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, $\dim M⩾4$, 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.