Ahlfors and conformal invariants
Annales Academiae Scientiarum Fennicae Series A I Mathematica
Commentationes in honorem Lars V. Ahlfors LXXX annos nato AHLFORS AND CONFORMAT INVAR,IANTS Albert .'Baernstein If* O. Introduction Conformal invariants include quantities such äs harmonic measures, hyperbolic distances and extremal lengths which do not change under conformal map- ping. It would be quite impossible to give a systematic introduction in a small space, or even to discuss most of Atrlfors's contributions to this subject. Thus, this article is in the nature of an excursion. Our
... excursion. Our point of departure will be the conjecture of Denjoy about asymptotic values of entire functions, which Ahlfors proved in 1928 by means of his "distortion theorem". We shall examine some work leading up to this proof, and the proofs of the conjecture itself by Ahlfors and by Carleman and Beurling which carne soon a,fter that of Ahlfors. Each of the three proofs spawned theories and problems which are still of great interest in complex function theory and beyond, and we shall follow some of these threads down to the present day. Perhaps the main theme of the subject is to find relations between conformal and Euclidean quantities. During our journeg we shall visit some problems that are now essentially settled, such as the relation between harmonic and Hausdorff measure in a simply connected domain (Section 6), a^nd others that are not, notably Painlevd's problem of geometrically characterizing null sets for bounded analytic functions (Section 7). For the "a,ngular derivative problem" (Section 5), the authorities differ as to whether the problem is or is not completely solved. These subjects have all been strongly influenced by the work of Ahlfors, either directly, or indirectly through tools such as extremal length in whose development he has played a major role. My choice of topics has been somewhat arbitrary. I regret especially having no space to discuss hyperbolic metrics or extremal lengths for multiple curve families. See [Mi], [We 2], urrd [I] for some intresting recent results. Also, I would have liked to discuss work related to Bloch's theorem, but there is time only to note that the Ahlfors-Grunsky conjecture of 1937 about the exact value of Bloch's constant has still been neither proved nor disproved.