$L^p$-Extremal Teichmüller mappings between Riemann surfaces are diffeomorphisms

We consider minimisers in the homotopy class of a homeomorphism $f_0:R\to S$ between analytically finite Riemann surfaces with minimal $L^p$- conformal energy $${\mathsf{E}}_p(f:R,S)={\int }_R{\text{IK}}^p(z,f)d{\sigma }_R(z)\mathrm{.}$$ The problem was first raised by Ahlfors in his celebrated proof of Teichmüller’s theorem-the case $p=\mathrm{\infty }$, but the existence, topological regularity and analytic regularity of these $L^p$ minimisers remained unknown for all $1<p<\mathrm{\infty }$. Ahlfors established weak existence for $p\ge 2$. Here we prove that for all p, $1\le p<\mathrm{\infty }$, such minimisers exist, are unique and are diffeomorphisms. They are quasiconformal but not diffeomorphic at $p=\mathrm{\infty }$.