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 andM. 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 (X) over bar is negative, or umbilic and nonlocally conformally flat. As a result, we replace the geometric restrictions given by Gonzalez and Qing (2013) and Gonzalez 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 Poincare-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.