# Weinstein neighborhood theorem for Lagrangians with Legendrian boundary

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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.

• If there is a symplectic cap, or a "symplectic collar", to $W$ in which the Lagrangian $L$ extends as a Lagrangian (now with the original Legendrian $\partial L$ as a hypersurface), you could just take the restriction of the Weinstein tubular neighborhood. $\leftarrow$ Might be something stupid. Oct 30, 2018 at 16:31
• @ChrisGerig the issue with this approach is that the Weinstein neighborhood map $\varphi:U' \to TL'$ of the closed extension $L'$ into the capped symplectic manifold $W'$ may not send $U' \cap W$ to $T^*L \subset T^*L'$. In other words, the domain of the restricted chart may not be correct. Nov 6, 2018 at 19:53

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.

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 $$T(\partial X)^\omega$$.

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 $$\varphi^*(\omega|_V) = \omega_{\text{std}}|_U$$.

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: $$$$\varphi|_L = \text{Id} \qquad \varphi^*(\omega|_V)|_L = \omega_{\text{std}}|_L \qquad T(\partial U)^{\omega_{\text{std}}} = T(\partial U)^{\varphi^*\omega}$$$$ Here $$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.

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: $$$$\label{eqn:fiber_integration_sigma} d\tau = 0 \qquad \tau|_X = 0 \qquad (\iota^*_{\partial X}\tau)|_\kappa = 0$$$$

Then there exists a $$k$$-form $$\sigma \in \Omega^k(U)$$ with the following properties. $$$$\label{eqn:fiber_integration_tau} d\sigma = \tau \qquad \sigma|_X = 0 \qquad (\iota^*_{\partial X}\sigma)|_\kappa = 0$$$$

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.

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 $$\kappa_0|_L = \kappa_1|_L$$.

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 $$1$$.

The proof of Lemma 2 is straight-forward.