Deformations of Annuli on Riemann surfaces with Smallest Mean Distortion

TL;DR

This paper investigates the deformation of annuli on Riemann surfaces with small mean distortion, extending the Nitsche conjecture to a broader class of metrics and identifying extremal harmonic maps that minimize distortion.

Contribution

It formulates the $ ho$-Nitsche conjecture for $ ho$-harmonic maps, determines extremal maps with minimal mean distortion, and shows these are Dirichlet minimizers within certain conditions.

Findings

$ ho$-Nitsche harmonic maps minimize mean distortion.

Extremal mappings are Dirichlet minimizers within the $ ho$-Nitsche condition.

Outside the $ ho$-Nitsche range, no energy minimizers exist among homeomorphisms.

Abstract

Let $A$ and $A'$ be two circular annuli and let $\rho$ be a radial metric defined in the annulus $A'$. Consider the class $\mathcal H_\rho$ of $\rho-$harmonic mappings between $A$ and $A'$. It is proved recently by Iwaniec, Kovalev and Onninen that, if $\rho=1$ (i.e. if $\rho$ is Euclidean metric) then $\mathcal H_\rho$ is not empty if and only if there holds the Nitsche condition (and thus is proved the J. C. C. Nitsche conjecture). In this paper we formulate an condition (which we call $\rho-$Nitsche conjecture) with corresponds to $\mathcal H_\rho$ and define $\rho-$Nitsche harmonic maps. We determine the extremal mappings with smallest mean distortion for mappings of annuli w.r. to the metric $\rho$. As a corollary, we find that $\rho-$Nitsche harmonic maps are Dirichlet minimizers among all homeomorphisms $h:A\to A'$. However, outside the $\rho$-Nitsche condition of the modulus of the annuli, within the class of homeomorphisms, no such energy minimizers exist. % However, %outside the $\rho-$Nitsche range of the modulus of the annuli, %within the class of homeomorphisms, no such energy minimizers exist. This extends some recent results of Astala, Iwaniec and Martin (ARMA, 2010) where it is considered the case $\rho=1$ and $\rho=1/|z|$.