Weakly asymptotically hyperbolic manifolds

2015/06/10 by Paul T. Allen, James Isenberg, Allen, Paul T. +5 · 3 citations
Mathematics · Physics and Astronomy · #53C21 #58J05 #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #gr-qc #math.DG #msc:53C21 #msc:58J05

paper · pdf · doi:10.48550/arxiv.1506.03399

Final version submitted to journal

arxiv created 2016/10/27 · arxiv updated 2016/10/28

Abstract

We introduce a class of "weakly asymptotically hyperbolic" geometries whose sectional curvatures tend to -1 and are C⁰, but are not necessarily C¹, 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.

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• The σ_k-Loewner-Nirenberg problem on Riemannian manifolds for k=(n)/(2) and beyond
• Green functions and a positive mass theorem for asymptotically hyperbolic 3-manifolds
• The shear-free condition and constant-mean-curvature hyperboloidal initial data

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