Scalar curvature and harmonic maps to S¹

For a harmonic map u : M³ → S¹ on a closed, oriented $3$-manifold, we establish the identity \[ 2 π ∫θ ∈ S^1 χ (Σ _θ) ≥ 1/2 ∫θ ∈ S^1 ∫Σ_θ ( { du }⁻² { Hess (u) }^2 + R_M ) \] relating the scalar curvature RM of M to the average Euler characteristic of the level sets Σ_θ = u⁻¹ θ. As our primary application, we extend the Kronheimer–Mrowka characterization of the Thurston norm on H₂ (M; Z) in terms of R⁻M L² and the harmonic norm to any closed $3$-manifold containing no nonseparating spheres. Additional corollaries include the Bray–Brendle–Neves rigidity theorem for the systolic inequality (min RM) sys₂ (M) ≤ 8π, and the well-known result of Schoen and Yau that T³ admits no metric of positive scalar curvature.