Timeline for Compact surfaces whose Gaussian curvature is a subharmonic function

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Dec 25, 2017 at 18:53	comment	added	Alexandre Eremenko		This answer proves that $\kappa$ must be constant. To answer the original question it remains to say that yes, on every compact surface, there is a metric of constant curvature. The sign of this constant depends only on the topology of the surface: + for the sphere, 0 for tori, and - for every other surface.
Dec 25, 2017 at 18:37	vote	accept	Ali Taghavi
Dec 25, 2017 at 12:57	history	answered	Ben McKay	CC BY-SA 3.0