# Length minimizing Hamiltonian paths for symplectically aspherical manifolds

Ely Kerman^{[1]}; François Lalonde^{[2]}

- [1] University of Toronto, Department of Mathematics, Toronto Ont. (Canada)
- [2] Université de Montréal, Département de Mathématiques et de Statistiques, Montréal, Québec (Canada)

Annales de l’institut Fourier (2003)

- Volume: 53, Issue: 5, page 1503-1526
- ISSN: 0373-0956

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topKerman, Ely, and Lalonde, François. "Length minimizing Hamiltonian paths for symplectically aspherical manifolds." Annales de l’institut Fourier 53.5 (2003): 1503-1526. <http://eudml.org/doc/116079>.

@article{Kerman2003,

abstract = {In this note we consider the length minimizing properties of Hamiltonian paths generated
by quasi-autonomous Hamiltonians on symplectically aspherical manifolds. Motivated by the
work of Polterovich and Schwarz, we study the role, in the Floer complex of the
generating Hamiltonian, of the global extrema which remain fixed as the time varies. Our
main result determines a natural condition which implies that the corresponding path
minimizes the positive Hofer length. We use this to prove that a quasi-autonomous
Hamiltonian generates a length minimizing path if it has under-twisted fixed global
extrema $P, Q$ and no contractible periodic orbits with period one and action
outside the interval $[\{\mathcal \{A\}\}(Q), \{\mathcal \{A\}\}(P)]$. This, in turn, allows us to produce
new examples of autonomous Hamiltonian flows which are length minimizing for all times.
These constructions are based on the geometry of coisotropic submanifolds. Finally, we
give a new proof of the fact that quasi-autonomous Hamiltonians generate length
minimizing paths over short time intervals.},

affiliation = {University of Toronto, Department of Mathematics, Toronto Ont. (Canada); Université de Montréal, Département de Mathématiques et de Statistiques, Montréal, Québec (Canada)},

author = {Kerman, Ely, Lalonde, François},

journal = {Annales de l’institut Fourier},

keywords = {Hofer's geometry; Hamiltonian diffeomorphism; Floer homology; length minimizing paths; coisotropic submanifolds},

language = {eng},

number = {5},

pages = {1503-1526},

publisher = {Association des Annales de l'Institut Fourier},

title = {Length minimizing Hamiltonian paths for symplectically aspherical manifolds},

url = {http://eudml.org/doc/116079},

volume = {53},

year = {2003},

}

TY - JOUR

AU - Kerman, Ely

AU - Lalonde, François

TI - Length minimizing Hamiltonian paths for symplectically aspherical manifolds

JO - Annales de l’institut Fourier

PY - 2003

PB - Association des Annales de l'Institut Fourier

VL - 53

IS - 5

SP - 1503

EP - 1526

AB - In this note we consider the length minimizing properties of Hamiltonian paths generated
by quasi-autonomous Hamiltonians on symplectically aspherical manifolds. Motivated by the
work of Polterovich and Schwarz, we study the role, in the Floer complex of the
generating Hamiltonian, of the global extrema which remain fixed as the time varies. Our
main result determines a natural condition which implies that the corresponding path
minimizes the positive Hofer length. We use this to prove that a quasi-autonomous
Hamiltonian generates a length minimizing path if it has under-twisted fixed global
extrema $P, Q$ and no contractible periodic orbits with period one and action
outside the interval $[{\mathcal {A}}(Q), {\mathcal {A}}(P)]$. This, in turn, allows us to produce
new examples of autonomous Hamiltonian flows which are length minimizing for all times.
These constructions are based on the geometry of coisotropic submanifolds. Finally, we
give a new proof of the fact that quasi-autonomous Hamiltonians generate length
minimizing paths over short time intervals.

LA - eng

KW - Hofer's geometry; Hamiltonian diffeomorphism; Floer homology; length minimizing paths; coisotropic submanifolds

UR - http://eudml.org/doc/116079

ER -

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