# From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifolds

@inproceedings{Cieliebak2012FromST, title={From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifolds}, author={Kai Cieliebak and Yakov M. Eliashberg}, year={2012} }

A beautiful and comprehensive introduction to this important field. --Dusa McDuff, Barnard College, Columbia University This excellent book gives a detailed, clear, and wonderfully written treatment of the interplay between the world of Stein manifolds and the more topological and flexible world of Weinstein manifolds. Devoted to this subject with a long history, the book serves as a super introduction to this area and also contains the authors' new results. --Tomasz Mrowka, MIT This book is…

## 289 Citations

Topological Methods in Stein Geometry

- Mathematics
- 2011

In this chapter we introduce more advanced topological methods to the study of geometric problems on Stein manifolds and to Oka theory. We begin by considering complex points of smooth real surfaces…

Topology of symplectic fillings of contact 3-manifolds

- Mathematics
- 2014

These are some lecture notes for my mini-course in the Gradu ate Workshop on 4-Manifolds, August 18-22, 2014 at the Simons Center for G eometry and Physics. Most of this manuscript is an adaptation…

Maximal contact and symplectic structures

- Mathematics, Computer ScienceJournal of Topology
- 2020

It is proved that all contact manifolds have symplectic caps, a general procedure for producing contact manifold with many Weinstein fillings is introduced, and a new proof of the existence of codimension two contact embeddings is given.

An algebraic approach to the algebraic Weinstein conjecture

- MathematicsJournal of Fixed Point Theory and Applications
- 2022

How does one measure the failure of Hochschild homology to commute with colimits? Here I relate this question to a major open problem about dynamics in contact manifolds — the assertion that Reeb…

On symplectic fillings of spinal open book decompositions I: Geometric constructions

- Mathematics
- 2018

A spinal open book decomposition on a contact manifold is a generalization of a supporting open book which exists naturally e.g. on the boundary of a symplectic filling with a Lefschetz fibration…

Stein Structures: Existence and Flexibility

- Mathematics
- 2014

This survey on the topology of Stein manifolds is an extract from the book of Cieliebak and Eliashberg (From Stein to Weinstein and Back—Symplectic Geometry of Affine Complex Manifolds, Colloquium…

On symplectic fillings of spinal open book decompositions II: Holomorphic curves and classification.

- Mathematics
- 2020

In this second paper of a two-part series, we prove that whenever a contact 3-manifold admits a uniform spinal open book decomposition with planar pages, its (weak, strong and/or exact) symplectic…

Non-Exact Symplectic Cobordisms Between Contact 3-Manifolds

- Mathematics
- 2010

We show that the pre-order defined on the category of contact manifolds by arbitrary symplectic cobordisms is considerably less rigid than its counterparts for exact or Stein cobordisms: in…

On the combinatorics of exact Lagrangian surfaces

- Mathematics
- 2016

We study Weinstein 4-manifolds which admit Lagrangian skeleta given by attaching disks to a surface along a collection of simple closed curves. In terms of the curves describing one such skeleton, we…

Geometric and algebraic presentations of Weinstein domains

- Mathematics
- 2019

We prove that geometric intersections between Weinstein handles induce algebraic relations in the wrapped Fukaya category, which we use to study the Grothendieck group. We produce a surjective map…