Length minimizing Hamiltonian paths for symplectically aspherical manifolds

Kerman, 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 -