On the Jacobian of the Douady-Earle extension

TL;DR

This paper studies the Jacobian of the Douady--Earle extension between hyperbolic surfaces, proving it equals one only for isometries and demonstrating sequences where the Jacobian can grow arbitrarily large.

Contribution

It characterizes the Jacobian of the Douady--Earle extension and constructs examples showing unbounded Jacobian behavior.

Findings

Jacobian equals one iff the extension is an isometry

Existence of sequences with unbounded Jacobian values

Provides insight into the geometric properties of the Douady--Earle extension

Abstract

Given an isotopy class between two closed hyperbolic surfaces, the Douady--Earle extension provides a unique analytic diffeomorphism representative. In this paper we investigate the Jacobian of the Douady--Earle extension map $F$. We prove that $|\operatorname{Jac} F| \equiv 1$ precisely when $F$ is an isometry. Moreover, we construct a sequence of hyperbolic surfaces $\{\Sigma_i\}$ together with a fixed domain surface $\Sigma_0$ for which the Douady--Earle extension maps $F_i:\Sigma_0\to\Sigma_i$ satisfy $\max_{x\in\Sigma_0} \operatorname{Jac} F_i \to +\infty$.