Conformal Metrics with Prescribed Fractional Scalar Curvature on Conformal Infinities with Positive Fractional Yamabe Constants

Abstract

Let (X, g(+)) be an asymptotically hyperbolic manifold with conformal infinity (M, [(h) over cap]). Our primary aim is to introduce the prescribed fractional scalar curvature problem on M and to provide its solutions under various geometric conditions on X and M. We also deduce the existence results for the fractional Yamabe problem in the end-point cases, e.g., n = 3, gamma = 1 2 and M is non-umbilic, etc. Finally, we prove that all solutions we find here are smooth on M.