Weakly asymptotically hyperbolic manifolds

TL;DR

This paper introduces a new class of weakly asymptotically hyperbolic manifolds with less regularity, studies curvature decay, identifies an invariant tensor as an obstruction, and extends Fredholm theory to these geometries.

Contribution

It defines weakly asymptotically hyperbolic manifolds, analyzes curvature decay rates, and extends elliptic operator results to this broader setting.

Findings

Curvature invariants decay at specific rates at infinity.

Existence of a conformally invariant obstruction tensor.

Any such metric is conformally related to a constant negative curvature metric.

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.