Almost Einstein and Poincaré-Einstein manifolds in Riemannian signature
A real function \(s\) on a Riemannian manifold \((M,g)\) is called an almost Einstein structure if the trace-free part of \[ \text{Hess}^g(s)\;+\;sP^g \] vanishes identically, where \(P^g\) denotes the Schouten tensor. This PDE is conformally covariant and for any solution \(s\) the metric \(s^{-2}g\) in the conformal class \([g]\) on \(M\) is Einstein. However, \(s\) might have zeros. Thus, in general, the underlying conformal manifold \((M,[g])\) is called almost Einstein (=AE). This extends the notion of Einstein spaces to conformal geometry. The idea for this comes from tractor calculus. In his work the author describes the singular set \(\Sigma\) for an AE structure \(s\). The notion of almost scalar constant spaces is introduced as well. In the negative scalar curvature case any singular set \(\Sigma\neq\emptyset\) is a smooth hypersurface and the model of AE manifolds is closely linked to the Poincaré-Einstein model. The author compares the conformal geometry of \(\Sigma\) and \(M\). Moreover, he derives curvature conditions for the existence of AE structures. This relates to Dirichlet type problems of finding AE structures on an ambient space for a given scale singularity \(\Sigma\). Also, the relation of the Fefferman-Graham ambient spaces for \(\Sigma\) and \(M\) is discussed.
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