Sobolev-class asymptotically hyperbolic manifolds and the Yamabe problem

2022/06/26 by Paul T. Allen, John M. Lee, Allen, Paul T. +3 · 1 citation
Mathematics · #46E35 (Secondary) #58J05 (Primary) 53C21 #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2206.12854

openalex publication_date 2022/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider asymptotically hyperbolic manifolds whose metrics have Sobolev-class regularity, and introduce several technical tools for studying PDEs on such manifolds. Our results employ two novel families of function spaces suitable for metrics potentially having a large amount of interior differentiability, but whose Sobolev regularity implies only a Hölder continuous conformal structure at infinity. We establish Fredholm theorems for elliptic operators arising from metrics in these families. To demonstrate the utility of our methods, we solve the Yamabe problem in this category. As a special case, we show that the asymptotically hyperbolic Yamabe problem is solvable so long as the metric admits a W^1,p conformal compactification, with p greater than the dimension of the manifold.

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