On the extremal sets of extremal quasiconformal mappings

On extremal quasiconformal mappings of non-landslide type

“…Let Δ be the open unit disk in the complex plane. For a quasiconformal mapping f on Δ, we let μ f = fz f z be the Beltrami coefficient of f and let K(f ) = 1 + |μ g (z)| 1 − |μ g (z)| < K(f ) − δ} has a positive measure for some δ > 0 (refer to [6], [11] for the proof). For the details of extremal quasiconformal mappings theory, we refer to [1], [2], [5], [7], [8], [9].…”

“…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]).…”

A note on extremal quasi-conformal mappings

“…has a positive measure for some δ > 0 (see [13] by Zhou et al or [7] by Reich). So it is natural to ask the following question: (A) Suppose that f is an arbitrarily given quasi-conformal mapping such that [f ] contains extremal mappings more than one.…”

“…A similar question as (A) with a little difference was also proposed in [13]. Now let us introduce some terminologies.…”

Existence of extremal Beltrami coefficients with nonconstant modulus

“…In the latter case, in fact there are infinitely many extremal Beltrami coefficients in [μ] T (Δ) (see [9], [2]). Moreover, in this setting there definitely exists an extremal Beltrami coefficient of nonconstant modulus in [μ] T (Δ) (see [7], [11], [12]).…”