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Weinstein's neighborhood theorem says that every Lagrangian has a standard neighborhood. The more precise statement goes like this.



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> **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 $L \subset T^\*L$.
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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.



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> **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 $L \subset T^\*L$.
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**Remark On Proof Of Theorem 1:** The basic result that the usual Lagrangian neighborhood theorem depends on is the following lemma (see [[1]](https://math.barnard.edu/sites/default/files/utahnotaug28_0.pdf) or McDuff-Salamon).



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> **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 $s \in S$.
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> 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 $\varphi^\*\omega\_1 = \omega\_0$.
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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.



Weinstein neighborhood theorem for Lagrangians with Legendrian boundary
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.



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> **Theorem:** (Weinstein Neighborhood With Boundary) Let $(X,\omega)$ be a symplectic manifold with boundary $\partial X$ and let $L \subset X$ be a properly embedded, Lagrangain sub-manifold with boundary $\partial L \subset \partial X$ transverse to $T(\partial X)^\omega$.
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> Then there exists a neighborhood $U \subset T^\*L$ of $L$ (as the zero section), a neighborhood $V \subset X$ of $L$ and a diffeomorphism $f:U \simeq V$ such that $\varphi^\*(\omega|\_V) = \omega\_{\text{std}}|\_U$.
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**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.



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> **Lemma 1:** (Fiber Integration With Boundary) Let $X$ be a compact manifold with boundary, $\pi:E \to X$ be a rank $k$ vector-bundle with metric and $\pi:U \to X$ be the closed disk bundle of $E$. Let $\kappa \subset T(\partial U)$ be a distribution on $\partial U$ invariant under fiber-wise scaling. Finally, suppose that $\tau \in \Omega^{k+1}(U)$ is a $(k+1)$-form such that: 
>  \begin{equation} \label{eqn:fiber\_integration\_sigma} d\tau = 0 \qquad \tau|\_X = 0 \qquad (\iota^\*\_{\partial X}\tau)|\_\kappa = 0\end{equation}
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> Then there exists a $k$-form $\sigma \in \Omega^k(U)$ with the following properties.
>  \begin{equation} \label{eqn:fiber\_integration\_tau} d\sigma = \tau \qquad \sigma|\_X = 0 \qquad (\iota^\*\_{\partial X}\sigma)|\_\kappa = 0 \end{equation}
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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.



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> **Lemma 2:** Let $U$ be a manifold and $L \subset U$ be a closed sub-manifold. Let $\kappa\_0,\kappa\_1$ be rank $1$ orientable distributions in $TU$ such that $\kappa\_i|\_L \cap TL = \{0\}$ and $\kappa\_0|\_L = \kappa\_1|\_L$. 
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> Then there exists a neighborhood $U' \subset U$ of $L$ and a family of smooth embeddings $\psi:U' s\times I \to U$ with the following four properties.
>  $$
> \psi\_t|\_{\partial L} = \text{Id} \qquad d(\psi\_t)\_u = \text{Id} \text{ for }u \in L \qquad \psi\_0 = \text{Id} \qquad [\psi\_1]\_\*(\kappa\_0) = \kappa\_1
> $$
>  Furthermore, we can take $\psi\_t$ to be $t$-independent for $t$ near $0$ and $1$.
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The proof of Lemma 2 is straight-forward.


