scholarly journals Comparison of spectral invariants in Lagrangian and Hamiltonian Floer theory

Filomat ◽  
2016 ◽  
Vol 30 (5) ◽  
pp. 1161-1174 ◽  
Author(s):  
Jovana Djuretic ◽  
Jelena Katic ◽  
Darko Milinkovic
Keyword(s):  
Periodic Orbits ◽  
Symplectic Manifold ◽  
Floer Homology ◽  
Lagrangian Submanifolds ◽  
Spectral Invariants ◽  
Floer Theory

We compare spectral invariants in periodic orbits and Lagrangian Floer homology case, for a closed symplectic manifold P and its closed Lagrangian submanifolds L, when ?|?2(P,L)=0, and ?|?2(P,L)=0. We define a product HF*(H)?HF*(H:L) ? HF*(H:L) and prove subadditivity of invariants with respect to this product.

scholarly journals Non-contractible periodic orbits in Hamiltonian dynamics on closed symplectic manifolds

2016 ◽  
Vol 152 (9) ◽  
pp. 1777-1799 ◽  
Author(s):  
Viktor L. Ginzburg ◽  
Başak Z. Gürel
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Symplectic Manifold ◽  
Floer Homology ◽  
Hamiltonian Dynamics ◽  
Symplectic Manifolds ◽  
Homotopy Classes ◽  
Hamiltonian Diffeomorphisms ◽  
Favorable Conditions ◽  
Hamiltonian Diffeomorphism

We study Hamiltonian diffeomorphisms of closed symplectic manifolds with non-contractible periodic orbits. In a variety of settings, we show that the presence of one non-contractible periodic orbit of a Hamiltonian diffeomorphism of a closed toroidally monotone or toroidally negative monotone symplectic manifold implies the existence of infinitely many non-contractible periodic orbits in a specific collection of free homotopy classes. The main new ingredient in the proofs of these results is a filtration of Floer homology by the so-called augmented action. This action is independent of capping and, under favorable conditions, the augmented action filtration for toroidally (negative) monotone manifolds can play the same role as the ordinary action filtration for atoroidal manifolds.

scholarly journals Spectral invariants for monotone Lagrangians

2018 ◽  
Vol 10 (03) ◽  
pp. 627-700 ◽  
Author(s):  
Rémi Leclercq ◽  
Frol Zapolsky
Keyword(s):  
Generating Functions ◽  
Floer Homology ◽  
Spectral Invariants ◽  
Seminal Paper ◽  
Floer Theory ◽  
Cotangent Bundles

Since spectral invariants were introduced in cotangent bundles via generating functions by Viterbo in the seminal paper [73], they have been defined in various contexts, mainly via Floer homology theories, and then used in a great variety of applications. In this paper we extend their definition to monotone Lagrangians, which is so far the most general case for which a “classical” Floer theory has been developed. Then, we gather and prove the properties satisfied by these invariants, and which are crucial for their applications. Finally, as a demonstration, we apply these new invariants to symplectic rigidity of some specific monotone Lagrangians.

scholarly journals Spectral invariants of distance functions

2016 ◽  
Vol 08 (04) ◽  
pp. 655-676 ◽  
Author(s):  
Suguru Ishikawa
Keyword(s):  
Symplectic Manifold ◽  
Symplectic Form ◽  
Floer Homology ◽  
Symplectic Manifolds ◽  
Distance Functions ◽  
Spectral Invariants ◽  
Euclidian Space ◽  
Closed Riemann Surface ◽  
Genus Formula ◽  
Spectral Invariant

Calculating the spectral invariant of Floer homology of the distance function, we can find new superheavy subsets in symplectic manifolds. We show if convex open subsets in Euclidian space with the standard symplectic form are disjointly embedded in a spherically negative monotone closed symplectic manifold, their complement is superheavy. In particular, the [Formula: see text] bouquet in a closed Riemann surface with genus [Formula: see text] is superheavy. We also prove some analogous properties of a monotone closed symplectic manifold. These can be used to extend Seyfaddni’s result about lower bounds of Poisson bracket invariant.

