Flexible Weinstein manifolds

Summary

This survey on flexible Weinstein manifolds, which is essentially an extract from [Cieliebak and Eliashberg 2012], provides to an interested reader a shortcut to theorems on deformations of flexible Weinstein structures and their applications.

1. Introduction

The notion of a Weinstein manifold was introduced in [Eliashberg and Gromov 1991], formalizing the symplectic handlebody construction from AlanWeinstein's paper [1991] and the Stein handlebody construction from [Eliashberg 1990]. Since then, the notion of a Weinstein manifold has become one of the central notions in symplectic and contact topology. The existence question for Weinstein structures on manifolds of dimension > 4 was settled in [Eliashberg 1990]. The past five years have brought two major breakthroughs on the uniqueness question: From [McLean 2009] and other work we know that, on any manifold of dimension > 4 which admits a Weinstein structure, there exist infinitely many Weinstein structures that are pairwise nonhomotopic (but formally homotopic). On the other hand, Murphy's h-principle for loose Legendrian knots [Murphy 2012] has led to the notion of flexible Weinstein structures, which are unique up to homotopy in their formal class. In this survey, which is essentially an extract from [Cieliebak and Eliashberg 2012], we discuss this uniqueness result and some of its applications.

1A. Weinstein manifolds and cobordisms.

Definition. A Weinstein structure on an open manifold V is a triple (ω, X, ∅ )Where

• •(ω is a symplectic form on V ,

• •∅:V→ℝ is an exhausting generalized Morse function,