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6. Liouville sectors

Some questions related to Liouville sectors and their symplectic invariants.
    1.     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 manifolds-with-convex-boundary 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 local-to-global criterion for verifying whether 1 is in the image of the open-closed map. But we don’t have any bounds on the action of the Hochschild class which hits 1.

                      Problem 6.5.

                      Is there a "local-to-global" principle for verifying "low-action 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 $\mathcal{W}(X)$?
                                        1. 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.