On the maximum of the Moebius invariant in the four disjoint domain problem

D. A. Kirillova

Far-Eastern State Social-Humanitarian Academy

Abstract: Let $r(D,a)$ denote the conformal radius of the domain $D$ with respect to the point $a$. In this paper we obtain the supremum of the product
$$ \prod_{k=1}^{4}\frac{r(D_{k},a_{k})}{|a_{k+1}-a_{k}|}, \quad a_{5}:=a_{1} $$
for all simply connected disjoint domains $D_{k}\subset\overline{\mathbb{C}}$ and points $a_{k}\in D_{k},k=1,\ldots,4$. Using the method of interior variations due to M. Schiffer we establish the form of quadratic differential associated with extremal partition problem $\prod\limits_{k=1}^{n}r(D_{k},a_{k})|a_{k+1}-a_{k}|^{-1}\to\sup$ for arbitrary $n\geqslant 3$. For $n=4$ we studed the circle domains and their boundaries for the corresponding quadratic differential.

Key words: conformal radius, Moebius invariants, extremal partitions, quadratic differential.

UDC: 517.54

MSC: Primary 30C70; Secondary 30C75

Received: 04.12.2009

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