Families of degenerating Poincar\'e-Einstein metrics on $\mathbb{R}^4$

TL;DR

This paper constructs the first continuous families of Poincaré-Einstein metrics on  that develop cusps and exhibit degenerations at infinity, expanding understanding of metric degenerations in geometric analysis.

Contribution

It provides the first examples of degenerating Poincare9-Einstein metrics on , including those with cusp formation and conformal infinity degenerations, using a novel ansatz.

Findings

First examples of cusping Poincare9-Einstein metrics on 

Families with degenerations at conformal infinity

Method based on Riemannian ansatz of Debever and Pleba
44ski-Demia
44ski

Abstract

We provide the first example of continuous families of Poincar\'e-Einstein metrics developing cusps on the trivial topology $\mathbb{R}^4$. We also exhibit families of metrics with unexpected degenerations in their conformal infinity only. These are obtained from the Riemannian version of an ansatz of Debever and Pleba\'nski-Demia\'nski. We additionally indicate how to construct similar examples on more complicated topologies.