Fractional Yamabe problem on locally flat conformal infinities of Poincare-Einstein manifolds

Abstract

We study in this paper the fractional Yamabe problem first considered by Gonzalez-Qing on the conformal infinity $(M^n,[h])$ of a Poincaré-Einstein manifold $(X^{n+1},g^{+})$ with either $n=2$ or $n\ge 3$ and $(M^n,[h])$ is locally flat - namely $(M,h)$ is locally conformally flat. However, as for the classical Yamabe problem, because of the involved quantization phenomena, the variational analysis of the fractional one exhibits also a local situation and a global one. Furthermore the latter global situation includes the case of conformal infinities of Poincaré-Einstein manifolds of dimension either 2 or of dimension greater than $2$ and which are locally flat, and hence the minimizing technique of Aubin- Schoen in that case clearly requires an analogue of the positive mass theorem of Schoen-Yau which is not known to hold. Using the algebraic topological argument of Bahri-Coron, we bypass the latter positive mass issue and show that any conformal infinity of a Poincaré-Einstein manifold of dimension either $n=2$ or of dimension $n\ge 3$ and which is locally flat admits a Riemannian metric of constant fractional scalar curvature.

The current version - as of July 2021 - corresponds to sections 5,6,7 of the previous one. We have split out the others to a separate paper 'Asymptotics of the Poisson kernel and Green's functions of the fractional conformal Laplacian'

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Cited by in corpus (2)

• Uniformization Theorems: Between Yamabe and Paneitz
• Conformal metrics with prescribed fractional scalar curvature on conformal infinities with positive fractional Yamabe constants