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The action functional on dual Legendrian submanifolds of the loop space of a contact three dimensional closed manifold
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Maalaoui, Ali. The action functional on dual Legendrian submanifolds of the loop space of a contact three dimensional closed manifold. Retrieved from https://doi.org/doi:10.7282/T34748FD

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TitleThe action functional on dual Legendrian submanifolds of the loop space of a contact three dimensional closed manifold
NameMaalaoui, Ali (author); Bahri, Abbas (chair); Chanillo, Sagun (internal member); Woodward, Christopher (internal member); Masmoudi, Nader (outside member); Rutgers University; Graduate School - New Brunswick
Date Created2013
Other Date2013-05 (degree)
SubjectMathematics, Three-manifolds (Topology), Manifolds (Mathematics)
Extentix, 122 p. : ill.
DescriptionThe main object of my dissertation is the study of the action functional of a contact form on a three dimensional manifold. This is part of a long program started by Professor A. Bahri [3], [4] in constructing a contact homology giving information about the number of periodic orbits of the Reeb vector field, that is an attempt to approach the Weinstein conjecture for 3-manifolds. Even though it appears to be proved by C. Taubes in a series of paper (see for instance [39] for more details). Given a closed 3-dimensional manifold, we prove an S1 homotopy equivalence between a subspace Cß of Legendrian curves and the free loop space. This space appears to be convenient from a variational point of view, in contact form geometry and used in the approach developed by A. Bahri. Indeed, it is the right space of variations on which we study the action functional. In a second part we study the Fredholm assumption for a modifed version of the action functional on the variational space Cß. That is, whether the functional is Fredholm or not. We take here the text-book case-study of a sequence of overtwisted contact forms on the 3-sphere introduced by Gonzalo and Varela [26]. We show that the Fredholm assumption does not hold. This is done by studying the dynamics of the contact form along a vector field on its kernel. We also prove the existence of a foliation stuck between the contact form and its Legendre dual in the part where they have opposite orientation. In the last part we present an explicit computation of the Bahri contact homology for a sequence of tight contact structures in the torus. We also extend this result to the case of torus bundles over S1. The homology that we find, allows us in particular to confirm the fact that the contact structures are not isotopic since it has different values for each structure.
NotePh.D.
NoteIncludes bibliographical references
NoteIncludes vita
Noteby Ali Maalaoui
Genretheses, ETD doctoral
Persistent URLhttps://doi.org/doi:10.7282/T34748FD
Languageeng
CollectionGraduate School - New Brunswick Electronic Theses and Dissertations
Organization NameRutgers, The State University of New Jersey
RightsThe author owns the copyright to this work.
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