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

Abstract

Assume that $(X,g^{+})$ is an asymptotically hyperbolic manifold, $(M,[\bar{h}])$ is its conformal infinity, $\rho$ is the geodesic boundary defining function associated to $\bar{h}$ and $\bar{g}={\rho }^2{g}^{+}$. 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.