On the Jacobian 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 $|\mathrm{Jac}F|\equiv 1$ 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 ${\max }_{x\in {\Sigma }_0}\mathrm{Jac}{F}_i\to +\infty$.

11 pages, 4 figures, comments are welcome