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