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

We study the fractional Yamabe problem first considered by Gonzalez-Qing [] on the conformal infinity (M-n, [h]) of a Poincare-Einstein manifold (Xn+1, g(+)) with either n=2 or n >= 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 Poincare-Einstein manifolds of dimension either n=2 or of dimension n >= 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 Poincare-Einstein manifold of dimension either n=2 or of dimension n >= 3 and which is locally flat admits a Riemannian metric of constant fractional scalar curvature.