Scalar curvature, mean curvature and harmonic maps to the circle

Abstract

We study harmonic maps from a 3-manifold with boundary to ${\mathbb{S}}^1$ and prove a special case of dihedral rigidity of three dimensional cubes whose dihedral angles are $\pi /2$. Furthermore we give some applications to mapping torus hyperbolic 3-manifolds.

13 pages, 2 figures; comments welcome