Existence theorems of the fractional Yamabe problem

TL;DR

This paper establishes several existence results for the fractional Yamabe problem on conformal infinities of asymptotically hyperbolic manifolds, under various geometric conditions, simplifying previous restrictions and extending to lower dimensions.

Contribution

It provides new existence theorems for the fractional Yamabe problem with less restrictive geometric assumptions, including cases with negative mean curvature points and lower-dimensional manifolds.

Findings

Existence results for fractional Yamabe problem with negative mean curvature points.

Existence under zero mean curvature with non-umbilic or non-conformally flat conditions.

Extension to lower-dimensional manifolds and Poincaré-Einstein manifolds.

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.