Reeb components of leafwise complex foliations and their symmetries I

TL;DR

This paper reviews the Hopf construction of Reeb components with leafwise complex structures and investigates their automorphism groups, revealing an infinite-dimensional structure in certain cases.

Contribution

It provides a detailed analysis of the automorphism groups of Reeb components with leafwise complex structures, especially in complex dimension one, highlighting their infinite-dimensional aspects.

Findings

Automorphism groups contain infinite-dimensional vector spaces.

The Hopf construction is central to understanding Reeb components.

Automorphisms are characterized for specific types of Reeb components.

Abstract

We review the standard Hopf construction of Reeb components with leafwise complex structure and almost determine the group of leafwise holomorphic smooth automorphisms for Reeb components of certain type in the case of complex leaf dimension one. In particular, it contains an infinite dimensional vector space.