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On the Minimal Number of Periodic Reeb Orbits on a Contact Manifold Jean Gutt

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On the Minimal Number of Periodic Reeb Orbits on a Contact Manifold Jean Gutt On the minimal number of periodic Reeb orbits on a contact manifold Jean Gutt To cite this version: Jean Gutt. On the minimal number of periodic Reeb orbits on a contact manifold. General Mathemat- ics [math.GM]. Université de Strasbourg, 2014. English. NNT : 2014STRAD018. tel-01016954v2 HAL Id: tel-01016954 https://tel.archives-ouvertes.fr/tel-01016954v2 Submitted on 16 Sep 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Th ese` de doctorat Discipline: Math´ematiques pr´esent´ee par Jean Gutt Th`ese pr´esent´ee en vue de l’obtention des grades de Docteur de l’Universit´ede Strasbourg & de Docteur en Sciences de l’Universit´elibre de Bruxelles On the minimal number of periodic Reeb orbits on a contact manifold dirig´ee par Fr´ed´eric Bourgeois & Alexandru Oancea Soutenue le 27 juin 2014 devant le jury compos´ede : M. Albers Peter Universit¨atM¨unster Rapporteur Mme Bertelson M´elanie Universit´elibre de Bruxelles Examinateur M. Bourgeois Fr´ed´eric Universit´elibre de Bruxelles Directeur de th`ese & Universit´eParis-Sud M. Colin Vincent Universit´edeNantes Rapporteur M. Damian Mihai Universit´edeStrasbourg Examinateur M. Fine Joel Universit´elibredeBruxelles Pr´esidentduJury M. Oancea Alexandru Universit´ePierre et Marie Curie Directeur de th`ese M. Opshtein Emmanuel Universit´ede Strasbourg Examinateur Acknowledgements First of all I would like to thank my supervisors Fr´ed´eric Bourgeois and Alexandru Oancea for their great guidance which helped shaping this thesis, for their patience when I went astray in my endeavour and for their encouragements when I got completely lost. Without their enlightening remarks and helpful discussions the task of writing this thesis would have been far more difficult. I am greatly indebted to Peter Albers and Vincent Colin who accepted to act as rap- porteur for this thesis. I am very grateful to M´elanie Bertelson, Mihai Damian, Joel Fine and Emmanuel Opshtein who accepted to be members of the jury. I also thank the former and current members of the mathematics departments of the Universit´eLibre de Bruxelles, the Universit´ede Strasbourg and the Universit´ePierre et Marie Curie for creating such a friendly, welcoming and positive environment. My work greatly benefited from discussions with Strom Borman, Samuel Lisi, Michael Hutchings, Jungsoo Kang, Yiming Long, Maksim Maydanskiy, Vinicius Ramos and Alexan- der Ritter. I am grateful to the Foundation for Scientific Research (FNRS-FRS) for its support and to the Coll`ege Doctoral Europ´een for providing me lodging in Strasbourg. I acknowledge partial support from the ERC via the grant StG-259118-STEIN, from the ERC via the grant StG-239781-ContactMath, from an ARC of the Communaut´efran¸caise de Belgique and from the Pˆole d’Attraction Interuniversitaire Dygest. This thesis would not have been fun without all the friends that supported me during these 4 years. Thanks to the persons who shared an office with me (and congratulations for not killing me for my stupid jokes), thanks also to the PhD students of Bruxelles, Strasbourg and Paris for their friendly support. Warm thanks to Germain for all the “game evenings” , and to Julian, Arthur, Gus, Arno, Ann, Carine. The caring and loving environment provided by my family was wonderful; special thanks to my parents, to my sister and to my grandmother who were always there for me. I would also like to thank my parents in law for everything they did for me. Last but not least, I thank from the bottom of my heart my beloved wife for her presence and patience. Contents 1 Background on symplectic homology 23 1.1 Floer Homology . 23 1.1.1 Parenthesis : A glimpse on signs . 29 1.2 Symplectic Homology . 32 1.2.1 Setup ................................... 33 1.2.2 Symplectic homology . 34 1.2.3 Parenthesis on the homotpy of homotopies Theorem . 38 1.2.4 Positive symplectic homology . 40 1.3 S1-equivariant symplectic homology . 43 1.3.1 S1-equivariant homology . 43 1.3.2 S1-equivariant symplectic homology . 44 1.3.3 Positive S1-equivariant symplectic homology . 51 1.4 An alternative presentation of the S1-equivariant symplectic homology . 53 1 2 SH S ,+ and periodic Reeb orbits 59 2.1 The multicomplex defining positive S1-equivariant homology . 59 2.1.1 Construction of admissible Hamiltonians from elements in . 60 Hstd 2.1.2 The complex for a subclass of special Hamiltonians . 