Ricci Curvature, Reeb Flows and Contact 3-Manifolds

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