On the Greenfield-Wallach and Katok conjectures

TL;DR

This paper reviews recent advances on the Greenfield-Wallach and Katok conjectures concerning special vector fields, culminating in a proof for three-dimensional cases by linking the problem to Reeb flows and contact geometry.

Contribution

It provides a proof of the conjectures in three dimensions by reducing the problem to Reeb flows and applying the Weinstein conjecture.

Findings

Proof of the conjectures in dimension three.

Reduction to Reeb flow case using recent work.

Application of Weinstein conjecture to settle contact case.

Abstract

We survey recent progress on the Greenfield-Wallach and Katok conjectures on globally hypoelliptic and cohomology free vector fields and derive a proof of the conjectures in dimension three. The argument is primarily based on recent work of F. and J. Rodriguez Hertz which allows us to reduce the question to the case of a Reeb flow for a contact form. The contact case is settled by invoking the Weinstein conjecture (which has been recently announced by C. Taubes).