Fumio Maitani

On rigidity of an end under conformal mappings preserving the infinite homology basis

Let f be a quasiconformal self-mapping of an open Riemann surface R. Assume that every cycle γ in R is homologous to f(γ) and f is conformal on an end D of R. Then we show that f is the identity mapping on D if the first homology group of D is infinitely generated.

Canonical functions on open riemann surfaces

The canonical functions are meromorphic functions with a finite numbcr of poles and their real parts are, roughly speaking, constant on each ideai boundary. compment of an open Riemann surface. The existence and geometrical: propties of such functions have been investigated mainly- far open Riernann surfaces of finite genus. In this paper we consider an open Riemann surface (of genus, possibly. ∞), and show that a canonical function with n poles (n<∞) on R, is (i) n-valent a.e.and (ii) the cluster set at every ideal boundary component of R is a vertical slit. The convering property (i) is proved by a generalized version of the classical Kobes lemma and its new application “Elevator lemma” and (ii) is shown by our Elevator lemma and the extremal length method.