라그라지안 휘트니트릭과 넓이조건

Abstract

Let Ｄ be an open disk in R² and consider the open subset Ｄ×R＾(n-1)×R＾(n-1) of R＾(2n) with the standard symplectic structure. Also let l_(0), l₁be two open arcs in Ｄ which are closed as subsets of Ｄ meeting each other transversely at exactly two points. We construct a Hamiltonian flow with compact support, Ψ_(u) : Ｄ×Ｒ＾(n-1)×Ｒ＾(n-1)→Ｄ×Ｒ＾(n-1)×Ｒ＾(n-1), 0 ≤ u ≤ 1 such that Ψ_(0) = 1 and . Ψ₁(l_(0)×Ｒ＾(n-1)×0)∩(l₁×0×Ｒ＾(n-1)) = φ. Construction of such Ψ_(u), 0 ≤ u ≤ 1, deserves called a Lagrangian Whitney trick. We will show that the condition is needed involving the areas of the regions Ｄ in divided by l_(0) and l₁for the construction.