Solving the Yamabe Problem on Closed Manifolds by Double Iteration Scheme

The author introduces a double iterative scheme to solve the Yamabe equation $ - \frac{4(n - 1)}{n - 2}\Delta_{g} u + S_{g} u = \lambda u^{\frac{n + 2}{n - 2}} $ on closed manifolds $ (M, g) $ with $ \dim M \geqslant 3 $. Thus $g$ admits a conformal change to a constant scalar curvature metric. In contrast to the traditional functional minimization, the Yamabe problem is fully solved in five cases classified by the sign of the scalar curvature $ S_{g} $ and the sign of the first eigenvalue of conformal Laplacian.