Unique extremality in the tangent space of the universal Teichmueller space

The primary purpose of this paper is to announce some results obtained by the authors, including characterizations of unique extremality and examples of unique extremal dilatation of nonconstant modulus. These results bring much light on the question of unique extremality. The examples actually are a surprise and give answer to a longstanding question posed by Teichmüller, Rech and the others.