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Examples around the strong Viterbo conjecture.
TLDR
In this article, it was shown that for monotone toric domains in arbitrary dimension, the Gromov width agrees with the first equivariant capacity, and that all normalized symplectic capacities agree on convex domains.Abstract:
A strong version of a conjecture of Viterbo asserts that all normalized symplectic capacities agree on convex domains. We review known results showing that certain specific normalized symplectic capacities agree on convex domains. We also review why all normalized symplectic capacities agree on $S^1$-invariant convex domains. We introduce a new class of examples called "monotone toric domains", which are not necessarily convex, and which include all dynamically convex toric domains in four dimensions. We prove that for monotone toric domains in four dimensions, all normalized symplectic capacities agree. For monotone toric domains in arbitrary dimension, we prove that the Gromov width agrees with the first equivariant capacity. We also study a family of examples of non-monotone toric domains and determine when the conclusion of the strong Viterbo conjecture holds for these examples. Along the way we compute the cylindrical capacity of a large class of "weakly convex toric domains" in four dimensions.read more
Citations
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Symplectic topology and hamiltonian dynamics
Ivar Ekeland,Helmut Hofer +1 more
TL;DR: On etudie des applications symplectiques non lineaires as discussed by the authors, construction d'une capacite symplectique, Problemes de plongement, and Probleme de rigidite
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ECH capacities and the Ruelle invariant
TL;DR: In this paper, it was shown that the error term in the ECH capacities converges to a constant determined by a Ruelle invariant, which measures the average rotation of the Reeb flow on the boundary.
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Symplectic homology of fiberwise convex sets and homology of loop spaces
Kei Irie,О. П. Шевченко +1 more
TL;DR: For any nonempty, compact and fiberwise convex set $K$ in $T^*\mathbb{R}^n, this article proved an isomorphism between symplectic homology and a certain relative homology of loop spaces.
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Quantum Polar Duality and the Symplectic Camel: a Geometric Approach to Quantization
TL;DR: The notion of quantum polarity was introduced in this paper, where the orthogonal projections of the covariance ellipsoid of a quantum state on the configuration and momentum spaces form what they call a dual quantum pair.
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Coisotropic Hofer-Zehnder capacities of convex domains and related results
Rongrong Jin,Guangcun Lu +1 more
TL;DR: In this article, the coisotropic Hofer-Zehnder capacity of bounded convex domains with special submanifolds and the leaf relation has been studied and derived.
References
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Journal ArticleDOI
Pseudo holomorphic curves in symplectic manifolds
TL;DR: In this article, the authors define a parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J).
Journal ArticleDOI
Functors and Computations in Floer homology with Applications Part II
TL;DR: In this article, it was shown that the Floer cohomology of the cotangent bundle is isomorphic to the cohomologies of the loop space of the base.
Journal ArticleDOI
The dynamics on three-dimensional strictly convex energy surfaces
TL;DR: In this paper, it was shown that a Hamiltonian flow on a three-dimensional strictly convex energy surface S C R4 possesses a global surface of section of disc type.
Journal ArticleDOI
Symplectic topology and Hamiltonian dynamics
Ivar Ekeland,Helmut Hofer +1 more
TL;DR: On etudie des applications symplectiques non lineaires as mentioned in this paper, construction d'une capacite symplectique, Problemes de plongement, and Probleme de rigidite