Ideal Liouville Domains - a cool gadget

TL;DR

This paper introduces ideal Liouville domains, a refined concept in symplectic geometry that simplifies the handling of Liouville forms and non-compactness issues, enhancing their applicability in contact geometry.

Contribution

It defines ideal Liouville domains, explores their fundamental properties, and demonstrates their significance in contact geometry, streamlining previous complexities.

Findings

Simplifies the structure of Liouville domains

Provides a new framework for contact geometry applications

Reduces technical difficulties related to non-compactness

Abstract

Liouville domains have become central objects in symplectic and contact geometry. However, the auxiliary data they involve --- namely, Liouville forms --- and the non-compactness of their completions generate some inconvenience. The notion of ideal Liouville domains is designed to suppress these awkward aspects and to let symplectic structures play the leading role. Ideal Liouville domains are compact manifolds with boundary whose interior carries a symplectic form satisfying some tameness condition along the boundary. Their definition and their basic properties are presented in the first part of these notes, while the second part discusses their relevance in contact geometry.