60 2.1.3 Perturbation of Morse-Bott Hamiltonians . 62 1 2.2 Computing SH S ,+ ................................ 63 2.2.1 The example of Brieskorn spheres . 68 2.2.2 Homology with non contractible orbits . 69 3 Structural properties of symplectic homology 71 3.1 Transfermorphism ................................ 71 3.1.1 Transfer morphism for S1-equivariant symplectic homology . 81 3.2 Invariance of symplectic homology . 83 3.2.1 Invariance of the homology of contact fillings . 90 4 OntheminimalnumberofperiodicReeborbits 93 4.1 Minimal number of periodic Reeb orbits on a hypersurface in R2n . 93 6 Contents 4.2 Extension of the definitions . 96 4.2.1 Positive symplectic homology . 96 4.2.2 Positive S1-equivariant symplectic homology . 100 4.3 Reeb orbits on hypersurfaces in negative line bundles . 100 5 Normal forms for symplectic matrices 107 5.1 Preliminaries ................................... 112 5.1.1 A few technical lemmas . 114 1 5.2 Normal forms for A V when λ / S . ...................... 117 | [λ] ∈ 5.3 Normal forms for A V when λ = 1. ..................... 120 | [λ] ± 5.3.1 If p = 2 k 1isodd ........................... 120 − 5.3.2 If p = 2 k iseven ............................. 121 iφ 1 5.4 Normal forms for A V when λ = e S 1 . 124 | [λ] ∈ \ {± } 5.4.1 If p = 2 k 1isodd ........................... 125 − 5.4.2 If p = 2 k iseven ............................. 127 6 Generalised Conley-Zehnder index 135 6.1 Introduction.................................... 135 6.2 TheConley-Zehnderindex. 138 6.3 The Robbin-Salamon index for a path of Lagrangians . 143 6.4 The Robbin-Salamon index for a path of symplectic matrices . 145 6.4.1 Generalized Conley-Zehnder index . 145 6.4.2 Another index defined by Robbin and Salamon . 149 6.5 Characterization of the Robbin-Salamon index . 151 6.6 A formula for the Robbin-Salamon index . 157 Bibliography 159 7 Prolegomenon As the title indicates, periodic Reeb orbits are at the centre of this work. The general framework is contact geometry. A contact structure on a manifold M of dimension 2 n 1 − is a hyperplane field ξ which is maximally non integrable; i.e if we write locally ξ = ker α where α is a differential 1-form, then α (dα )n 1 = 0 everywhere. If α is globally defined, ∧ − 6 we say that ξ is coorientable and we call α a contact form . In this thesis we consider only coorientable contact structures. The 1-form α is not unique; for any function f : M R, f → the 1-form e α defines the same contact structure. The Reeb vector field Rα associated to a contact form α is the unique vector field on M characterized by: ι(Rα)dα = 0 and α(Rα) = 1. Since this vector field does not vanish anywhere, there are no fixed points of its flow. Periodic orbits are thus the most noticeable objects in the flow. Poincar´epointed out their interest in his “trait´ede la m´ecanique c´eleste”: “Ce qui nous rend ces solutions p´eriodiques si pr´ecieuses, c’est qu’elles sont, pour ainsi dire, la seule br`eche par o`unous puissions essayer de p´en´etrer dans une place jusqu’ici r´eput´ee inabordable.” Does there exist a periodic Reeb orbit for any contact form on any manifold? 2n 1 The answer to this question is negative as shown by the following example. Let R − be endowed with the “standard” contact form n 1 − α = dz xidy i. − i X=1 The Reeb vector field is given by Rα = ∂z, and there are no periodic Reeb orbits. The same question for a compact oriented manifold endowed with a contact 1-form is still open. It is called the Weinstein conjecture. Some partial results are known. Taubes answered positively this question for manifolds in dimension 3. Theorem 0.0.1 ([Tau07]) If Y is a closed oriented three-manifold with a contact form, then the associated Reeb vector field has a closed orbit. In higher dimensions, only partial results have been obtained. Let us mention a first result due to Viterbo: 1 Contents Theorem 0.0.2 ([Vit87]) If Σ (R2n, ω ) is a compact hypersurface of “contact type”, ⊂ 0 then Σ has at least one periodic Reeb orbit. More recently Albers and Hofer proved: Theorem 0.0.3 ([AH09]) Let (M, ξ ) be a closed “PS-overtwisted 1” contact manifold. Then the Reeb vector field associated to any contact form α inducing ξ has a contractible periodic orbit. One could also quote many particular results related for example to Hamiltonian dynamics. Another natural question is to ask how many periodic Reeb orbits can exist. Since any periodic orbit can be iterated any number of times, it is more reasonable to ask how 2n 1 many geometrically distinct periodic orbits can exist.
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