Symplectic embeddings and the lagrangian bidisk

@article{Ramos2017SymplecticEA,
title={Symplectic embeddings and the lagrangian bidisk},
author={Vinicius G. B. Ramos},
journal={Duke Mathematical Journal},
year={2017},
volume={166},
pages={1703-1738}
}
In this paper we obtain sharp obstructions to the symplectic embedding of the lagrangian bidisk into four-dimensional balls, ellipsoids and symplectic polydisks. We prove, in fact, that the interior of the lagrangian bidisk is symplectomorphic to a concave toric domain using ideas that come from billiards on a round disk. In particular, we answer a question of Ostrover. We also obtain sharp obstructions to some embeddings of ellipsoids into the lagrangian bidisk.

Figures from this paper

Symplectic embeddings into disk cotangent bundles
• Mathematics
• 2021
In this paper, we compute the embedded contact homology (ECH) capacities of the disk cotangent bundles D∗S2 and D∗RP 2. We also find sharp symplectic embeddings into these domains. In particular, we
On the rigidity of lagrangian products
• Mathematics
Journal of Symplectic Geometry
• 2019
Motivated by work of the first author, this paper studies symplectic embedding problems of lagrangian products that are sufficiently symmetric. In general, lagrangian products arise naturally in the
Symplectic embeddings of the ℓp-sum of two discs
• Mathematics
• 2021
In this paper, we study symplectic embedding questions for the lp-sum of two discs in ℝ4, when 1 ≤ p ≤∞. In particular, we compute the symplectic inner and outer radii in these cases, and show how ...
Symplectic embeddings from concave toric domains into convex ones
Embedded contact homology gives a sequence of obstructions to four-dimensional symplectic embeddings, called ECH capacities. In "Symplectic embeddings into four-dimensional concave toric domains",
Symplectic embeddings of the $\ell_p$-sum of two discs
• Mathematics
• 2019
In this paper we study symplectic embedding questions for the $\ell_p$-sum of two discs in ${\mathbb R}^4$, when $1 \leq p \leq \infty$. In particular, we compute the symplectic inner and outer radii
On the Ekeland–Hofer symplectic capacities of the real bidisc
• Mathematics
Pacific Journal of Mathematics
• 2020
In $\mathbb{C}^2$ with the standard symplectic structure we consider the bidisc $D^2\times D^2$ constructed as the product of two open real discs of radius $1$. We compute explicit values for the
Symplectic Embeddings of Toric Domains
This work examines a special class of symplectic manifolds known as toric domains. The main problem under consideration is embedding one toric domain into another in a way that preserves the
On symplectic capacities and their blind spots
• Mathematics
• 2021
In this paper we settle three basic questions concerning the Gutt-Hutchings capacities from [9] which are conjecturally equal to the Ekeland-Hofer capacities from [7, 8]. Our primary result settles a
The Viterbo's capacity conjectures for convex toric domains and the product of a $1$-unconditional convex body and its polar
• Mathematics
• 2020
In this note, we show that the strong Viterbo conjecture holds true on any convex toric domain, and that the Viterbo's volume-capacity conjecture holds for the product of a $1$-unconditional convex
Examples around the strong Viterbo conjecture
• Mathematics
Journal of Fixed Point Theory and Applications
• 2022
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

References

SHOWING 1-10 OF 16 REFERENCES
Symplectic embeddings into four‐dimensional concave toric domains
• Mathematics
• 2013
ECH (embedded contact homology) capacities give obstructions to symplectically embedding one symplectic four‐manifold with boundary into another. We compute the ECH capacities of a large family of
Symplectic embeddings from concave toric domains into convex ones
Embedded contact homology gives a sequence of obstructions to four-dimensional symplectic embeddings, called ECH capacities. In "Symplectic embeddings into four-dimensional concave toric domains",
Symplectic embeddings of 4‐dimensional ellipsoids
We show how to reduce the problem of symplectically embedding one 4-dimensional rational ellipsoid into another to a problem of embedding disjoint unions of balls into ℂP 2 . For example, the problem
When symplectic topology meets Banach space geometry
In this paper we survey some recent works that take the first steps toward establishing bilateral connections between symplectic geometry and several other fields, namely, asymptotic geometric
J-Holomorphic Curves and Symplectic Topology
• Mathematics
• 2004
The theory of $J$-holomorphic curves has been of great importance since its introduction by Gromov in 1985. In mathematics, its applications include many key results in symplectic topology. It was
Symplectic embedding of real bi-disc
Let z j  = x j  + iy j  ∈ ℂ (j = 1, 2) and let Ω = {(z 1, z 2) ∈ ℂ2 : |x 1|2 + |x 2|2 < 1, |y 1|2 + |y 2|2 < 1} be a real bi-disc in ℂ2. In this article we will find the sharp lower bound for R such
Bounds for Minkowski Billiard Trajectories in Convex Bodies
• Mathematics
• 2011
In this paper we use the Ekeland-Hofer-Zehnder symplectic capacity to provide several bounds and inequalities for the length of the shortest periodic billiard trajectory in a smooth convex body in
Pseudo holomorphic curves in symplectic manifolds
Definitions. 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). The image C=f(S)C V is called
Quantitative embedded contact homology
Define a "Liouville domain" to be a compact exact symplectic manifold with contact-type boundary. We use embedded contact homology to assign to each four-dimensional Liouville domain (or subset