JournalsjemsVol. 23, No. 9pp. 3017–3073

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

Assume that (X,g+)(X, g^+) is an asymptotically hyperbolic manifold, (M,[hˉ])(M, [\bar{h}]) is its conformal infinity, ρ\rho is the geodesic boundary defining function associated to hˉ\bar{h} and gˉ=ρ2g+\bar{g} = \rho^2 g^+. For any γ\gamma in (0,1)(0,1), we prove that the solution set of the γ\gamma-Yamabe problem on MM is compact in C2(M)C^2(M) provided that convergence of the scalar curvature R[g+]R[g^+] of (X,g+)(X, g^+) to n(n+1)-n(n+1) is sufficiently fast as ρ\rho tends to 0 and the second fundamental form on MM never vanishes. Since most of the arguments in the blow-up analysis performed here are insensitive to the geometric assumption imposed on XX, 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