scholarly journals Orderability, contact non-squeezing, and Rabinowitz Floer homology

2018 ◽  
Vol 16 (6) ◽  
pp. 1481-1547
Author(s):  
Peter Albers ◽  
Will J. Merry
Keyword(s):  
Floer Homology ◽  
Rabinowitz Floer Homology

scholarly journals ESTIMATES AND COMPUTATIONS IN RABINOWITZ–FLOER HOMOLOGY

2009 ◽  
Vol 01 (04) ◽  
pp. 307-405 ◽  
Author(s):  
ALBERTO ABBONDANDOLO ◽  
MATTHIAS SCHWARZ
Keyword(s):  
Maximum Principle ◽  
Energy Level ◽  
Exact Sequence ◽  
Floer Homology ◽  
Energy Functional ◽  
Convex Compact ◽  
Closed Geodesics ◽  
Action Functional ◽  
Uniform Estimates ◽  
Rabinowitz Floer Homology

The Rabinowitz–Floer homology of a Liouville domain W is the Floer homology of the Rabinowitz free period Hamiltonian action functional associated to a Hamiltonian whose zero energy level is the boundary of W. This invariant has been introduced by K. Cieliebak and U. Frauenfelder and has already found several applications in symplectic topology and in Hamiltonian dynamics. Together with A. Oancea, the same authors have recently computed the Rabinowitz–Floer homology of the cotangent disk bundle D* M of a closed Riemannian manifold M, by means of an exact sequence relating the Rabinowitz–Floer homology of D* M with its symplectic homology and cohomology. The first aim of this paper is to present a chain level construction of this exact sequence. In fact, we show that this sequence is the long homology sequence induced by a short exact sequence of chain complexes, which involves the Morse chain complex and the Morse differential complex of the energy functional for closed geodesics on M. These chain maps are defined by considering spaces of solutions of the Rabinowitz–Floer equation on half-cylinders, with suitable boundary conditions which couple them with the negative gradient flow of the geodesic energy functional. The second aim is to generalize this construction to the case of a fiberwise uniformly convex compact subset W of T* M whose interior part contains a Lagrangian graph. Equivalently, W is the energy sublevel associated to an arbitrary Tonelli Lagrangian L on TM and to any energy level which is larger than the strict Mañé critical value of L. In this case, the energy functional for closed geodesics is replaced by the free period Lagrangian action functional associated to a suitable calibration of L. An important issue in our analysis is to extend the uniform estimates for the solutions of the Rabinowitz–Floer equation — both on cylinders and on half-cylinders — to Hamiltonians which have quadratic growth in the momenta. These uniform estimates are obtained by the Aleksandrov integral version of the maximum principle. In the case of half-cylinders, they are obtained by an Aleksandrov-type maximum principle with Neumann conditions on part of the boundary.

scholarly journals Continuation homomorphism in Rabinowitz Floer homology for symplectic deformations

2011 ◽  
Vol 151 (3) ◽  
pp. 471-502 ◽  
Author(s):  
YOUNGJIN BAE ◽  
URS FRAUENFELDER
Keyword(s):  
Exact Sequence ◽  
Isoperimetric Inequality ◽  
Short Exact Sequence ◽  
Loop Space ◽  
Floer Homology ◽  
Critical Value ◽  
Alternative Proof ◽  
Rabinowitz Floer Homology ◽  
Space Homology ◽  
Loop Space Homology

AbstractWill J. Merry computed Rabinowitz Floer homology above Mañé's critical value in terms of loop space homology in [14] by establishing an Abbondandolo–Schwarz short exact sequence. The purpose of this paper is to provide an alternative proof of Merry's result. We construct a continuation homomorphism for symplectic deformations which enables us to reduce the computation to the untwisted case. Our construction takes advantage of a special version of the isoperimetric inequality which above Mañé's critical value holds true.

scholarly journals Translated points and Rabinowitz Floer homology

2013 ◽  
Vol 13 (1) ◽  
pp. 201-214 ◽  
Author(s):  
Peter Albers ◽  
Will J. Merry
Keyword(s):  
Floer Homology ◽  
Rabinowitz Floer Homology

scholarly journals Vanishing of Rabinowitz Floer homology on negative line bundles

2016 ◽  
Vol 285 (1-2) ◽  
pp. 493-517 ◽  
Author(s):  
Peter Albers ◽  
Jungsoo Kang
Keyword(s):  
Floer Homology ◽  
Line Bundles ◽  
Rabinowitz Floer Homology

scholarly journals Translated points on hypertight contact manifolds

2018 ◽  
Vol 10 (02) ◽  
pp. 289-322
Author(s):  
Matthias Meiwes ◽  
Kathrin Naef
Keyword(s):  
Floer Homology ◽  
Contact Manifold ◽  
Contact Form ◽  
Contact Manifolds ◽  
Rabinowitz Floer Homology

A contact manifold admitting a supporting contact form without contractible Reeb orbits is called hypertight. In this paper we construct a Rabinowitz–Floer homology associated to an arbitrary supporting contact form for a hypertight contact manifold [Formula: see text], and use this to prove versions of a conjecture of Sandon [17] on the existence of translated points and to show that positive loops of contactomorphisms give rise to non-contractible Reeb orbits.

scholarly journals $S^1$-equivariant Rabinowitz--Floer homology

2016 ◽  
Vol 45 (3) ◽  
pp. 293-323 ◽  
Author(s):  
Urs FRAUENFELDER ◽  
Felix SCHLENK
Keyword(s):  
Floer Homology ◽  
Rabinowitz Floer Homology

scholarly journals LEAF-WISE INTERSECTIONS AND RABINOWITZ FLOER HOMOLOGY

2010 ◽  
Vol 02 (01) ◽  
pp. 77-98 ◽  
Author(s):  
PETER ALBERS ◽  
URS FRAUENFELDER
Keyword(s):  
Critical Points ◽  
Floer Homology ◽  
Symplectic Manifolds ◽  
Action Functional ◽  
Multiplicity Results ◽  
Contact Type ◽  
Existence And Multiplicity ◽  
Rabinowitz Floer Homology ◽  
Intersection Points ◽  
Rabinowitz Action Functional

In this paper we explain how critical points of a particular perturbation of the Rabinowitz action functional give rise to leaf-wise intersection points in hypersurfaces of restricted contact type. This is used to derive existence and multiplicity results for leaf-wise intersection points in hypersurfaces of restricted contact type in general exact symplectic manifolds. The notion of leaf-wise intersection points was introduced by Moser [16].

Sign in / Sign up

Export Citation Format

Share Document