Minimal Atlases of Closed Symplectic Manifolds

Abstract

Abstract. We study the number of Darboux charts needed to cover a closed connected symplectic manifold (M, ω) and effectively estimate this number from below and from above in terms of the Lusternik-Schnirelmann category of M and the Gromov width of (M, ω).

Quantum Unsharpness and Symplectic Rigidity

“…It is currently unknown whether the asymptotic behavior ∼ N −2 in inequality (3) is optimal: this is a direction of an ongoing research. Let us mention also that according to [40,Theorem 3.6], every closed 2n-dimensional symplectic manifold can be covered by 2n + 1 displaceable subsets.…”

Minimal symplectic atlases of Hermitian symmetric spaces

“…Using the explicit computation of the volume of a classical domain (Ω, ω 0 ) given by L. K. Hua in [9], we are able to prove the following corollary, which extends the computation of S B for the Grassmannians given in [20] to any classical irreducible HSSCT. Before stating the corollary, we recall that a classical irreducible HSSCT is one of the following quotients of compacts Lie groups:…”

“…where ⌊x⌋ denote the maximal integer smaller than or equal to x. The following theorem summarizes the results about minimal atlases obtained in [20] that we need in the proof of Theorem 1.…”

“…Theorem A (Rudyak-Schlenk [20]). Let (M, ω) be a compact connected 2n-dimensional symplectic manifold.…”

Lagrangian product tori in symplectic manifolds

“…Since each element of the linear symplectic group Sp(2n; Ê) can be realized as linearization of a Hamiltonian symplectomorphism preserving the point ϕ(0), we can also assume that dϕ(0) = dψ(0). There is a symplectic isotopy [10] or the proof of Lemma 2.2 in [22]. Therefore, we may assume that ϕ = ψ on B 2n (b ′ ).…”