Unique extremality, local extremality and extremal non-decreasable dilatations

Abstract

Given a quasi-symmetric self-homeomorphism h of the unit circle S l , let Q(h) be the set of all quasiconformal mappings with the boundary correspondence h. In this paper, it is shown that there exists certain quasi-symmetric homeomorphism h, such that Q(h) satisfies either of the conditions,(1) Q{h) admits a quasiconformal mapping that is both uniquely locallyextremal and uniquely extremal-non-decreasable instead of being uniquely extremal; (2) Q(h) contains infinitely many quasiconformal mappings each of which has an extremal non-decreasable dilatation. An infinitesimal version of this result is also obtained.

Some characterizations of unique extremality

“…The answer is negative. A nonuniquely extremal Beltrami differential μ given by Theorem 1 (2) in [14] is locally extremal and actually satisfies (A) and (B). Of course, μ is of nonconstant modulus.…”

Nondecreasable and Weakly Nondecreasable Dilatations

“…Zhou and Chen [18] studied some special nondecreasable dilatations. The author [14] proved that a Teichmüller class may contain an infinite number of nondecreasable extremal dilatations. The existence of a nondecreasable extremal in a class is generally unknown.…”

On non-uniqueness of weakly non-decreasable extremal dilatations

“…Since Theorem 2 in [4] indicates that there exists [µ] Z ∈ Z(∆) such that [µ] Z contains infinitely many extremals but only one non-decreasable extremal, we have the following corollary. Corollary 1.…”

“…In [3], Shen and Chen proved the following theorem. The author [4] proved that an infinitesimal Teichmüller class may contain infinitely many non-decreasable extremal dilatations. The existence of a non-decreasable extremal in a class is generally unknown.…”