Lipschitz regularity of energy-minimal mappings between doubly connected Riemann surfaces

2021/08/15 by Kalaj, David
#Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2108.06787

Abstract

Let M and N be doubly connected Riemann surfaces with \mathscrC^1,α boundaries and with nonvanishing conformal metrics σ and \wp respectively, and assume that \wp is a smooth metric with bounded Gauss curvature K and finite area. Assume that \Ho_ρ(M, N) is the class of all \mathscrW^1,2 bomeomorphisms between M and N and assume that E^\wp: \Ho_ρ(M, N)→ R is the Dirichlet-energy functional, where \Ho_ρ(M, N) is the closure of \Ho_ρ(M, N) in \mathscrW^1,2(M,N). By using a result of Iwaniec, Kovalev and Onninen in \citeduke that the minimizer, is locally Lipschitz, we prove that the minimizer, of the energy functional E^\wp, which is not a diffeomorphism in general, is a globally Lipschitz mapping of M onto N.

Related

• Energy-minimal diffeomorphisms between doubly connected Riemann surfaces
• Lipschitz property of minimisers between double connected surfaces
• Kellogg's theorem for diffeomophic minimisers of Dirichlet energy\n between doubly connected Riemann surfaces
• Minimisers and Kellogg's theorem
• A note on the smoothness of energy-minimizing incompressible deformations