Unicity theorem for holomorphic maps on open Riemann surfaces with the conformal metrics

Abstract

In this paper, we give a unicity theorem of two holomorphic maps on two open Riemann surfaces with conformal metrics, which share a family of hypersurfaces {Q_j}^q_j=1 in Pⁿ(C) located in N-subgeneral position. Furthermore, motivated by the value distribution properties of the Gauss maps for K-quasiregular harmonic surfaces in [5], we give a unicity result for K-quasiregular harmonic surfaces in R³.