Asymptotic Behaviour of the Conformal Representation of a Jordan Domain with a Small Hole in Schauder Spaces

Abstract

We consider a suitably normalized Riemann map g[ζ] of the plane annulus A(r[ζ],1) ≡ z ∈ ℂ: r[ζ] 0 is a real parameter. We analyze the behaviour of the corresponding g[ζ] as e tends to 0. More precisely, we show that the nonlinear operator which takes the quadruple (w, ∈, ξ, ζo) to the corresponding triple of functions $$(r^{-1}[\zeta] g [\zeta]^{(-1)}o\zeta^{i}, g[\zeta]^{(-1)}o\zeta^{o},\in^{-1}r [\zeta])$$ can be continued real analytically around a singular quadruple (w, 0, ξ, ζo) corresponding to an annular domain with an interior degenerate curve. As a corollary, one can deduce information on the behaviour of the relative capacity of the domain enclosed by ζi = w + ∈ξ with respect to that enclosed by ζo as ∈ tends to 0.