6. Liouville sectors
Some questions related to Liouville sectors and their symplectic invariants.
Let $\bar{X}$ be a Liouville manifold, $F\subset \partial_\infty \bar{X}$ a Liouville hypersurface. Then the complement of a standard neighborhood of $F$ is a Liouville sector.
In the case where $F$ is a page of an open book decomposition of $\partial_\infty\bar{X}$, then $\partial_\infty X=F\times[0,1],$ which is fully stopped: it has a Reeb vector field with no closed orbits.Problem 6.1.
Are there any fully stopped compact contact manifoldswithconvexboundary other than $F\times[0,1]$? 
Suppose that instead of the Liouville hypersurface $F$, we are given only its core $\Lambda\subset\partial_\infty\bar{X}$.
Problem 6.2.
Can we recover $F$ from $\Lambda$? Can we do so uniquely? 
Problem 6.3.
Find a geometric description of the adjoint of the functor $\mathcal{W}(X)\to\mathcal{W}(X')$ for an inclusion $X\hookrightarrow X'$ of Liouville sectors. 
Problem 6.4.
Construct the embedding $\mathcal{W}(X)\otimes\mathcal{W}(X')\to\mathcal{W}(X\times X')$ for $X,X'$ Liouville sectors. 
Work of [arXiv:1706.03152] gives a localtoglobal criterion for verifying whether 1 is in the image of the openclosed map. But we donâ€™t have any bounds on the action of the Hochschild class which hits 1.
Problem 6.5.
Is there a "localtoglobal" principle for verifying "lowaction generation"? 
Problem 6.6.
What is the generation criterion for $\mathcal{W}(X)$, $X$ a Liouville sector? 
Problem 6.7.
Given a Liouville hypersurface $F\subset\partial_\infty \bar{X}$, is there a Lefschetz fibration $\pi:\bar{X}\to\mathbb{C}$ for which $F$ is a Liouville subdomain of the fiber of $\pi$? 
Objects of $\mathcal{W}(X)$ which are given by immersed Lagrangians sometimes become embedded after a stabilization to $\mathcal{W}(X\times T^*[0,1]).$ We would like to understand this process more systematically.
Problem 6.8.
Is there any precise sense in which $\text{colim}_n\mathcal{W}(X\times T^*B^n)$ is better behaved than $\mathcal{W}(X)$? 
Problem 6.9.
Suppose that $X$ is a Liouville manifold with smooth Lagrangian core $L$. Is $X$ equivalent to $T^*L$ as exact symplectic manifolds? Can we define the "fibers" as objects of $\mathcal{W}(X)$?
Remark. We know what the images of these fibers are in $\mathcal{W}(X\times T^*[0,1])$, but since $X$ is not necessarily Weinstein, this might not be equivalent to $\mathcal{W}(X)$.

Cite this as: AimPL: Arborealization of singularities of Lagrangian skeleta, available at http://aimpl.org/arborlagrange.