The Yamabe problem asks if, given a Riemannian manifold $(M,g_0)$, it is possible to find a conformal metric $g$ on $M$ with constant scalar curvature. I would like to know if there is some "easy" proof that this is true in the case of closed oriented surfaces. (Easy should mean, for example, without assuming Uniformization or using Ricci flow).
2026-03-26 19:35:52.1774553752
Easy solution to Yamabe problem for surfaces
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For zero and negative Yamabe cases the proof is relatively simple. For the full resolution one needs to use the positive mass theorem.