Nonlinear Riemann-Hilbert problem for bordered Riemann surfaces

Abstract

Let Σ be a bordered Riemann surface with genus g and m boundary components. Let { γ z } z ∈∂Σ be a smooth family of smooth Jordan curves in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] which all contain the point 0 in their interior. Then there exists a holomorphic function f ( z ) on Σ smooth up to the boundary with at most 2 g + m - 1 zeros on Σ such that f ( z ) ∈ γ z for every z ∈ ∂Σ.