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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