Liouville polarizations and the rigidity of their Lagrangian skeleta in dimension $4$

TL;DR

This paper introduces Liouville polarizations for open symplectic manifolds, leading to new results in symplectic embeddings, Lagrangian intersections, and Legendrian barriers, advancing understanding of symplectic and contact geometry in dimension 4.

Contribution

It presents a novel class of polarizations and demonstrates their applications in symplectic embedding problems, Lagrangian intersection theory, and contact geometry.

Findings

Solved a symplectic embedding question by Sackel-Song-Varolgunes-Zhu and Brendel.

Discovered new Lagrangian non-removable intersections at small scales.

Identified a new phenomenon of Legendrian barriers in contact geometry.

Abstract

The main theme of this paper is the introduction of a new type of polarizations, suited for some open symplectic manifolds, and their applications. These applications include symplectic embedding results that answer a question by Sackel-Song-Varolgunes-Zhu and Brendel, new Lagrangian non-removable intersections at small scales, and a novel phenomenon of Legendrian barriers in contact geometry.