Weakly asymptotically hyperbolic manifolds

Abstract

We introduce a class of "weakly asymptotically hyperbolic" geometries whose sectional curvatures tend to \(-1\) and are \(C^0\), but are not necessarily \(C^1\), conformally compact. We subsequently investigate the rate at which curvature invariants decay at infinity, identifying a conformally invariant tensor which serves as an obstruction to "higher order decay" of the Riemann curvature operator. Finally, we establish Fredholm results for geometric elliptic operators, extending the work of Rafe Mazzeo and John M. Lee to this setting. As an application, we show that any weakly asymptotically hyperbolic metric is conformally related to a weakly asymptotically hyperbolic metric of constant negative curvature.