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 deduce 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 is either non-umbilic or umbilic but non-locally conformally flat. As a result, we replace the geometric restrictions given by Gonz\'alez-Qing (Analysis and PDE, 2013) and Gonz\'alez-Wang (arXiv:1503.02862) with simpler ones. Also, inspired by Marques (Comm. Anal. Geom., 2007) and Almaraz (Pacific J. Math., 2010), we study lower-dimensional manifolds. Finally, the situation when $X$ is Poincar\'e-Einstein, $M$ is either locally conformally flat or 2-dimensional is covered under the validity of the positive mass theorem for the fractional conformal Laplacians.