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2018 Existence theorems of the fractional Yamabe problem
Seunghyeok Kim, Monica Musso, Juncheng Wei
Anal. PDE 11(1): 75-113 (2018). 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 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.

Citation

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Seunghyeok Kim. Monica Musso. Juncheng Wei. "Existence theorems of the fractional Yamabe problem." Anal. PDE 11 (1) 75 - 113, 2018. https://doi.org/10.2140/apde.2018.11.75

Information

Received: 22 March 2016; Revised: 10 May 2017; Accepted: 10 August 2017; Published: 2018
First available in Project Euclid: 20 December 2017

zbMATH: 1377.53056
MathSciNet: MR3707291
Digital Object Identifier: 10.2140/apde.2018.11.75

Subjects:
Primary: 53C21
Secondary: 35R11, 53A30

Rights: Copyright © 2018 Mathematical Sciences Publishers

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Vol.11 • No. 1 • 2018
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