1. The Weinstein conjecture in the presence of submanifolds having a Legendrian foliation
- Author
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Ana Rechtman, Klaus Niederkrüger, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Unité de Mathématiques Pures et Appliquées (UMPA-ENSL), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Lyon (ENS Lyon), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), and École normale supérieure de Lyon (ENS de Lyon)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Pure mathematics ,010102 general mathematics ,Holomorphic function ,Weinstein conjecture ,Dynamical Systems (math.DS) ,01 natural sciences ,Contractible space ,Connected sum ,Foliation ,[MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG] ,010104 statistics & probability ,34J45, 53D10, 53D35 ,Mathematics - Symplectic Geometry ,Reeb vector field ,FOS: Mathematics ,Symplectic Geometry (math.SG) ,Geometry and Topology ,Mathematics - Dynamical Systems ,0101 mathematics ,Orbit (control theory) ,Mathematics::Symplectic Geometry ,Analysis ,Real projective space ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
Helmut Hofer introduced in '93 a novel technique based on holomorphic curves to prove the Weinstein conjecture. Among the cases where these methods apply are all contact 3--manifolds $(M,\xi)$ with $\pi_2(M) \ne 0$. We modify Hofer's argument to prove the Weinstein conjecture for some examples of higher dimensional contact manifolds. In particular, we are able to show that the connected sum with a real projective space always has a closed contractible Reeb orbit., Comment: 11 pages, 2 figures
- Published
- 2011
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