Communications in Analysis and Geometry

Volume 26 (2018)

Number 1

Weakly asymptotically hyperbolic manifolds

Pages: 1 – 61

DOI: https://dx.doi.org/10.4310/CAG.2018.v26.n1.a1

Authors

Paul T. Allen (Department of Mathematical Sciences, Lewis and Clark College, Portland, Oregon, U.S.A.)

James Isenberg (Department of Mathematics, University of Oregon, Eugene, Or., U.S.A.)

John M. Lee (Department of Mathematics, University of Washington, Seattle, Wa., U.S.A.)

Iva Stavrov Allen (Department of Mathematical Sciences, Lewis and Clark College, Portland, Oregon, U.S.A.)

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 [Elliptic theory of differential edge operators, I Comm. Partial Differential Equations 16 (1991), no. 10, 1615–1664] and John M. Lee [Fredholm operators and Einstein metrics on conformally compact manifolds, Mem. Amer. Math. Soc. 183 (2006), no. 864, vi+83] 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.

Full Text (PDF format)

Received 18 June 2015

Published 31 January 2018