (1983) no. 1, pp. 135-157 | DOI | MR | Zbl

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