2024/10/30 by Gaven Martin, Cong Yao, Martin, Gaven +1 · 1 citation
Mathematics · Physics and Astronomy · #30C #Advanced Differential Equations and Dynamical Systems #Complex Variables (math.CV) #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2410.22667
openalex publication_date 2024/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider minimisers of the p-exponential conformal energy for homeomorphisms f:R → S of finite distortion \IK(z,f) between analytically finite Riemann surfaces in a fixed homotopy class [f0],\mEp(f:R,S)=∫R exp(p\IK(z,f)) dσ(z). Homeomorphic minimisers exist should the barrier be a homeomorphism of finite energy, \mEp(f0,R,S)<∞. In general this problem is not variational, however the Euler-Lagrange equations show the inverses h=f-1 of sufficiently regular stationary solutions have an associated holomorphic quadratic differential -- the Ahlfors-Hopf differential, Φ=exp(p\IK(w,h)) hwh_\wbar dσR(h). From the Riemann-Roch theorem and an approximation technique, we show the variational equations hold for extremal mappings. We take this as a starting point for higher regularity to show that if h:Ω→Ω is a Sobolev homeomorphism between planar domains with holomorphic Ahlfors-Hopf differential, then h is a diffeomorphism. It will follow that h is harmonic in a metric induced by its own (smooth) distortion. We develop equations for the Beltrami coefficient of h, establishing a connection between degenerate elliptic non-linear Beltrami equations and these harmonic mappings. On the surface we conclude that minimisers fp∈ [f0] of \mEp(f:R,S) are diffeomorphisms and are unique stationary points. This now links two different approaches to Teichmüller theory; the classical theory of extremal quasiconformal maps and the harmonic mapping theory. As p→∞ we show fp→ f_∞ to recover the unique extremal quasiconformal mapping . This extremal quasiconformal mapping is not a diffeomorphism (unless it is conformal) and fp degenerates on a divisor. As p→0 we recover the harmonic diffeomorphism in [f0] and Shoen-Yau's results.