Nonlinear Riemann-Hilbert problem for bordered Riemann surfaces

Let Sigma be a bordered Riemann surface with genus g and m boundary components. Let {gamma(z)}(Zis an element ofpartial derivativeSigma) be a smooth family of smooth Jordan curves in C which all contain the point 0 in their interior. Then there exists a holomorphic function f(z) on Sigma smooth up to the boundary with at most 2g + m - 1 zeros on Sigma such that f(z) is an element of gammaz for every z is an element of partial derivativeSigma.