Boundary behaviors of the Poincaré density and its derivatives near a nonisolated boundary point

Abstract

Let P_Ω(z)|dz| be the Poincaré metric element with the constant Gauss curvature −4 of a hyperbolic domain Ω in the complex plane C. We find some boundary properties of the Poincaré density P_Ω and its complex partial derivatives (P_Ω)_z, (P_Ω)_zz and (P_Ω)_z\bar{z}, in terms of the distance δ_Ω(z) of z∈Ω and the boundary of Ω in C. For the proof we make use of the sharp, lower estimates of P_Ω(K) of a domain Ω(K)⊂C such that K=C{\backslash}Ω(K) is a non-degenerate continuum. Several properties of the function p(z, K), z∈Ω(K), are proposed.