Scalar curvature and harmonic one-forms on three-manifolds with boundary

For a homotopically energy-minimizing map u : N-3 -> S(1 )on a compact, oriented 3-manifold N with boundary, we establish an identity relating the average Euler characteristic of the level sets u(-1){theta} to the scalar curvature of N and the mean curvature of the boundary partial derivative N. As an application, we obtain some natural geometric estimates for the Thurston norm on 3-manifolds with boundary, generalizing results of Kronheimer-Mrowka and the second named author from the closed setting. By combining these techniques with results from minimal surface theory, we obtain moreover a characterization of the Thurston norm via scalar curvature and the harmonic norm for general closed, oriented three-manifolds, extending Kronheimer and Mrowka's characterization for irreducible manifolds to arbitrary topologies.