Geometry of Liouville sectors and the maximum principle

TL;DR

This paper develops a new framework of Floer data for Liouville sectors, enabling the use of the maximum principle to analyze pseudoholomorphic curves and related structures in symplectic geometry.

Contribution

It introduces sectorial almost complex structures and Hamiltonians compatible with the maximum principle, answering a key question about confinement results in wrapped Fukaya categories and symplectic cohomology.

Findings

Established a new class of Floer data compatible with the maximum principle.

Proved existence of a pseudoconvex pair $(ta,J)$ with specific taming and exhaustion properties.

Unified the analysis of wrapped Fukaya categories and symplectic cohomology without estimates.

Abstract

We introduce a new package of Floer data of $\lambda$-sectorial almost complex structures $J$ and sectorial Hamiltonians $H$ on the Liouville sectors introduced by Ganatra-Pardon-Shende the pairs of which are amenable to the maximum principle for the analysis of pseudoholomorphic curves relevant to the studies of wrapped Fukaya categories and of symplectic cohomology. It is also amenable to the strong maximum principle in addition when paired with cylindrical Lagrangian boundary conditions. The present work answers to a question raised by Ganatra-Pardon-Shende concerning a characterization of almost complex structures and Hamiltonians in that all the relevant confinement results in the studies of wrapped Fukaya category, symplectic cohomology and closed-open (and open-closed) maps between them can be uniformly established via the maximum principle through tensorial calculations, Hamiltonian calculus and sign considerations without making any estimates. Along the way, we prove the existence of a pseudoconvex pair $(\psi,J)$ such that \emph{$J$ is $d\lambda$-tame and $\psi$ is an exhaustion function of $\operatorname{nbhd}(\partial_\infty M \cup \partial M)$ that also satisfies the equation $-d\psi \circ J = \lambda$ thereon for any Liouville sector with corners $(M,\lambda)$}.