Minimizers of a weighted maximum of the Gauss curvature

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Abstract

On a Riemann surface [`(S)] with smooth boundary, we consider Riemannian metrics conformal to a given background metric. Let κ be a smooth, positive function on [`(S)]. 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.
Original languageEnglish
Pages (from-to)199-207
Number of pages9
JournalAnnals of Global Analysis and Geometry
Volume41
Issue number2
DOIs
Publication statusPublished - 2012

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