A note on extremal quasi-conformal mappings

Abstract

An extremal quasi-conformal mapping f of a domain D is said to be of non-landslide type if the set E f (δ) := {z ∈ D : |μ f (z)| μ ∞ − δ} has no interior points for any δ > 0. In this paper, we construct a quasi-conformal mapping f of the unit disc D such that its Teichmüller equivalence class [f ] contains infinitely many extremal mappings of non-landslide type. The relation between extremal mappings of non-landslide type and locally extremal mappings is also discussed.Keywords quasi-conformal mappings, extremal quasi-conformal mappings, locally extremal quasi-conformal mappings MSC(2000): Primary 37F30, Secondary 30F70Citation: Li Z. A note on extremal quasi-conformal mappings.

Extremal Beltrami Differentials of Non-Landslide Type

“…Let D be the unit disk in the complex plane. In [5], Li investigated non-uniqueness of extremal Beltrami differentials of non-landslide type in a Teichmüller equivalence class in T(D) and claimed the following theorem. The proof of the theorem by Li is lengthy and complicated.…”

“…Let E ⊂ D\D r be a compact subset of positive measure with empty interior. Define In [5], Li posed the following problem.…”

Nondecreasable and Weakly Nondecreasable Dilatations

“…The concept of nonlandslide was first introduced by Li in [6] for extremal Beltrami differentials. Here, we generalize the definition for general Beltrami differentials.…”

On extremal quasiconformal mappings of non-landslide type

“…However, we still do not know the answer to Problem 1. In [4], Li proposed the following problem. Furthermore, using the methods in the proof of Theorem 1.2, as a generalization of Theorem 1.1, we get the following theorem.…”

“…Otherwise, an extremal quasiconformal mapping f is of non-landslide type when the set E f (δ) has no interior points for any δ > 0. So it is natural to ask the following question (refer to [4], [11]).…”