Fractional Yamabe Problem on Locally Flat Conformal Infinities of Poincaré-Einstein Manifolds

Abstract

Abstract We study the fractional Yamabe problem first considered by Gonzalez-Qing [36] on the conformal infinity $(M^{n}, \;[h])$ of a Poincaré-Einstein manifold $(X^{n+1}, \;g^{+})$ with either $n=2$ or $n\geq 3$ and $(M^{n}, \;[h])$ 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 a local situation and also a global one. The latter global situation includes the case of conformal infinities of Poincaré-Einstein manifolds of dimension either $n=2$ or of dimension $n\geq 3$ and which are locally flat, and hence the minimizing technique of Aubin [4] and Schoen [48] in that case clearly requires an analogue of the positive mass theorem of Schoen-Yau [49], which is not known to hold. Using the algebraic topological argument of Bahri-Coron [8], 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 $n\geq 3$ and which is locally flat admits a Riemannian metric of constant fractional scalar curvature.