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

We obtain the evolution equations for the Riemann tensor, the Ricci tensor and the scalar curvature induced by the mean curvature flow. The evolution for the scalar curvature is similar to the Ricci flow, however, negative, rather than positive, curvature is preserved. Our results are valid in any dimension.

We consider constant mean curvature surfaces with finite topology, properly embedded in three-space in the sense of Alexandrov. Such surfaces with three ends and genus zero were constructed and completely classified by the authors [GKS2, GKS1]. Here we extend the arguments to the case of an arbitrary number of ends, under the assumption that the asymptotic axes of the ends lie in a common plane...