scholarly journals -rigidity of Lagrangian submanifolds and punctured holomorphic disks in the cotangent bundle

2021 ◽  
Vol 157 (11) ◽  
pp. 2433-2493
Author(s):  
Cedric Membrez ◽  
Emmanuel Opshtein
Keyword(s):  
Cotangent Bundle ◽  
Symplectic Geometry ◽  
Homology Class ◽  
Lagrangian Submanifolds ◽  
Area Spectrum ◽  
Zero Section ◽  
Integral Homology ◽  
Cotangent Bundles ◽  
Maslov Class ◽  
Holomorphic Disks

Abstract Our main result is the $\mathbb {\mathcal {C}}^{0}$ -rigidity of the area spectrum and the Maslov class of Lagrangian submanifolds. This relies on the existence of punctured pseudoholomorphic disks in cotangent bundles with boundary on the zero section, whose boundaries represent any integral homology class. We discuss further applications of these punctured disks in symplectic geometry.

scholarly journals Hyperkähler metrics near Lagrangian submanifolds and symplectic groupoids

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Maxence Mayrand
Keyword(s):  
Symplectic Manifold ◽  
Cotangent Bundle ◽  
Lagrangian Submanifolds ◽  
Poisson Manifold ◽  
Lagrangian Submanifold ◽  
Zero Section ◽  
Poisson Structures ◽  
Symplectic Structures ◽  
Symplectic Groupoids ◽  
Hyperkähler Metrics

Abstract The first part of this paper is a generalization of the Feix–Kaledin theorem on the existence of a hyperkähler metric on a neighborhood of the zero section of the cotangent bundle of a Kähler manifold. We show that the problem of constructing a hyperkähler structure on a neighborhood of a complex Lagrangian submanifold in a holomorphic symplectic manifold reduces to the existence of certain deformations of holomorphic symplectic structures. The Feix–Kaledin structure is recovered from the twisted cotangent bundle. We then show that every holomorphic symplectic groupoid over a compact holomorphic Poisson surface of Kähler type has a hyperkähler structure on a neighborhood of its identity section. More generally, we reduce the existence of a hyperkähler structure on a symplectic realization of a holomorphic Poisson manifold of any dimension to the existence of certain deformations of holomorphic Poisson structures adapted from Hitchin’s unobstructedness theorem.

scholarly journals Lagrangian exotic spheres

2016 ◽  
Vol 08 (03) ◽  
pp. 375-397 ◽  
Author(s):  
Tobias Ekholm ◽  
Thomas Kragh ◽  
Ivan Smith
Keyword(s):  
Cotangent Bundle ◽  
Zero Section ◽  
Homotopy Formula ◽  
Connected Sum ◽  
Cotangent Bundles ◽  
Sum Formula ◽  
Lagrangian Embedding ◽  
Exotic Spheres

Let [Formula: see text]. We prove that the cotangent bundles [Formula: see text] and [Formula: see text] of oriented homotopy [Formula: see text]-spheres [Formula: see text] and [Formula: see text] are symplectomorphic only if [Formula: see text], where [Formula: see text] denotes the group of oriented homotopy [Formula: see text]-spheres under connected sum, [Formula: see text] denotes the subgroup of those that bound a parallelizable [Formula: see text]-manifold, and where [Formula: see text] denotes [Formula: see text] with orientation reversed. We further show that if [Formula: see text] and [Formula: see text] admits a Lagrangian embedding in [Formula: see text], then [Formula: see text]. The proofs build on [1] and [18] in combination with a new cut-and-paste argument; that also yields some interesting explicit exact Lagrangian embeddings, for instance of the sphere [Formula: see text] into the plumbing [Formula: see text] of cotangent bundles of certain exotic spheres. As another application, we show that there are re-parametrizations of the zero-section in the cotangent bundle of a sphere that are not Hamiltonian isotopic (as maps rather than as submanifolds) to the original zero-section.

scholarly journals Exact Lagrangian submanifolds in simply-connected cotangent bundles

2007 ◽  
Vol 172 (1) ◽  
pp. 1-27 ◽  
Author(s):  
Kenji Fukaya ◽  
Paul Seidel ◽  
Ivan Smith
Keyword(s):  
Lagrangian Submanifolds ◽  
Cotangent Bundles ◽  
Simply Connected

SOME RESULTS ON GENERALIZED CALABI–YAU MANIFOLDS

2006 ◽  
Vol 03 (05n06) ◽  
pp. 1273-1292
Author(s):  
PAOLO DE BARTOLOMEIS ◽  
ADRIANO TOMASSINI
Keyword(s):  
Lagrangian Submanifolds ◽  
Lagrangian Submanifold ◽  
Dimensional Case ◽  
Maslov Class ◽  
Calabi Yau Manifolds

We consider generalized Calabi–Yau manifolds and we give a formula for the Maslov class of a Lagrangian submanifold of a generalized Calabi–Yau manifold. In particular, we characterize the Lagrangian submanifolds with vanishing Maslov class. In the 6-dimensional case, we refine our definition. Finally, we construct some examples.

scholarly journals Quasi Semi-Border Singularities

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 495
Author(s):  
Fawaz Alharbi ◽  
Suliman Alsaeed
Keyword(s):  
Spectral Sequence ◽  
Equivalence Relation ◽  
Symplectic Geometry ◽  
Lagrangian Submanifolds ◽  
Natural Extension ◽  
Main Method ◽  
Equivalent Relation ◽  
Sequence Method ◽  
Artificial Nature

We obtain a list of simple classes of singularities of function germs with respect to the quasi m-boundary equivalence relation, with m ≥ 2 . The results obtained in this paper are a natural extension of Zakalyukin’s work on the new non-standard equivalent relation. In spite of the rather artificial nature of the definitions, the quasi relations have very natural applications in symplectic geometry. In particular, they are used to classify singularities of Lagrangian projections equipped with a submanifold. The main method that is used in the classification is the standard Moser’s homotopy technique. In addition, we adopt the version of Arnold’s spectral sequence method, which is described in Lemma 2. Our main results are Theorem 4 on the classification of simple quasi classes, and Theorem 5 on the classification of Lagrangian submanifolds with smooth varieties. The brief description of the main results is given in the next section.

scholarly journals THE MASLOV CLASS OF LAGRANGIAN TORI AND QUANTUM PRODUCTS IN FLOER COHOMOLOGY

2010 ◽  
Vol 02 (01) ◽  
pp. 57-75 ◽  
Author(s):  
LEV BUHOVSKY
Keyword(s):  
Spectral Sequence ◽  
Floer Cohomology ◽  
Lagrangian Torus ◽  
Cup Product ◽  
Maslov Class ◽  
Lagrangian Tori ◽  
Holomorphic Disks

We use Floer cohomology to prove the monotone version of a conjecture of Audin: the minimal Maslov number of a monotone Lagrangian torus in ℝ2n is 2. Our approach is based on the study of the quantum cup product on Floer cohomology and in particular the behavior of Oh's spectral sequence with respect to this product. As further applications, we prove existence of holomorphic disks with boundaries on Lagrangians as well as new results on Lagrangian intersections.

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