A class of compact Poincare-Einstein manifolds: properties and construction

TL;DR

This paper introduces a geometric construction method for generating and classifying compact Poincare-Einstein and almost Einstein manifolds, revealing their structure and relationships with conformal and hypersurface geometries.

Contribution

It develops an explicit construction principle that exhaustively classifies certain almost Einstein and Poincare-Einstein manifolds, linking them to conformal structures and hypersurfaces.

Findings

Construction yields families of compact Poincare-Einstein manifolds.

Classification results show the construction's exhaustiveness.

Almost Einstein structures relate to constant mean curvature hypersurfaces.

Abstract

We develop a geometric and explicit construction principle that generates classes of Poincare-Einstein manifolds, and more generally almost Einstein manifolds. Almost Einstein manifolds satisfy a generalisation of the Einstein condition; they are Einstein on an open dense subspace and, in general, have a conformal scale singularity set that is a conformal infinity for the Einstein metric. In particular, the construction may be applied to yield families of compact Poincare-Einstein manifolds, as well as classes of almost Einstein manifolds that are compact without boundary. We obtain classification results which show that the construction essentially exhausts a class of almost Einstein (and Poincare-Einstein) manifold. We develop the general theory of fixed conformal structures admitting multiple compatible almost Einstein structures. We also show that, in a class of cases, these are canonically related to a family of constant mean curvature totally umbillic embedded hypersurfaces.