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 [20] and John M. Lee [17] 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 scalar curvature.

Low regularity Poincaré–Einstein metrics

“…By applying the inverse function theorem, we correct these approximate solutions to obtain actual Einstein metrics with the same conformal infinities. Finally, we show that the resulting Einstein metrics are in the class of "weakly asymptotically hyperbolic" metrics introduced in [1], which implies that they have C 1,1 conformal compactifications. This paper is structured as follows.…”

“…Function Spaces. In this section we review the function spaces from [8] and [1] that we will need in the subsequent analysis. The main result of this section is the extension of the regularization procedure from [1] to Lipschitz spaces.…”

“…We will use two scales of Hölder spaces of sections of E from [1], which we call plain and fancy. For k ≥ 0 and α ∈ [0, 1], the plain Hölder spaces C k,α (M ; E) are defined by the norm…”

“…The details of the construction ensure that the regularity of the metric cannot be improved to C 2 by any coordinate or conformal change. Next, by using the regularization technique and intermediate spaces introduced by Allen, Isenberg, Stavrov Allen and the second author in [1], we produce approximate solutions to the linearized Einstein equation with C 1,1 regularity. By applying the inverse function theorem, we correct these approximate solutions to obtain actual Einstein metrics with the same conformal infinities.…”

“…In the next section we describe the function spaces we will need in the subsequent analysis. We extend the definitions and regularization procedure given in [1] to Lipschitz spaces. In Section 3 we describe our extension map from C 1,1 boundary metrics to weakly asymptotically hyperbolic metrics in the interior.…”