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A new proof of the Riemann-Poincaré uniformization theorem.

Within the framework of the prescribing curvature problem for Riemannian 2-manifolds, we give a new proof of the Riemann-Poincaré Uniformization Theorem. The approach is variational and the method is based on a lemma of Brézis [2]. As a significant feature, we have been able to reveal an intimate bond between the Differential Geometry of a Riemannian surface and a space of functions defined on it. M.S.C. 2000: 32Q30, 53C20, 49J40; 32Q05, 32Q10.