TY - JOUR
T1 - Rigid constellations of closed Reeb orbits
AU - Kerman, Ely
N1 - Publisher Copyright:
© The Author 2017.
PY - 2017/11/1
Y1 - 2017/11/1
N2 - We use Hamiltonian Floer theory to recover and generalize a classic rigidity theorem of Ekeland and Lasry. That theorem can be rephrased as an assertion about the existence of multiple closed Reeb orbits for certain tight contact forms on the sphere that are close, in a suitable sense, to the standard contact form. We first generalize this result to Reeb flows of contact forms on prequantization spaces that are suitably close to Boothby-Wang forms. We then establish, under an additional nondegeneracy assumption, the same rigidity phenomenon for Reeb flows on any closed contact manifold. A natural obstruction to obtaining sharp multiplicity results for closed Reeb orbits is the possible existence of fast closed orbits. To complement the existence results established here, we also show that the existence of such fast orbits cannot be precluded by any condition which is invariant under contactomorphisms, even for nearby contact forms.
AB - We use Hamiltonian Floer theory to recover and generalize a classic rigidity theorem of Ekeland and Lasry. That theorem can be rephrased as an assertion about the existence of multiple closed Reeb orbits for certain tight contact forms on the sphere that are close, in a suitable sense, to the standard contact form. We first generalize this result to Reeb flows of contact forms on prequantization spaces that are suitably close to Boothby-Wang forms. We then establish, under an additional nondegeneracy assumption, the same rigidity phenomenon for Reeb flows on any closed contact manifold. A natural obstruction to obtaining sharp multiplicity results for closed Reeb orbits is the possible existence of fast closed orbits. To complement the existence results established here, we also show that the existence of such fast orbits cannot be precluded by any condition which is invariant under contactomorphisms, even for nearby contact forms.
KW - Floer homology
KW - Reeb flows
KW - periodic orbits
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U2 - 10.1112/S0010437X17007448
DO - 10.1112/S0010437X17007448
M3 - Article
AN - SCOPUS:85054169693
SN - 0010-437X
VL - 153
SP - 2394
EP - 2444
JO - Compositio Mathematica
JF - Compositio Mathematica
IS - 11
ER -