Scalar curvature and harmonic maps to $S^1$

Abstract

For a harmonic map $u:M^3\to S^1$ on a closed, oriented $3$--manifold, we establish the identity $$2π\int_{θ\in S^1}χ(Σ_θ)\geq \frac{1}{2}\int_{θ\in S^1}\int_{Σ_θ}(|du|^{-2}|Hess(u)|^2+R_M)$$ relating the scalar curvature $R_M$ 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 $H_2(M;\mathbb{Z})$ in terms of $\|R_M^-\|_{L^2}$ 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 $(\min R_M)sys_2(M)\leq 8π$, and the well--known result of Schoen and Yau that $T^3$ admits no metric of positive scalar curvature.