Yamabe problem

Remarks

The scalar curvature $\tilde R$ of $\tilde g$ is related to the scalar curvature $R$ of $g$ by $$-\frac{4(n-1)}{n-2} \Delta_g u + R u = \tilde R \, u^{(n+2)/(n-2)},$$ where $\Delta_g$ is the Laplace–Beltrami operator of $g$, the analogue of the Laplacian. So the problem asks for a positive solution of this semilinear elliptic equation with $\tilde R$ constant. The exponent $(n+2)/(n-2)$ is the critical Sobolev exponent, and this is why the direct variational method does not apply at once. The answer is yes. Yamabe proposed a proof that Neil Trudinger found to contain a gap. Trudinger, Thierry Aubin and Richard Schoen completed the proof, with Schoen settling the remaining cases with the positive mass theorem. In dimension $2$ the analogous statement is a consequence of the uniformization theorem.