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

Abstract

Assume that (X, g(+)) is an asymptotically hyperbolic manifold, (M, [(h) over bar]) is its conformal infinity, rho is the geodesic boundary defining function associated to (h) over bar and (g) over bar = rho(2)g(+). For any gamma in (0, 1), we prove that the solution set of the gamma-Yamabe problem on M is compact in C-2(M) provided that convergence of the scalar curvature R[g(+)] of (X, g(+)) to -n(n + 1) is sufficiently fast as rho 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.