Assume that is an asymptotically hyperbolic manifold, is its conformal infinity, is the geodesic boundary defining function associated to and . For any in , we prove that the solution set of the -Yamabe problem on is compact in provided that convergence of the scalar curvature of to is sufficiently fast as tends to 0 and the second fundamental form on never vanishes. Since most of the arguments in the blow-up analysis performed here are insensitive to the geometric assumption imposed on , our proof also provides a general scheme toward other possible compactness theorems for the fractional Yamabe problem.
Cite this article
Seunghyeok Kim, Monica Musso, Juncheng Wei, A compactness theorem for the fractional Yamabe problem, Part I: The nonumbilic conformal infinity. J. Eur. Math. Soc. 23 (2021), no. 9, pp. 3017–3073DOI 10.4171/JEMS/1068