Ricci Curvature, Reeb Flows and Contact 3-Manifolds

TL;DR

This paper explores the realization of functions as Ricci curvatures of Reeb vector fields on contact 3-manifolds, using topological methods to understand the conditions and limitations of such realizations.

Contribution

It introduces a topological approach to realize admissible functions as Ricci curvatures of Reeb vector fields, highlighting the role of contact topology in resolving singularities.

Findings

Every admissible function can be realized as Ricci curvature of a singular metric.

Realization depends on contact topological data and involves resolving singularities.

The problem of resolving singularities remains partially understood.

Abstract

Given a contact 3-manifold we consider the problem of when a given function can be realized as the Ricci curvature of a Reeb vector field for the contact structure. We will use topological tools to show that every admissible function can be realized as such Ricci curvature for a singular metric which is an honest compatible metric away from a measure zero set. However, we will see that resolving such singularities depends on contact topological data and is yet to be fully understood.