Scalar curvature and circle-valued harmonic maps

We introduce a new tool for relating the scalar curvature of a Riemannian manifold to its global geometry and topology, based on the study of level sets of harmonic functions and harmonic maps to the circle. We will explain how these ideas lead to simple new proofs and improvements upon some well-known results in three-manifold geometry and general relativity, previously studied primarily via minimal surface and Dirac operator methods.