Effects of geometric constraints on a steady flow potential are described by an elliptic-hyperbolic generalization of the harmonic map equations. Sufficient conditions are given for global triviality. MSC2000 : 58E20, 58E99, 75N10.

We give results on the following questions about a topologically tame hyperbolic 3-manifold M : 1. Does M have nonzero L-harmonic 1-forms? 2. Does zero lie in the spectrum of the Laplacian acting on Λ(M)/Ker(d)?

We prove that a quasiconformal map of the sphere S admits a harmonic quasi-isometric extension to the hyperbolic space H, thus confirming the well known Schoen Conjecture in dimension 3.

Let (M, g) be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show thatM is a hyperbolic manifold of constant sectional curvature −h 2 4 , provided M is asymptotically harmonic of constant h > 0.

We study the Gauss map of minimal surfaces in the Heisenberg group Nil3 endowed with a left-invariant Riemannian metric. We prove that the Gauss map of a nowhere vertical minimal surface is harmonic into the hyperbolic plane H. Conversely, any nowhere antiholomorphic harmonic map into H is the Gauss map of a nowhere vertical minimal surface. Finally, we study the image of the Gauss map of compl...

We show that all harmonic morphisms from 3-dimensional Minkowski space with values in a surface have a Weierstrass representation involving the complex numbers or the hyperbolic numbers depending on the signature of the codomain. We deduce that there is a nontrivial globally defined submersive harmonic morphism from Minkowski 3-space to a surface, in contrast to the Riemannian case. We show tha...