On the Jacobian of the Douady-Earle extension

Chris Connell; Yuping Ruan; Shi Wang

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arXiv · 2512.15032

On the Jacobian of the Douady-Earle extension

Chris Connell·Yuping Ruan·Shi Wang

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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 $\max_{x\in\Sigma_0} \operatorname{Jac} F_i \to +\infty$.

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Chris Connell, Yuping Ruan, Shi Wang. 2025-12-17. On the Jacobian of the Douady-Earle extension. https://arxiv.org/abs/2512.15032

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