The Yamabe problem with singularities

Abstract

Let $(M,g)$ be a compact Riemannian manifold of dimension $n\ge 3$. Under some assumptions, we prove that there exists a positive function $\phi$ solution of the following Yamabe type equation Δφ+ hφ= \tilde h φ^{\frac{n+2}{n-2}} where $h\in L^p(M)$, $p>n/2$ and $\tilde{h}\in \mathbb{R}$. We give the regularity of $\phi$ with respect to the value of $p$. Finally, we consider the results in geometry when $g$ is a singular Riemannian metric and $h=\frac{n-2}{4(n-1)}{R}_g$, where $R_g$ is the scalar curvature of $g$.