scholarly journals Symmetrical symplectic capacity with applications

2012 ◽  
Vol 32 (6) ◽  
pp. 2253-2270 ◽  
Author(s):  
Chungen Liu ◽  
◽  
Qi Wang
Keyword(s):  
Symplectic Capacity

Hofer–Zehnder symplectic capacity for two-dimensional manifolds

1993 ◽  
Vol 123 (5) ◽  
pp. 945-950 ◽  
Author(s):  
Mei-Yue Jiang
Keyword(s):  
Symplectic Form ◽  
Cotangent Bundle ◽  
Dimensional Manifold ◽  
Two Dimensional ◽  
Symplectic Capacity

SynopsisIn this paper, we show that the Hofer-Zehnder symplectic capacity of a two-dimensional manifold is same as the area of the manifold. Using this fact, we also get a result on the symplectic capacity of the cotangent bundle of the torus with its canonical symplectic form.

Symplectic embeddings from R2n into some manifolds

2000 ◽  
Vol 130 (1) ◽  
pp. 53-61 ◽  
Author(s):  
M.-Y. Jiang
Keyword(s):  
Symplectic Manifold ◽  
Symplectic Form ◽  
Bounded Domains ◽  
Two Dimensional ◽  
Symplectic Capacity ◽  
Elementary Methods

We show by elementary methods that there are symplectic embeddings from standard (R2n, ω0) into (Σ x R2n−2, ω ⊕ ω0) and (T2n−2k x R2k, ω ⊕ ω0), where (Σ, ω) is a closed two-dimensional symplectic manifold, and (T2n−2k, ω) is the torus with a constant symplectic form ω. Some estimates of Gromov's symplectic capacity are given for bounded domains in these manifolds.

scholarly journals Equality Cases in Viterbo’s Conjecture and Isoperimetric Billiard Inequalities

2018 ◽  
Vol 2020 (7) ◽  
pp. 1957-1978
Author(s):  
Alexey Balitskiy
Keyword(s):  
Convex Body ◽  
Symplectic Capacity ◽  
Special Cases

Abstract We apply the billiard technique to deduce some results on Viterbo’s conjectured inequality between the volume of a convex body and its symplectic capacity. We show that the product of a permutohedron and a simplex (properly related to each other) delivers equality in Viterbo’s conjecture. Using this result as well as previously known equality cases, we prove some special cases of Viterbo’s conjecture and interpret them as isoperimetric-like inequalities for billiard trajectories.

scholarly journals C0-characterization of symplectic and contact embeddings and Lagrangian rigidity

2019 ◽  
Vol 30 (09) ◽  
pp. 1950035
Author(s):  
Stefan Müller
Keyword(s):  
Lagrangian Submanifolds ◽  
Sufficient Conditions ◽  
Intersection Theory ◽  
Holomorphic Curve ◽  
Contact Manifolds ◽  
Symplectic Capacity ◽  
Contact Topology ◽  
Shape Invariant ◽  
Necessary And Sufficient

We present a novel [Formula: see text]-characterization of symplectic embeddings and diffeomorphisms in terms of Lagrangian embeddings. Our approach is based on the shape invariant, which was discovered by Sikorav and Eliashberg, intersection theory and the displacement energy of Lagrangian submanifolds, and the fact that non-Lagrangian submanifolds can be displaced immediately. This characterization gives rise to a new proof of [Formula: see text]-rigidity of symplectic embeddings and diffeomorphisms. The various manifestations of Lagrangian rigidity that are used in our arguments come from [Formula: see text]-holomorphic curve methods. An advantage of our techniques is that they can be adapted to a [Formula: see text]-characterization of contact embeddings and diffeomorphisms in terms of coisotropic (or pre-Lagrangian) embeddings, which in turn leads to a proof of [Formula: see text]-rigidity of contact embeddings and diffeomorphisms. We give a detailed treatment of the shape invariants of symplectic and contact manifolds, and demonstrate that shape is often a natural language in symplectic and contact topology. We consider homeomorphisms that preserve shape, and propose a hierarchy of notions of Lagrangian topological submanifold. Moreover, we discuss shape-related necessary and sufficient conditions for symplectic and contact embeddings, and define a symplectic capacity from the shape.

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