The tension equation with holomorphic coefficients, harmonic mappings and rigidity

The tension equation for a mapping is the non-linear second-order equation Solutions are "harmonic" mappings. Here, we give a complete description of the solution space of mappings of degree to this equation when is entire. Each solution is a quasiconformal surjection and when the set of normalized solutions is endowed with the Teichmuller metric, the solution space is isometric to the hyperbolic plane. More generally, for harmonic mappings between domains in , with defining a flat metric, we stablish a very strong maximum principle for the distortion - up to multiplicative factor , real and harmonic, the Beltrami coefficient of is quasiregular - and thus open and discrete when non-constant. This follows from the remarkable fact that the Beltrami coefficient of the inverse of a harmonic mapping itself satisfies a nonlinear homogeneous Beltrami equation.