Gluing constructions for asymptotically hyperbolic manifolds with constant scalar curvature

TL;DR

This paper demonstrates methods to deform asymptotically hyperbolic manifolds with constant scalar curvature into exact Kottler solutions in the asymptotic region, under various smallness, decay, or genericity conditions.

Contribution

It introduces gluing constructions that allow precise deformation of asymptotically hyperbolic manifolds to exact solutions in the asymptotic region, expanding the toolkit for geometric analysis.

Findings

Deformation is possible under smallness conditions in dimensions n≥3.

Fast decay conditions enable deformation in dimensions n≥5.

A genericity condition allows deformation in dimensions n≥9.

Abstract

We show that asymptotically hyperbolic initial data satisfying smallness conditions in dimensions $n\ge 3$, or fast decay conditions in $n\ge 5$, or a genericity condition in $n\ge 9$, can be deformed, by a deformation which is supported arbitrarily far in the asymptotic region, to ones which are exactly Kottler ("Schwarzschild- adS") in the asymptotic region.