Singular points on compactifications of a Riemann surface
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.
scientific article; zbMATH DE number 3977389
scientific article