Suppose I have a smooth map $u: \mathbb{R}\times [0,1]\rightarrow TP$ , where $P$ is a symplectic manifold with an almosct complex compactible structure , consider $k>1$, such that $u\in L^2_{k}(\mathbb{R}\times [0,1],P)$ and $u(\tau,i)\subset L_i$ for $i=0,1$ where $L_i$ are lagrangian submanifolds , and that $\lim_{\tau\rightarrow \infty}u(\tau,t)=x\in L_1\cap L_0$ and $\lim_{\tau\rightarrow -\infty}u(\tau,t)=y\in L_1\cap L_0$. Now consider $\xi \in L_{k}^2(\mathbb{R}\times [0,1],P)$ vector fields along $u$ and $\eta \in L_{k-1}^2(\mathbb{R}\times [0,1],P)$
Will it in fact be true that $\int_{0}^1 \langle \xi,\eta\rangle|_{-\infty}^{\infty}dt=0 ?$
I have tried using sobolev inequalities here and see if this would force me to have that $\lim_{\tau \rightarrow \infty}\xi=0$ but I couldn't manage to get anything useful.
If anyone knows another approach to this I would appreciate. Thanks in advance.