MathOverflow Asked by Julian Chaidez on November 3, 2021
$require{AMScd}$
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 $dlambda$ 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 $dsigma = omega_1 – omega_0$ and then you integrate the vector-field $Z_t$ satisfying:
$$iota_{Z_t}omega_t = -sigmaquadtext{where}quadomega_t = (1-t)omega_0 + tomega_1$$
This gives you a family of diffeomorphisms with $varphi^*_tomega_t = omega_0$ 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 $T(partial U)^{dlambda}$ and $T(partial V)^{dlambda}$ 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.
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: 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.
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} dtau = 0 qquad tau|_X = 0 qquad (iota^*_{partial X}tau)|_kappa = 0end{equation}
Then there exists a $k$-form $sigma in Omega^k(U)$ with the following properties. begin{equation} label{eqn:fiber_integration_tau} dsigma = tau qquad sigma|_X = 0 qquad (iota^*_{partial X}sigma)|_kappa = 0 end{equation}
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 $psi:U' stimes I to U$ 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.
Answered by Julian Chaidez on November 3, 2021
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