Unique extremality

Extremal problems for quasiconformal maps of punctured plane domains

“…Set µ = Belt(f ). One of the two main results in [3] (see also [4], [11], [10], [12]) is: Proposition 1.1. The mapping f is a uniquely extremal quasiconformal map if and only if the linear functional λ :…”

“…It is known (see [3], [4], [10], [12]) that the affine map is uniquely extremal if and only if there is a sequence φ n from A 1 (M ) which satisfies the first two conditions of Proposition 1.2 and that in particular δ(φ n ) → 0 when n → ∞. Although this last statement provides the necessary and sufficient conditions it is usually easier to construct a sequence where the δ functional is bounded and the third condition of the above proposition is satisfied rather then a sequence where the delta functional tends to zero.…”

Book Review: Quasiconformal Teichmüller theory

“…Chapter 9, "Teichmüller uniqueness", is based on [15], joint work of the junior author with three colleagues, unfortunately omitted from the bibliography. The chapter begins with a discussion of critical examples, mostly the work of K. Strebel [52], [53] (see also [46]), concerning unique extremality of quasiconformal maps.…”

“…Perhaps most annoying is the inadvertent omission of [15] from the bibliography. This important paper is the basis of about half of Chapter 9; the omission is compounded by the fact that significant portions of that manuscript are reproduced without even the correction of numbering of lemmas.…”

Infinitesimally extremal Beltrami differentials of non-landslide type

“…The reason is that each extremal in [µ] Z can be of non-constant modulus (see [1,12]). Therefore, we have a direct corollary.…”