Maximal contact and symplectic structures

TL;DR

This paper develops a method for constructing maximal Weinstein and contact structures by gluing along subdomains, leading to new exotic symplectic and contact manifolds with diverse properties.

Contribution

It introduces a gluing procedure for Weinstein domains and contact structures, enabling the creation of maximal structures and new exotic examples in high-dimensional symplectic and contact geometry.

Findings

Constructed maximal Weinstein domains containing given Weinstein subdomains.

Produced exotic cotangent bundles with many Lagrangians that are formally but not Hamiltonian isotopic.

Extended results on symplectic caps and contact embeddings to high dimensions.

Abstract

We introduce a procedure for gluing Weinstein domains along Weinstein subdomains. By gluing along flexible subdomains, we show that any finite collection of high-dimensional Weinstein domains with the same topology are Weinstein subdomains of a `maximal' Weinstein domain also with the same topology. As an application, we produce exotic cotangent bundles containing many closed regular Lagrangians that are formally Lagrangian isotopic but not Hamiltonian isotopic and also give a new construction of exotic Weinstein structures on Euclidean space. We describe a similar construction in the contact setting which we use to produce `maximal' contact structures and extend several existing results in low-dimensional contact geometry to high-dimensions. We prove that all contact manifolds have symplectic caps, introduce a general procedure for producing contact manifolds with many Weinstein fillings, and give a new proof of the existence of codimension two contact embeddings.