Dirichlet's Principle, Uniqueness of Harmonic Maps and Extremal QC Mappings

Miodrag Mateljević

This expository paper consists of the various uniqueness theorems which follow, in general, from the length-area principle of Grotzsch. The structure of this paper is as follows. In Section I we give the main ideas and basic results. In the subsections A, B, C, D and E we discuss connections between the Grotzsch principle, Teichmiiller's approach, the Main Inequality and Dirichlet's principle. In the subsections F and G we consider extremal problems for quasiconformal mappings. In particular, we give short review of new results and solve some problems, which originally were subject of investigation of Teichmiiller, Reich, Strebel and the other mathematicians.