Remarks on the Conditions for Unique Extremality of Quasiconformal Mappings

Edgar Reich

In work dating to 1981, two different sufficient conditions for unique extremality of quasiconformal mappings, referred to here as Theorem~A and Theorem~B were found. In the case of Theorem~A, the condition is now known to also be necessary. In spite of this, Theorem~B has been found to be the more useful of the two conditions in delicate situations. However, the question of whether the conditions of Theorem~B are necessary is still open. We discuss some matters relevant to this problem. As a new illustration of Theorem~B, we show that it can be used to give a self-contained proof of a result of Strebel's.