$L^p$-Extremal Teichm\"uller mappings between Riemann surfaces are diffeomorphisms

Abstract

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 \[ \mathsf{E}_p(f:R,S)=\int_R \IK^p(z,f)\; d\sigma_R(z). \] The problem was first raised by Ahlfors in his celebrated proof of Teichm\"uller's theorem-the case $p=\infty$, but the existence, topological regularity and analytic regularity of these $L^p$ minimisers remained unknown for all $1<p<\infty$. Ahlfors established weak existence for $p\geq 2$. Here we prove that for all p, $1\leq p<\infty$, such minimisers exist, are unique and are diffeomorphisms. They are quasiconformal but not diffeomorphic at $p=\infty$.