Diffeomorphic solutions of Ahlfors-Hopf equations

2025/10/22 by Gaven Martin, Cong Yao, Martin, Gaven +1
Materials Science · Mathematics · #30C62 31A05 49J10 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Nonlinear Partial Differential Equations #Quasicrystal Structures and Properties

paper · pdf · doi:10.48550/arxiv.2510.19375

openalex publication_date 2025/10/22 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

Here we advance the study of boundary the value problem for extremal functions of mean distortion and the associated Teichmüller spaces interpolating between the classical examples of extremal quasiconformal mappings, and the more recent approach through harmonic mappings (of extreme Dirichlet energy). In this paper we focus on the Alhfors-Hopf differential Φ=A(\mathbbK(w,h))h_w h_w η(h), where h=f^-1 is the pseudo-inverse of an extremal mapping f for the problem inf_f:\mathbbD→\mathbbD∫_\mathbbD A(\mathbbK(z,f)) dz, \mathbbK(z,f) = \frac|f_z|²+|f_z|²|f_z|²-|f_z|². where the infimum is taken over those homeomorphisms of finite distortion f:\mathbbD→\mathbbD with f|\mathbbS=f₀, typically a quasisymmetric barrier function. The inner-variational equations, an analogue of the Euler-Lagrange equations, show Φ is holomorphic at an extremal. Exploiting this Ahlfors-Hopf differential, we prove that an extreme point f is a local diffeomorphism in \mathbbD, resolving some conjectures in [16].

Citations

• The exponential Teichmüller theory: Ahlfors--Hopf differentials and diffeomorphisms
• On the uniqueness of extremal mappings of finite distortion
• The Teichmüller problem for L^p-means of distortion
• Higher regularity and uniqueness for inner variational equations
• The L^p Teichmüller theory: Existence and regularity of critical points
• Non-variational extrema of exponential Teichmüller spaces
• Monotone Hopf-Harmonics

Related

• The exponential Teichmüller theory: Ahlfors--Hopf differentials and diffeomorphisms
• On the uniqueness of extremal mappings of finite distortion
• The L^p Teichmüller theory: Existence and regularity of critical points
• Higher regularity and uniqueness for inner variational equations
• Extension of boundary homeomorphisms to mappings of finite distortion