Teichmüller's theorem in higher dimensions and its applications

2020/06/02 by Golberg, Anatoly, Sugawa, Toshiyuki, Vuorinen, Matti · 1 citation
#30C75 #31B15 #Complex Variables (math.CV) #FOS: Mathematics #Primary 30C65 #Secondary 26B35

paper · doi:10.48550/arxiv.2006.01565

Abstract

For a given ring (domain) in ℝⁿ we discuss whether its boundary components can be separated by an annular ring with modulus nearly equal to that of the given ring. In particular, we show that, for all n≥ 3 , the standard definition of uniformly perfect sets in terms of Euclidean metric is equivalent to the boundedness of moduli of separating rings. We also establish separation theorems for a "half" of a ring. As applications of those results, we will prove boundary Hölder continuity of quasiconformal mappings of the ball or the half space in ℝⁿ.

Cited by

• Modulus estimates and cavitation in higher dimensions

Related

• Modulus estimates of semirings with applications to boundary extension problems
• Distortion of quasiconformal mappings with identity boundary values
• On quasiconformal maps with identity boundary values
• A Commentary on Teichmüller's paper ``Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Flächen''
• A Commentary on Teichmüller's paper "Untersuchungen über konforme und quasikonforme Abbildungen" (Investigations on conformal and quasiconformal mappings) (to appear in Vol. VII of the Handbook of Teichmüller theory