symplectic capacity
Recently Published Documents


TOTAL DOCUMENTS

13
(FIVE YEARS 0)

H-INDEX

3
(FIVE YEARS 0)

2020 ◽  
Vol 40 (8) ◽  
pp. 4705-4765
Author(s):  
Rongrong Jin ◽  
◽  
Guangcun Lu


2019 ◽  
Vol 30 (09) ◽  
pp. 1950035
Author(s):  
Stefan Müller

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.



2019 ◽  
Vol 30 (3) ◽  
pp. 437-443
Author(s):  
E. D. Gluskin
Keyword(s):  


2018 ◽  
Vol 2020 (7) ◽  
pp. 1957-1978
Author(s):  
Alexey Balitskiy

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.



2012 ◽  
Vol 153 (2) ◽  
pp. 261-279 ◽  
Author(s):  
HANSJÖRG GEIGES ◽  
KAI ZEHMISCH

AbstractWe study holomorphic spheres in certain symplectic cobordisms and derive information about periodic Reeb orbits in the concave end of these cobordisms from the non-compactness of the relevant moduli spaces. We use this to confirm the strong Weinstein conjecture (predicting the existence of null-homologous Reeb links) for various higher-dimensional contact manifolds, including contact type hypersurfaces in subcritical Stein manifolds and in some cotangent bundles. The quantitative character of this result leads to the definition of a symplectic capacity.



2012 ◽  
Vol 272 (3-4) ◽  
pp. 1291-1320 ◽  
Author(s):  
Kei Irie


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


Author(s):  
M.-Y. Jiang

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.



Sign in / Sign up

Export Citation Format

Share Document