A compactness theorem of the fractional Yamabe problem, Part I: The non-umbilic conformal infinity

2018/08/15 by Seunghyeok Kim, Monica Musso, Kim, Seunghyeok +3 · 3 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1808.04951

openalex publication_date 2018/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Assume that (X, g⁺) is an asymptotically hyperbolic manifold, (M, [h]) is its conformal infinity, ρ is the geodesic boundary defining function associated to h and g = ρ² g⁺. For any γ∈ (0,1), we prove that the solution set of the γ-Yamabe problem on M is compact in C²(M) provided that convergence of the scalar curvature R[g⁺] of (X, g⁺) to -n(n+1) is sufficiently fast as ρ tends to 0 and the second fundamental form on M never vanishes. Since most of the arguments in blow-up analysis performed here is irrelevant 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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