Existence theorems of the fractional Yamabe problem

Kim, Seunghyeok; Musso, Monica; Wei, Juncheng

doi:10.2140/apde.2018.11.75

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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

DOI: 10.2140/apde.2018.11.75

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 $\bar{X}$ 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.

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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

Vol. 11, No. 1, 2018