Sobolev-class asymptotically hyperbolic manifolds and the Yamabe problem

TL;DR

This paper develops new analytical tools for Sobolev-class asymptotically hyperbolic manifolds and successfully solves the Yamabe problem within this regularity framework, broadening the scope of geometric analysis.

Contribution

Introduces novel function spaces and establishes Fredholm theorems for elliptic operators on Sobolev-class asymptotically hyperbolic manifolds, enabling the solution of the Yamabe problem under weaker regularity assumptions.

Findings

Established Fredholm properties for elliptic operators on Sobolev-class manifolds.

Proved the Yamabe problem is solvable with $W^{1,p}$ conformal compactifications for $p > ext{dimension}$.

Developed technical tools for PDE analysis on low-regularity geometric structures.

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 H\"older 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.