Existence theorems of the fractional Yamabe problem

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 González-Qing (2013) [19] and González-Wang (2015) [20] with simpler ones. Also, inspired by Marques (2007) [37] and Almaraz (2010) [1], we study lower-dimensional manifolds. Finally, the situation when X is Poincaré-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. 2010 Mathematics Subject Classification. Primary: 53C21, Secondary: 35R11, 53A30.