Existence theorems of the fractional Yamabe problem

Abstract

Let $X$ be an asymptotically hyperbolic manifold and $M$ its conformal infinity. This paper is devoted to deducing several existence results of the fractional Yamabe problem on $M$ under various geometric assumptions on $X$ and $M$. Firstly, we handle when the boundary $M$ has a point at which the mean curvature is negative. Secondly, we re-encounter the case when $M$ has zero mean curvature and satisfies one of the following conditions: nonumbilic, umbilic and a component of the covariant derivative of the Ricci tensor on $\overline{X}$ is negative, or umbilic and nonlocally conformally flat. As a result, we replace the geometric restrictions given by González and Qing (2013) and González and Wang (2017) with simpler ones. Also, inspired by Marques (2007) and Almaraz (2010), we study lower-dimensional manifolds. Finally, the situation when $X$ is Poincaré–Einstein and $M$ is either locally conformally flat or 2-dimensional is covered under a certain condition on a Green’s function of the fractional conformal Laplacian.