Abstract
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.
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Zhang, H. Evolution of curvatures on a surface with boundary to prescribed functions. manuscripta math. 149, 153–170 (2016). https://doi.org/10.1007/s00229-015-0769-z
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DOI: https://doi.org/10.1007/s00229-015-0769-z