A complete conformal metric of preassigned negative Gaussian curvature for a punctured hyperbolic Riemann surface

2004/06/28 by Rukmini Dey
Mathematics · #math.AP #math.DG

paper · pdf

published as Proc. Indian Acad. Sci. (Math. Sci.), Vol. 114, No. 2, May 2004, pp. 141-151 · 11 pages, no figures, no tables

arxiv created 2004/06/28 · arxiv updated 2009/12/01

Abstract

Let h be a complete metric of Gaussian curvature K₀ on a punctured Riemann surface of genus g ≥ 1 (or the sphere with at least three punctures). Given a smooth negative function K with K=K₀ in neighbourhoods of the punctures we prove that there exists a metric conformal to h which attains this function as its Gaussian curvature for the punctured Riemann surface. We do so by minimizing an appropriate functional using elementary analysis.

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