Moduli, capacity, BV-functions on the Riemann surfaces

Abstract

Let R is a Riemann surface, glued from finitely or countably many domains in the extended complex plane so that the following conditions are satisfied: each point in R projects onto a point w = prW in one on the glued domains, each point in R has a neighbourhood which is a univalent disk, or multivalent disk with the unique ramification point at the centre of disk. We study elementary properties of functions of bounded variation and sets of finite perimeter in an open set Q ⊂ R {W ∈ R: W is a ramification point or prW = ∞}. Further, by using Ziemer’s technique, we obtain the main result

\(C\left( {{F_{0,}}{F_1},G} \right) \cdot M\left( {{F_{0,}}{F_1},G} \right) = 1\)

. Here G is an open set with the compact closure on R, F₀ and F₁ are disjoint compact sets in the closure of G, C(F₀, F₁, G) is the conformal capacity of the condenser (F₀, F₁, G), M(F₀, F₁, G) is the conformal module of the family of all curves that separate F₀ from F₁ in G.