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 González-Qing (Analysis and PDE, 2013) and González-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 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.

34 pages

References in corpus (3)

• A Quartic Conformally Covariant Differential Operator for Arbitrary Pseudo-Riemannian Manifolds (Summary)
• Sign of Green's function of Paneitz operators and the Q curvature
• Some energy inequalities involving fractional GJMS operators

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• Asymptotics of the Poisson kernel and Green's functions of the fractional conformal Laplacian
• Uniqueness and some related estimates for Dirichlet problem with fractional Laplacian
• Subelliptic Nonlocal Brezis-Nirenberg Problems on Stratified Lie Groups
• Multiplicity of singular solutions to the fractional Yamabe problem on spheres