  The tension equation for a mapping $f:{\mathbb C}\to {\mathbb C}$ is the
nonlinear second order equation \[ \Delta f +\varphi(f) f_z f_{\bar z} = 0\]
Solutions are "harmonic" mappings. Here we give a complete description of the
solution space of mappings of degree 1 to this equation when $\varphi$ is
entire. Each solution is a quasiconformal surjection and when the set of
normalised solutions is endowed with the Teichm\"uller metric, the solution
space is isometric to the hyperbolic plane. More generally, for harmonic
mappings $f:\Omega \to (\tilde{\Omega},\rho)$ between domains in ${\mathbb C}$,
with $\rho(w)|dw|$ defining a flat metric we stablish a very strong maximum
principle for the distortion - up to multiplicative factor $e^{iv}$, $v$ real
and harmonic, the Beltrami coefficient of $f^{-1}$ is quasiregular - and thus
open and discrete when nonconstant. This follows from the remarkable fact that
the Beltrami coefficient of the inverse of a harmonic mapping itself satisfies
a nonlinear homogeneous Beltrami equation.
