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Singular points on compactifications of a Riemann surface. (English) Zbl 0605.30045

In this well-written paper, the authors extend the notion of capacity to a general compactification \(R^*\) of a hyperbolic Riemann surface R. The new definition involves consideration of Dirichlet integrals, and is called the Dirichlet capacity. Some of the basic theory of Dirichlet capacity on \(R^*\) is subsequently developed. As it turns out, in the case of the Kuramochi compactification, the Dirichlet capacity is a constant multiple of the Kuramochi capacity.
A point \(p\in R^*\) is called singular if the Dirichlet capacity of p is positive. If \(R^*_{{\mathcal R}}\) and \(R^*_{{\mathcal K}}\) represent the Royden and Kuramochi compactifications respectively, and \(\pi\) : \(R^*_{{\mathcal R}}\to R^*_{{\mathcal K}}\) is the natural projection, the main results states that: A point \(q\in R^*_{{\mathcal K}}-R\) is singular if and only if \(\pi^{-1}(q)\) contains a singular point. Moreover, for any \(q\in R^*_{{\mathcal K}}-R\), \(\pi^{-1}(q)\) contains at most one singular point.
Reviewer: J.L.Schiff

MSC:

30F99 Riemann surfaces
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[1] C. Constantinescu und A. Cornea,Ideale Ränder Riemannscher Flächen, Springer-Verlag, 1963. · Zbl 0112.30801
[2] M. Glasner and M. Nakai,Singular points on the Royden harmonic boundary, Complex Variables, to appear. · Zbl 0625.30047
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