Abstract
A compactness theorem for the fractional Yamabe problem, Part I: The nonumbilic conformal infinity
Highlights
Given any n ∈ N, let (Xn+1, g+) be an asymptotically hyperbolic manifold with conformal infinity (M n, [h])
According to [31, 56, 38, 11, 28], there exists a family of self-adjoint conformal covariant pseudo-differential operators P γ[g+, h] on M in general whose principal symbols are the same as those of (−∆h)γ
If (X, g+) is Poincare-Einstein and γ ∈ N, the operator P γ[g+, h] coincides with the GJMS operator constructed by Graham et al [30] via the ambient metric; refer to Graham and Zworski [31]
Summary
Given any n ∈ N, let (Xn+1, g+) be an asymptotically hyperbolic manifold with conformal infinity (M n, [h]). There exists an asymptotically hyperbolic manifold Xn+1 that can be realized as a small perturbation of the Poincare half-space, for which the solution set of the γ-Yamabe problem is non-compact provided that n ≥ 24 or 25 according to the magnitude of γ ∈ (0, 1); refer to [41]. In this example, the conformal infinity M is the totally geodesic (in particular, umbilic) boundary of X. Is compact for any q ∈ [1, 2∗n,γ ) and a smooth bounded domain Ω ⊂ Rn
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
