scholarly journals On the existence of Hamiltonian stationary Lagrangian submanifolds in symplectic manifolds

2011 ◽  
Vol 133 (4) ◽  
pp. 1067-1092 ◽  
Author(s):  
Dominic Joyce ◽  
Yng-Ing Lee ◽  
Richard Schoen
Keyword(s):  
Lagrangian Submanifolds ◽  
Symplectic Manifolds ◽  
Hamiltonian Stationary

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 Reduced dynamics and Lagrangian submanifolds of symplectic manifolds

2014 ◽  
Vol 47 (22) ◽  
pp. 225203 ◽  
Author(s):  
E García-Toraño Andrés ◽  
E Guzmán ◽  
J C Marrero ◽  
T Mestdag
Keyword(s):  
Lagrangian Submanifolds ◽  
Symplectic Manifolds

scholarly journals Rigid subsets of symplectic manifolds

2009 ◽  
Vol 145 (03) ◽  
pp. 773-826 ◽  
Author(s):  
Michael Entov ◽  
Leonid Polterovich
Keyword(s):  
Symplectic Manifold ◽  
Lagrangian Submanifolds ◽  
Symplectic Manifolds ◽  
Smooth Functions ◽  
Torus Actions ◽  
Moment Maps ◽  
Quantum Homology ◽  
Singular Sets ◽  
Hamiltonian Torus Actions ◽  
Commutative Subalgebras

AbstractWe show that there is an hierarchy of intersection rigidity properties of sets in a closed symplectic manifold: some sets cannot be displaced by symplectomorphisms from more sets than the others. We also find new examples of rigidity of intersections involving, in particular, specific fibers of moment maps of Hamiltonian torus actions, monotone Lagrangian submanifolds (following the works of P. Albers and P. Biran-O. Cornea) as well as certain, possibly singular, sets defined in terms of Poisson-commutative subalgebras of smooth functions. In addition, we get some geometric obstructions to semi-simplicity of the quantum homology of symplectic manifolds. The proofs are based on the Floer-theoretical machinery of partial symplectic quasi-states.

Symplectic displacement energy for Lagrangian submanifolds

1993 ◽  
Vol 13 (2) ◽  
pp. 357-367 ◽  
Author(s):  
Leonid Polterovich
Keyword(s):  
Lagrangian Submanifolds ◽  
Symplectic Manifolds ◽  
Holomorphic Curves ◽  
Displacement Energy ◽  
Symplectic Invariant

AbstractRecently H. Hofer defined a new symplectic invariant which has a beautiful dynamical meaning. In the present paper we study this invariant for Lagrangian submanifolds of symplectic manifolds. Our approach is based on Gromov's theory of pseudo-holomorphic curves.

scholarly journals Existence of pseudoheavy fibers of moment maps

2020 ◽  
pp. 2050047
Author(s):  
Morimichi Kawasaki ◽  
Ryuma Orita
Keyword(s):  
Lagrangian Submanifolds ◽  
Symplectic Manifolds ◽  
Partial Answer ◽  
Moment Maps

In this paper, we introduce the notion of pseudoheaviness of closed subsets of closed symplectic manifolds and prove the existence of pseudoheavy fibers of moment maps. In particular, we generalize Entov and Polterovich’s theorem, which ensures the existence of non-displaceable fibers. As its application, we provide a partial answer to a problem posed by them, which asks the existence of heavy fibers. Moreover, we obtain a family of singular Lagrangian submanifolds in [Formula: see text] with various rigidities.

scholarly journals Contact Dynamics: Legendrian and Lagrangian Submanifolds

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2704
Author(s):  
Oğul Esen ◽  
Manuel Lainz Valcázar ◽  
Manuel de León ◽  
Juan Carlos Marrero
Keyword(s):  
Lagrangian Submanifolds ◽  
Contact Manifold ◽  
Symplectic Manifolds ◽  
Contact Dynamics ◽  
Legendre Transformation ◽  
Lagrangian Dynamics ◽  
Symplectic Diffeomorphisms ◽  
Legendrian Submanifold ◽  
Legendrian Submanifolds

We are proposing Tulczyjew’s triple for contact dynamics. The most important ingredients of the triple, namely symplectic diffeomorphisms, special symplectic manifolds, and Morse families, are generalized to the contact framework. These geometries permit us to determine so-called generating family (obtained by merging a special contact manifold and a Morse family) for a Legendrian submanifold. Contact Hamiltonian and Lagrangian Dynamics are recast as Legendrian submanifolds of the tangent contact manifold. In this picture, the Legendre transformation is determined to be a passage between two different generators of the same Legendrian submanifold. A variant of contact Tulczyjew’s triple is constructed for evolution contact dynamics.

scholarly journals On forms, cohomology and BV Laplacians in odd symplectic geometry

2021 ◽  
Vol 111 (2) ◽  
Author(s):  
R. Catenacci ◽  
C. A. Cremonini ◽  
P. A. Grassi ◽  
S. Noja
Keyword(s):  
Symplectic Form ◽  
Symplectic Geometry ◽  
Lagrangian Submanifolds ◽  
Symplectic Manifolds ◽  
Wedge Product ◽  
De Rham

AbstractWe study the cohomology of the complexes of differential, integral and a particular class of pseudo-forms on odd symplectic manifolds taking the wedge product with the symplectic form as a differential. We thus extend the result of Ševera and the related results of Khudaverdian–Voronov on interpreting the BV odd Laplacian acting on half-densities on an odd symplectic supermanifold. We show that the cohomology classes are in correspondence with inequivalent Lagrangian submanifolds and that they all define semidensities on them. Further, we introduce new operators that move from one Lagragian submanifold to another and we investigate their relation with the so-called picture changing operators for the de Rham differential. Finally, we prove the isomorphism between the cohomology of the de Rham differential and the cohomology of BV Laplacian in the extended framework of differential, integral and a particular class of pseudo-forms.

Sign in / Sign up

Export Citation Format

Share Document