×

Möbius modulus of ring domains in \(\overline{\mathbb{R}}^n\). (English) Zbl 1134.30304

A ring \(R=R(C_1,C_2)\subset\overline{\mathbb R}^n\) is a domain whose complement is the union of two disjoint non-degenerate compact connected sets \(C_1\) and \(C_2\). A ring \(R(C_1,C_2)\) is said to separate the sets \(E\) and \(F\) if \(E\subset C_1\) and \(F\subset C_2\). The author introduces a new Möbius invariant modulus for ring domains \(R\) in \(\mathbb R^n\) as follows: \[ \text{mod}_MR=\max_{u,v\in\overline{\mathbb R}^n}\min_{x\in C_1,y\in C_2 }\left| \log\frac{| u-y|| x-v|}{| u-x|| y-v|}\right|. \]
The author obtains the following sharp estimate.
Theorem. Let \(a,b,c,d\in\overline{\mathbb{R}}^n\) be distinct points. Then \[ \max_R\text{mod}_MR=\text{arccosh}(\sigma(a,b,c,d)) \] , where the maximum ist taken over all rings \(R\) which separate the two sets \(\{a,b\}\) and \(\{c,d\}\).
This proves a conjecture of M. Vuorinen [Q. J. Math., Oxf. II. Ser. 43, No. 172, 501–514 (1992; Zbl 0766.30014)].

MSC:

30C35 General theory of conformal mappings
30C75 Extremal problems for conformal and quasiconformal mappings, other methods

Citations:

Zbl 0766.30014
PDF BibTeX XML Cite
Full Text: EuDML EMIS