A New proof of the Yamabe Conjecture on closed manifolds
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Abstract
We apply an iterative scheme to solve the Yamabe equation $ \Box_{g}u : = - \frac{4(n - 1)}{n - 2}\Delta_{g} u + S_{g} u = \lambda u^{\frac{n + 2}{n - 2}} $ in a small Riemannian domain $ (\Omega, g) $ with Dirichlet condition $ u = c > 0 $ on the smooth boundary $ \partial \Omega $. This is equivalent to finding a metric conformal to $g$ with constant scalar curvature. The iterative scheme solves the Yamabe equation uniformly for all cases. We then apply a similar iterative scheme, using the results on $ (\Omega, g) $,
the barrier method and a perturbation method, to solve the Yamabe equation on closed manifolds $ (M, g) $ with $ \dim M \geqslant 3 $. Thus $g$ admits a conformal change to a constant scalar curvature metric, which is the classical Yamabe Conjecture. In contrast to the traditional method, the Yamabe problem on a closed manifold is completely solved in five cases classified by the sign of the scalar curvature $ S_{g} $ and the sign of the first eigenvalue of the conformal Laplacian $\Box_g$. Instead of using calculus of variations arguments, we focus on local analysis; neither the Weyl tensor nor the positive mass theorem is used, and the proof is essentially dimension independent. To treat the most difficult case $ \lambda > 0 $, we construct lower and upper solutions to solve a perturbed Yamabe equation $ \Box_{g} u_{\beta} + \beta u_{\beta} = \left( \lambda_{\beta} - \kappa \right) u_{\beta}^{\frac{n + 2}{n - 2}} $ with $ \kappa > 0 $, $ \beta < 0 $ and some specific choices of $ \lambda_{\beta} $. We then successfully take the limit $ \beta \rightarrow 0 $ to solve the Yamabe equation.
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Attribution 4.0 International