A compactness theorem for the fractional Yamabe problem, Part I: The nonumbilic conformal infinity

Assume that {X; g+} is an asymptotically hyperbolic manifold, .M; TN hU/ is its conformal infinity, ρ is the geodesic boundary defining function associated to Nh and N g = ρ2g+. For any in .0; 1/, we prove that the solution set of the -Yamabe problem onM is compact in C2.M/ provided that convergence of the scalar curvature R[g+] of {X; g+} to -n.n C 1/ is sufficiently fast as ρ tends to 0 and the second fundamental form on M never vanishes. Since most of the arguments in the blow-up analysis performed here are insensitive to the geometric assumption imposed on X, our proof also provides a general scheme toward other possible compactness theorems for the fractional Yamabe problem.