The singular Weinstein conjecture

In this article, we investigate Reeb dynamics on bm-contact manifolds, previously introduced in [37], which are contact away from a hypersurface Z but satisfy certain transversality conditions on Z. The study of these contact structures is motivated by that of contact manifolds with boundary. The se...

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Detalles Bibliográficos
Autores: Miranda, E., Oms, C.
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2021
País:España
Institución:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
Repositorio:Recercat. Dipósit de la Recerca de Catalunya
OAI Identifier:oai:recercat.cat:2072/531265
Acceso en línea:http://hdl.handle.net/2072/531265
Access Level:acceso abierto
Palabra clave:b-Contact forms
b-Symplectic geometry
Overtwisted contact structures
Periodic orbits
Reeb dynamics
Singular Weinstein conjecture
51
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spelling The singular Weinstein conjectureMiranda, E.Oms, C.b-Contact formsb-Symplectic geometryOvertwisted contact structuresPeriodic orbitsReeb dynamicsSingular Weinstein conjecture51In this article, we investigate Reeb dynamics on bm-contact manifolds, previously introduced in [37], which are contact away from a hypersurface Z but satisfy certain transversality conditions on Z. The study of these contact structures is motivated by that of contact manifolds with boundary. The search of periodic Reeb orbits on those manifolds thereby starts with a generalization of the well-known Weinstein conjecture. Contrary to the initial expectations, examples of compact bm-contact manifolds without periodic Reeb orbits outside Z are provided. Furthermore, we prove that in dimension 3, there are always infinitely many periodic orbits on the critical set if it is compact. We prove that traps for the bm-Reeb flow exist in any dimension. This investigation goes hand-in-hand with the Weinstein conjecture on non-compact manifolds having compact ends of convex type. In particular, we extend Hofer's arguments to open overtwisted contact manifolds that are R+-invariant in the open ends, obtaining as a corollary the existence of periodic bm-Reeb orbits away from the critical set. The study of bm-Reeb dynamics is motivated by well-known problems in fluid dynamics and celestial mechanics, where those geometric structures naturally appear. In particular, we prove that the dynamics on positive energy level-sets in the restricted planar circular three body problem are described by the Reeb vector field of a b3-contact form that admits an infinite number of periodic orbits at the critical set. © 2021 The Author(s)11251151, PID2019-103849GB-I00; National Science Foundation, NSF: DMS-1440140; Agence Nationale de la Recherche, ANR: ANR-10-LABX-0098; Fonds National de la Recherche Luxembourg, FNR; Agència de Gestió d'Ajuts Universitaris i de Recerca, AGAUR; Institució Catalana de Recerca i Estudis Avançats, ICREA; Agencia Estatal de Investigación, AEI: 2017SGR932; Fondation Sciences Mathématiques de Paris, FSMPAcademic Press Inc.2021info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersion51 p.application/pdfhttp://hdl.handle.net/2072/531265RECERCAT (Dipòsit de la Recerca de Catalunya)reponame:Recercat. Dipósit de la Recerca de Catalunyainstname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)InglésAdvances in MathematicsL'accés als continguts d'aquest document queda condicionat a l'acceptació de les condicions d'ús establertes per la següent llicència Creative Commons: http://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccessoai:recercat.cat:2072/5312652026-05-29T05:05:01Z
dc.title.none.fl_str_mv The singular Weinstein conjecture
title The singular Weinstein conjecture
spellingShingle The singular Weinstein conjecture
Miranda, E.
b-Contact forms
b-Symplectic geometry
Overtwisted contact structures
Periodic orbits
Reeb dynamics
Singular Weinstein conjecture
51
title_short The singular Weinstein conjecture
title_full The singular Weinstein conjecture
title_fullStr The singular Weinstein conjecture
title_full_unstemmed The singular Weinstein conjecture
title_sort The singular Weinstein conjecture
dc.creator.none.fl_str_mv Miranda, E.
Oms, C.
author Miranda, E.
author_facet Miranda, E.
Oms, C.
author_role author
author2 Oms, C.
author2_role author
dc.subject.none.fl_str_mv b-Contact forms
b-Symplectic geometry
Overtwisted contact structures
Periodic orbits
Reeb dynamics
Singular Weinstein conjecture
51
topic b-Contact forms
b-Symplectic geometry
Overtwisted contact structures
Periodic orbits
Reeb dynamics
Singular Weinstein conjecture
51
description In this article, we investigate Reeb dynamics on bm-contact manifolds, previously introduced in [37], which are contact away from a hypersurface Z but satisfy certain transversality conditions on Z. The study of these contact structures is motivated by that of contact manifolds with boundary. The search of periodic Reeb orbits on those manifolds thereby starts with a generalization of the well-known Weinstein conjecture. Contrary to the initial expectations, examples of compact bm-contact manifolds without periodic Reeb orbits outside Z are provided. Furthermore, we prove that in dimension 3, there are always infinitely many periodic orbits on the critical set if it is compact. We prove that traps for the bm-Reeb flow exist in any dimension. This investigation goes hand-in-hand with the Weinstein conjecture on non-compact manifolds having compact ends of convex type. In particular, we extend Hofer's arguments to open overtwisted contact manifolds that are R+-invariant in the open ends, obtaining as a corollary the existence of periodic bm-Reeb orbits away from the critical set. The study of bm-Reeb dynamics is motivated by well-known problems in fluid dynamics and celestial mechanics, where those geometric structures naturally appear. In particular, we prove that the dynamics on positive energy level-sets in the restricted planar circular three body problem are described by the Reeb vector field of a b3-contact form that admits an infinite number of periodic orbits at the critical set. © 2021 The Author(s)
publishDate 2021
dc.date.none.fl_str_mv 2021
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/2072/531265
url http://hdl.handle.net/2072/531265
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Advances in Mathematics
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv 51 p.
application/pdf
dc.publisher.none.fl_str_mv Academic Press Inc.
publisher.none.fl_str_mv Academic Press Inc.
dc.source.none.fl_str_mv RECERCAT (Dipòsit de la Recerca de Catalunya)
reponame:Recercat. Dipósit de la Recerca de Catalunya
instname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
instname_str Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
reponame_str Recercat. Dipósit de la Recerca de Catalunya
collection Recercat. Dipósit de la Recerca de Catalunya
repository.name.fl_str_mv
repository.mail.fl_str_mv
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