On the Jacobian of the Douady-Earle extension

2025/12/17 by Connell, Chris, Ruan, Yuping, Wang, Shi
Mathematics · #Analytic and geometric function theory #Holomorphic and Operator Theory #Geometric and Algebraic Topology

paper · doi:10.48550/arxiv.2512.15032

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 |Jac F| ≡ 1 precisely when F is an isometry. Moreover, we construct a sequence of hyperbolic surfaces \Σ_i\ together with a fixed domain surface Σ₀ for which the Douady--Earle extension maps F_i:Σ₀→Σ_i satisfy max_x∈Σ0 Jac F_i → +∞.

Citations

• The Cayley hyperbolic space and volume entropy rigidity
• Canonical Metrics on the Moduli Space of Riemann Surfaces I
• The Degree Theorem in higher rank
• Minimal stretch maps between hyperbolic surfaces

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• Hyperbolic isometries of the fine curve graph of higher genus surfaces
• Diffeomorphismes harmoniques du plan hyperbolique
• Ahlfors-Weill Extensions for a Class of Minimal Surfaces
• Guiding Isotopies and Holomorphic Motions