The Yamabe problem with singularities

Details The Yamabe problem with singularities The Yamabe problem with singularities

2008-04-10

arxiv.org/abs/0804.1717v2

Mathematics

Analysis of PDEs

Scientific paper

10.1016/j.bulsci.2007.09.004

Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$. Under some assumptions, we prove that there exists a positive function $\varphi$ solution of the following Yamabe type equation \Delta \varphi+ h\varphi= \tilde h \varphi^{\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 $\varphi$ 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 $g$.

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