Scalar curvature and harmonic maps to $S^1$

TL;DR

This paper establishes a relationship between scalar curvature and the topology of level sets of harmonic maps from 3-manifolds to S^1, extending known characterizations of the Thurston norm and rigidity results.

Contribution

It introduces a new identity linking scalar curvature to Euler characteristics of level sets of harmonic maps, extending Thurston norm characterization and rigidity theorems.

Findings

Derived an integral identity relating scalar curvature and Euler characteristics.

Extended Kronheimer--Mrowka's Thurston norm characterization to broader 3-manifolds.

Reinforced known rigidity results for scalar curvature on specific 3-manifolds.

Abstract

For a harmonic map $u:M^3\to S^1$ on a closed, oriented $3$--manifold, we establish the identity $$2\pi \int_{\theta\in S^1}\chi(\Sigma_{\theta})\geq \frac{1}{2}\int_{\theta\in S^1}\int_{\Sigma_{\theta}}(|du|^{-2}|Hess(u)|^2+R_M)$$ relating the scalar curvature $R_M$ of $M$ to the average Euler characteristic of the level sets $\Sigma_{\theta}=u^{-1}\{\theta\}$. 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 $(\min R_M)sys_2(M)\leq 8\pi$, and the well--known result of Schoen and Yau that $T^3$ admits no metric of positive scalar curvature.