The exponential Teichm\"uller theory: Ahlfors--Hopf differentials and diffeomorphisms

We consider minimisers of the $p$-exponential conformal energy for homeomorphisms $f:R \to S$ of finite distortion $\IK(z,f)$ between analytically finite Riemann surfaces in a fixed homotopy class $[f_0]$,\[ \mE_p(f:R,S)=\int_R \exp(p\IK(z,f))\; d\sigma(z). \] Homeomorphic minimisers exist should the barrier be a homeomorphism of finite energy, $\mE_p(f_0,R,S)<\infty$. In general this problem is not variational, however the Euler-Lagrange equations show the inverses $h=f^{-1}$ of sufficiently regular stationary solutions have an associated holomorphic quadratic differential -- the Ahlfors-Hopf differential, \[\Phi=\exp(p\IK(w,h))\,h_w\overline{h_\wbar}\,d\sigma_R(h). \] From the Riemann-Roch theorem and an approximation technique, we show the variational equations hold for extremal mappings. We take this as a starting point for higher regularity to show that if $h:\Omega\to\tilde{\Omega}$ is a Sobolev homeomorphism between planar domains with holomorphic Ahlfors-Hopf differential, then $h$ is a diffeomorphism. It will follow that $h$ is harmonic in a metric induced by its own (smooth) distortion. We develop equations for the Beltrami coefficient of $h$, establishing a connection between degenerate elliptic non-linear Beltrami equations and these harmonic mappings. On the surface we conclude that minimisers $f_p\in [f_0]$ of $\mE_p(f:R,S)$ are diffeomorphisms and are unique stationary points. This now links two different approaches to Teichm\"uller theory; the classical theory of extremal quasiconformal maps and the harmonic mapping theory. As $p\to\infty$ we show $f_p\to f_\infty$ to recover the unique extremal quasiconformal mapping . This extremal quasiconformal mapping is not a diffeomorphism (unless it is conformal) and $f_p$ degenerates on a divisor. As $p\to0$ we recover the harmonic diffeomorphism in $[f_0]$ and Shoen-Yau's results.