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Weinstein's neighborhood theorem says that every Lagrangian has a standard neighborhood. The more precise statement goes like this.
Theorem 1: (Lagrangian Neighborhood Theorem) Let $(X,\omega)$ be a symplectic manifold and $L \subset X$ be a closed Lagrangian. Then there exists a neighborhood $U$ of $L$ in $X$ and a symplectomorphism $\varphi:U \simeq V \subset T^*L$ taking $L$ identically to the zero-section $L \subset T^*L$.
Now let $(W,\lambda)$ be a Liouville domain. That is, $W$ is a compact manifold with boundary, and $\lambda$ is a $1$-form on $W$ such that $d\lambda$ is symplectic and $\lambda|_{\partial W}$ is a contact form. Furthermore, let $L \subset W$ be a compact Lagrangian sub-manifold with Legendrian boundary $\partial L \subset \partial W$.
My question is whether the following version of the neighborhood theorem holds in this setting. It seems to me that if it is true, then it should be standard, but I can't find a reference.
Theorem 2 (Maybe?): There exists a neighborhood $U$ of $L$ in $W$ and a symplectomorphism of manifolds with boundary $\varphi:U \simeq V \subset T^*L$ taking $L$ identically to the zero-section $L \subset T^*L$.
Remark On Proof Of Theorem 1: The basic result that the usual Lagrangian neighborhood theorem depends on is the following lemma (see [1] or McDuff-Salamon).
Lemma: Let $X$ be a manifold with closed sub-manifold $S \subset X$, and let $\omega_0, \omega_1$ be two symplectic forms on $X$. Suppose that $\omega_0 = \omega_1$ on the fiber $T_sX$ for any $s \in S$.
Then there exists neighborhoods $N_0$ and $N_1$ of $S$ and a symplectomorphism $\varphi:N_0 \to N_1$ with $\varphi|_S = \text{Id}$ and $\varphi^*\omega_1 = \omega_0$.
The proof is a version of the usual Moser trick. You find a $1$-form $\sigma$ in a neighborhood with $d\sigma = \omega_1 – \omega_0$ and then you integrate the vector-field $Z_t$ satisfying:
$$\iota_{Z_t}\omega_t = -\sigma\quad\text{where}\quad\omega_t = (1-t)\omega_0 + t\omega_1$$
This gives you a family of diffeomorphisms with $\varphi^*_t\omega_t = \omega_0$ and you're done. If you try to run this proof on a sub-manifold $S \subset X$ with $\partial S \subset \partial X$, you run into the issue that $Z_t$ needs to be parallel to the boundary $\partial X$ in order for the flow to be well-defined. If I'm not mistaken, the criterion for this to be the case is:
$$ T(\partial X)^{\omega_t} \subset \ker(\sigma) \text{ on }\partial X $$
Here $T(\partial X)^{\omega_t}$ is the characteristic foliation on $\partial X$ with respect to $\omega_t$, i.e. the symplectic perp to the tangent space to $\partial X$. It isn't clear to me that you can even accomplish the above inclusion for $\sigma$, or that you can upgrade $\sigma$ to a family $\sigma_t$ with this property.
Speculation On Validity Of Theorem 2: On a conceptual level, I can't decide whether or not Theorem 2 is too optimistic. Here is what makes me skeptical about it.
Theorem 2 would imply not only that the boundaries $\partial U \simeq \partial V$ of $U$ and $V$ were contactomorphic, but also that the characteristic foliations $T(\partial U)^{d\lambda}$ and $T(\partial V)^{d\lambda}$ on $U$ and $V$ near $\partial L$ were the same. The characteristic foliation of a contact hypersurface is generally very sensitive to the embedding of said hypersurface, and from that perspective a standard neighborhood in the vein of Theorem 2 would be a bit surprising.
I haven't pursued this idea enough to produce a counter-example unfortunately.
Best Answer
Theorem 2 is true, verbatim. I will give an outline of the proof here, since the details make it kind of long. If you would like a detailed write-up and you don't want to do it yourself, DM or email me.
The actual statement that is true is more general: you do not need the boundary $\partial X$ to be contact or for the boundary $\partial L$ to be Legendrian.
Proof: The proof has two steps. First, we construct neighborhoods $U \subset T^*L$ and $V \subset X$ of $L$, and a diffeomorphism $\varphi:U \simeq V$ such that: \begin{equation} \varphi|_L = \text{Id} \qquad \varphi^*(\omega|_V)|_L = \omega_{\text{std}}|_L \qquad T(\partial U)^{\omega_{\text{std}}} = T(\partial U)^{\varphi^*\omega} \end{equation} Here $T(\partial U)^{\omega_{\text{std}}} \subset T(\partial U)$ is the symplectic perpendicular to $T(\partial U)$ with respect to $\omega_{\text{std}}$ (and similarly for $T(\partial U)^{\varphi^*\omega}$. Second, we apply Lemma 1 (below) and a Moser-type argument to conclude the result.
For the first part, the proof proceeds like this. First you pick a metric on $L$ and use the exponential map in the usual way, to get a diffeomorphism $\varphi:U \simeq V$ with $U \subset T^*L$, $V \subset X$ and $\varphi^*\omega_{\text{std}} = \omega$ along $L$. Then we use Lemma 2 below to modify $\varphi$ in a collar neighborhood of $\partial U$ to satisdy $T(\partial U)^{\omega_{\text{std}}} = T(\partial U)^{\varphi^*\omega}$.
The second part is basically identical to the usual Moser argument.
The proof of Lemma 1 just involves examining the proof of the version of the Poincare Lemma in McDuff-Salamon, and checking that the primitive constructed there satisfies the 3rd property. Note that to apply this lemma, you need to show that the characteristic foliation of $\partial(T^*L) \subset T^*L$ is invariant under fiber-scaling, but this is a quite easy Lemma.
The proof of Lemma 2 is straight-forward.