Scalar curvature, mean curvature and harmonic maps to the circle

TL;DR

This paper investigates harmonic maps from 3-manifolds with boundary to the circle, establishing dihedral rigidity for certain cubes and exploring applications to hyperbolic 3-manifolds.

Contribution

It proves a specific case of dihedral rigidity for 3D cubes with right angles and applies the results to mapping torus hyperbolic 3-manifolds.

Findings

Proved dihedral rigidity for cubes with dihedral angles of π/2

Established properties of harmonic maps to the circle in 3-manifolds

Applied results to hyperbolic 3-manifold mappings

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.