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A Riemannian metric invariant under Möbius transformations in \({\mathbb{R}}^ n\). (English) Zbl 0687.31002
Complex analysis, Proc. 13th Rolf Nevanlinna-Colloq., Joensuu/Finl. 1987, Lect. Notes Math. 1351, 223-235 (1988).
[For the entire collection see Zbl 0645.00003.]
For x, \(y\in R^ n\), \(n\geq 3\), let \(G(x,y)=\| x-y\|^{2-n}\) as usual. For a domain \(\Omega \subset R^ n\), let \(m^ x\) denote the greatest harmonic minorant in \(\Omega\) of the function \(y\mapsto G(y,x)\) restricted to \(\Omega\). Then the Riemannian metric in question is \(ds=(m^ x(x))^{1/(n-2)}\| dx\|\). Let \(G_{\Omega}\) denote the Green function of \(\Omega\). The author proves:
1\(\circ\) The quotient \(G_{\Omega}/G\) is invariant under Möbius transformations of \(\Omega\),
2\(\circ\) ds is conformally invariant,
3\(\circ\) ds is equivalent to the quasi-hyperbolic metric defined by F. W. Gehring and B. P. Palka [J. Anal. Math. 30, 172-199 (1976; Zbl 0349.30019)], provided \(\Omega\) satisfies a uniform cone condition at the boundary,
4\(\circ\) the case of the quarter space, \(Q_ n=\{(x_ 1,...,x_ n)\in R^ n:\quad x_ 1>0,\quad x_ 2>0\}\), is examined, where the explicit form of ds allows the calculation of certain sectional curvatures as well as an integral representation of positive solutions of the associated Laplace-Beltrami equation.
Reviewer: E.J.Akutowicz

MSC:
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
35A08 Fundamental solutions to PDEs
53C20 Global Riemannian geometry, including pinching