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

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JEP_2022__9__807_0,
     author = {Alberto Abbondandolo and Christian Lange and Marco Mazzucchelli},
     title = {Higher systolic inequalities for 3-dimensional~contact manifolds},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {807--851},
     publisher = {\'Ecole polytechnique},
     volume = {9},
     year = {2022},
     doi = {10.5802/jep.195},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.195/}
}

TY  - JOUR
AU  - Alberto Abbondandolo
AU  - Christian Lange
AU  - Marco Mazzucchelli
TI  - Higher systolic inequalities for 3-dimensional contact manifolds
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2022
SP  - 807
EP  - 851
VL  - 9
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.195/
DO  - 10.5802/jep.195
LA  - en
ID  - JEP_2022__9__807_0
ER  -

%0 Journal Article
%A Alberto Abbondandolo
%A Christian Lange
%A Marco Mazzucchelli
%T Higher systolic inequalities for 3-dimensional contact manifolds
%J Journal de l’École polytechnique — Mathématiques
%D 2022
%P 807-851
%V 9
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.195/
%R 10.5802/jep.195
%G en
%F JEP_2022__9__807_0

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/

Bibliographie
Cité par

[AB19] A. Abbondandolo & G. Benedetti - “On the local systolic optimality of Zoll contact forms”, 2019 | arXiv

[ABHSa18] A. Abbondandolo, B. Bramham, U. L. Hryniewicz & P. A. S. Salomão - “Sharp systolic inequalities for Reeb flows on the three-sphere”, Invent. Math. 211 (2018) no. 2, p. 687-778 | DOI | MR | Zbl

[ABHSa19] A. Abbondandolo, B. Bramham, U. L. Hryniewicz & P. A. S. Salomão - “Contact forms with large systolic ratio in dimension three”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 19 (2019) no. 4, p. 1561-1582 | MR | Zbl

[AG21] B. Albach & H. Geiges - “Surfaces of section for Seifert fibrations”, Arnold Math. J. 7 (2021) no. 4, p. 573-597 | DOI | MR | Zbl

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

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

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

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

[BK21] G. Benedetti & J. Kang - “A local contact systolic inequality in dimension three”, J. Eur. Math. Soc. (JEMS) 23 (2021) no. 3, p. 721-764 | DOI | MR | Zbl

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

[BP94] M. Bialy & L. Polterovich - “Geodesics of Hofer’s metric on the group of Hamiltonian diffeomorphisms”, Duke Math. J. 76 (1994) no. 1, p. 273-292 | DOI | MR | Zbl

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

[CGM20] D. Cristofaro-Gardiner & M. Mazzucchelli - “The action spectrum characterizes closed contact 3-manifolds all of whose Reeb orbits are closed”, Comment. Math. Helv. 95 (2020) no. 3, p. 461-481 | DOI | MR | Zbl

[CK94] C. B. Croke & B. Kleiner - “Conjugacy and rigidity for manifolds with a parallel vector field”, J. Differential Geom. 39 (1994) no. 3, p. 659-680 | MR | Zbl

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

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

[GG82] D. Gromoll & K. Grove - “On metrics on $S^{2}$ all of whose geodesics are closed”, Invent. Math. 65 (1981/82) no. 1, p. 175-177 | DOI | MR | Zbl

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

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

[GL18] H. Geiges & C. Lange - “Seifert fibrations of lens spaces”, Abh. Math. Sem. Univ. Hamburg 88 (2018) no. 1, p. 1-22, Correction: Ibid. 91 (2021), no. 1, p. 145–150 | DOI | MR | Zbl

[JN83] M. Jankins & W. D. Neumann - Lectures on Seifert manifolds, Brandeis Lect. Notes, vol. 2, Brandeis University, Waltham, MA, 1983

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

[KL21] M. Kegel & C. Lange - “A Boothby-Wang theorem for Besse contact manifolds”, Arnold Math. J. 7 (2021) no. 2, p. 225-241 | DOI | MR | Zbl

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

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

[LM04] P. Lisca & G. Matić - “Transverse contact structures on Seifert 3-manifolds”, Algebraic Geom. Topol. 4 (2004), p. 1125-1144 | DOI | MR | Zbl

[LS21] C. Lange & T. Soethe - “Sharp systolic inequalities for rotationally symmetric $2$ -orbifolds”, 2021 | arXiv

[MR20] M. Mazzucchelli & M. Radeschi - “On the structure of Besse convex contact spheres”, 2020 | arXiv

[MS17] D. McDuff & D. Salamon - Introduction to symplectic topology, Oxford Graduate Texts in Math., Oxford University Press, Oxford, 2017 | DOI

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

[RW17] M. Radeschi & B. Wilking - “On the Berger conjecture for manifolds all of whose geodesics are closed”, Invent. Math. 210 (2017) no. 3, p. 911-962 | DOI | MR | Zbl

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

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

[Tau07] C. H. Taubes - “The Seiberg-Witten equations and the Weinstein conjecture”, Geom. Topol. 11 (2007), p. 2117-2202 | DOI | MR | Zbl

[Wad75] A. W. Wadsley - “Geodesic foliations by circles”, J. Differential Geom. 10 (1975) no. 4, p. 541-549 | MR | Zbl

[Wei73] A. Weinstein - “Normal modes for nonlinear Hamiltonian systems”, Invent. Math. 20 (1973), p. 47-57 | DOI | MR | Zbl

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

Cité par Sources :