Abstract
On a Riemann surface \({\overline{\Sigma}}\) with smooth boundary, we consider Riemannian metrics conformal to a given background metric. Let κ be a smooth, positive function on \({\overline{\Sigma}}\). If K denotes the Gauss curvature, then the L ∞-norm of K/κ gives rise to a functional on the space of all admissible metrics. We study minimizers subject to an area constraint. Under suitable conditions, we construct a minimizer with the property that |K|/κ is constant. The sign of K can change, but this happens only on the nodal set of the solution of a linear partial differential equation.
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Moser, R., Schwetlick, H. Minimizers of a weighted maximum of the Gauss curvature. Ann Glob Anal Geom 41, 199–207 (2012). https://doi.org/10.1007/s10455-011-9278-9
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DOI: https://doi.org/10.1007/s10455-011-9278-9