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  "body": "\n## Problem\n\nLet $(M, g)$ be a compact [[Riemannian manifold|7CBT4SM83Y]] without boundary, of dimension $n \\ge 3$.\nIs there a smooth positive function $u$ on $M$ such that the metric $\\tilde g = u^{4/(n-2)} g$ has constant [[scalar curvature|ZHX2VYQDZ3]]?\n\n## Remarks\n\nThe scalar curvature $\\tilde R$ of $\\tilde g$ is related to the scalar curvature $R$ of $g$ by\n$$-\\frac{4(n-1)}{n-2} \\Delta_g u + R u = \\tilde R \\, u^{(n+2)/(n-2)},$$\nwhere $\\Delta_g$ is the Laplace–Beltrami operator of $g$, the analogue of the [[Laplacian|0E0B6RA8K0]].\nSo the problem asks for a positive solution of this semilinear elliptic equation with $\\tilde R$ constant.\nThe exponent $(n+2)/(n-2)$ is the critical Sobolev exponent, and this is why the direct variational method does not apply at once.\nThe answer is yes.\nYamabe proposed a proof that Neil Trudinger found to contain a gap.\nTrudinger, Thierry Aubin and Richard Schoen completed the proof, with Schoen settling the remaining cases with the positive mass theorem.\nIn dimension $2$ the analogous statement is a consequence of the uniformization theorem.\n\n## Explanations\n\n### L1 human\n\nOn a curved space one can stretch lengths by a factor that varies from point to point, without changing angles.\nThe Yamabe problem asks whether such a stretching can always make the scalar curvature, an average of the curvature at each point, the same everywhere.\nIt is a higher-dimensional version of the fact that every surface can be given a metric of constant curvature, and the answer is again yes.\n",
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