Vol. 11, No. 1, 2018

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Existence theorems of the fractional Yamabe problem

Seunghyeok Kim, Monica Musso and Juncheng Wei

Vol. 11 (2018), No. 1, 75–113
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 $X$ 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 satisfies one of the following conditions: nonumbilic, umbilic and a component of the covariant derivative of the Ricci tensor on $\overline{X}$ is negative, or umbilic and nonlocally conformally flat. As a result, we replace the geometric restrictions given by González and Qing (2013) and González 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 Poincaré–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.

Keywords
fractional Yamabe problem, conformal geometry, existence
Mathematical Subject Classification 2010
Primary: 53C21
Secondary: 35R11, 53A30
Milestones
Received: 22 March 2016
Revised: 10 May 2017
Accepted: 10 August 2017
Published: 17 September 2017
Authors
 Seunghyeok Kim Department of Mathematics College of Natural Sciences Hanyang University Seoul South Korea Monica Musso Facultad de Matemáticas Pontificia Universidad Católica de Chile Santiago Chile Juncheng Wei Department of Mathematics University of British Columbia Vancouver, BC Canada