scholarly journals A COMPARISON OF HOFER'S METRICS ON HAMILTONIAN DIFFEOMORPHISMS AND LAGRANGIAN SUBMANIFOLDS

2003 ◽  
Vol 05 (05) ◽  
pp. 803-811 ◽  
Author(s):  
YARON OSTROVER
Keyword(s):  
Symplectic Manifold ◽  
Isometric Embedding ◽  
Lagrangian Submanifolds ◽  
Canonical Embedding ◽  
Hamiltonian Diffeomorphisms

We compare Hofer's geometries on two spaces associated with a closed symplectic manifold (M,ω). The first space is the group of Hamiltonian diffeomorphisms. The second space ℒ consists of all Lagrangian submanifolds of M × M which are exact Lagrangian isotopic to the diagonal. We show that in the case of a closed symplectic manifold with π2(M) = 0, the canonical embedding of Ham (M) into ℒ, f ↦ graph (f) is not an isometric embedding, although it preserves Hofer's length of smooth paths.

scholarly journals Symplectic deformations of Floer homology and non-contractible periodic orbits in twisted disc bundles

2019 ◽  
Vol 23 (01) ◽  
pp. 1950084
Author(s):  
Wenmin Gong
Keyword(s):  
Magnetic Field ◽  
Periodic Orbits ◽  
Homotopy Class ◽  
Linear Growth ◽  
Floer Homology ◽  
Hamiltonian Function ◽  
Universal Cover ◽  
Zero Section ◽  
Free Homotopy Class ◽  
The Given

In this paper, we establish the existence of periodic orbits belonging to any [Formula: see text]-atoroidal free homotopy class for Hamiltonian systems in the twisted disc bundle, provided that the compactly supported time-dependent Hamiltonian function is sufficiently large over the zero section and the magnitude of the weakly exact [Formula: see text]-form [Formula: see text] admitting a primitive with at most linear growth on the universal cover is sufficiently small. The proof relies on showing the invariance of Floer homology under symplectic deformations and on the computation of Floer homology for the cotangent bundle endowed with its canonical symplectic form. As a consequence, we also prove that, for any non-trivial atoroidal free homotopy class and any positive finite interval, if the magnitude of a magnetic field admitting a primitive with at most linear growth on the universal cover is sufficiently small, the twisted geodesic flow associated to the magnetic field has a periodic orbit on almost every energy level in the given interval whose projection to the underlying manifold represents the given free homotopy class. This application is carried out by showing the finiteness of the restricted Biran–Polterovich–Salamon capacity.

scholarly journals Minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces

2019 ◽  
Vol 6 (1) ◽  
pp. 303-319
Author(s):  
Yoshihiro Ohnita
Keyword(s):  
Symmetric Space ◽  
Floer Homology ◽  
Lagrangian Submanifolds ◽  
Symplectic Manifolds ◽  
Lagrangian Submanifold ◽  
Riemannian Symmetric Space ◽  
Canonical Embedding ◽  
Real Form ◽  
Isotropy Representation ◽  
Totally Geodesic

AbstractAn R-space is a compact homogeneous space obtained as an orbit of the isotropy representation of a Riemannian symmetric space. It is known that each R-space has the canonical embedding into a Kähler C-space as a real form, and thus a compact embedded totally geodesic Lagrangian submanifold. The minimal Maslov number of Lagrangian submanifolds in symplectic manifolds is one of invariants under Hamiltonian isotopies and very fundamental to study the Floer homology for intersections of Lagrangian submanifolds. In this paper we show a Lie theoretic formula for the minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces, and provide some examples of the calculation by the formula.

scholarly journals Spectral invariants in Lagrangian Floer homology of open subset

2017 ◽  
Vol 53 ◽  
pp. 220-267
Author(s):  
Jelena Katić ◽  
Darko Milinković ◽  
Jovana Nikolić
Keyword(s):  
Open Subset ◽  
Floer Homology ◽  
Spectral Invariants
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