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

### Seunghyeok Kim

Hanyang University, Seoul, Republic of Korea### Monica Musso

University of Bath, UK### Juncheng Wei

University of British Columbia, Vancouver, Canada

## 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.

## 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. 3017–3073

DOI 10.4171/JEMS/1068