Geodesic
On the maximum of the Moebius invariant in the four disjoint domain problem
Dalʹnevostočnyj matematičeskij žurnal, Tome 10 (2010) no. 1, pp. 41-49 
D. A. Kirillova. On the maximum of the Moebius invariant in the four disjoint domain problem. Dalʹnevostočnyj matematičeskij žurnal, Tome 10 (2010) no. 1, pp. 41-49. http://geodesic.mathdoc.fr/item/DVMG_2010_10_1_a4/
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     title = {On the maximum of the {Moebius} invariant in the four disjoint domain problem},
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     url = {http://geodesic.mathdoc.fr/item/DVMG_2010_10_1_a4/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

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