The Yamabe problem with singularities

TL;DR

This paper proves existence and regularity results for solutions to a Yamabe-type equation on compact manifolds with singular metrics, extending classical results to cases with less regular curvature functions.

Contribution

It establishes existence and regularity of solutions to a Yamabe problem with singular metrics and less regular scalar curvature functions.

Findings

Existence of positive solutions under certain integrability conditions.

Regularity of solutions depends on the integrability exponent p.

Application to geometries with singular Riemannian metrics.

Abstract

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 $g$.