A Local Method for Compact and Non-compact Yamabe Problems

Let $(M, g)$ be a compact manifold or a complete non-compact manifold without boundary, $\dim M \geqslant 4$, and not locally conformally flat. In this article, we introduce a new local method to resolve the Yamabe problem on compact manifold for dimensions at least $4$, and the Yamabe problem on non-compact complete manifolds without boundary, which are pointwise conformal to subsets of some compact manifolds. In particular, the new local method applies to the hard cases--the Yamabe constants are positive. Our local method also generalizes Brezis and Nirenberg's nonlinear eigenvalue problem to subsets of manifolds.