Evolution of curvatures on a surface with boundary to prescribed functions

Let (M, g ₀) be a compact Riemann surface with boundary and with negative Euler characteristic. Let f(x) be a strictly negative smooth function on \({\bar{M}}\) and denote by \({\sigma(x)}\) the value of f in the interior and \({\zeta(x)}\) the value of f on the boundary. By studying the evolution of curvatures on M, we prove that there exist a constant \({\lambda_\infty}\) and a conformal metric \({g_\infty}\) such that \({\lambda_\infty\sigma(x)}\) and \({\lambda_\infty\zeta(x)}\) can be realized as the Gaussian curvature and boundary geodesic curvature of \({g_\infty}\) respectively.