Non-compact Einstein manifolds with symmetry

TL;DR

This paper proves a rigidity result for negatively curved Einstein manifolds with symmetry, showing the nilradical acts polarly and extends to minimal Einstein submanifolds, leading to a proof of the Alekseevskii conjecture.

Contribution

It establishes the polarly acting nilradical and minimal Einstein extensions, confirming the Alekseevskii conjecture for homogeneous Einstein manifolds with negative scalar curvature.

Findings

Nilradical acts polarly on the manifold

N-orbits extend to minimal Einstein submanifolds

Homogeneous Einstein manifolds with negative scalar curvature are diffeomorphic to Euclidean space

Abstract

For Einstein manifolds with negative scalar curvature admitting an isometric action of a Lie group G with compact, smooth orbit space, we show the following rigidity result: The nilradical N of G acts polarly, and the N-orbits can be extended to minimal Einstein submanifolds. As an application, we prove the Alekseevskii conjecture: Any homogeneous Einstein manifold with negative scalar curvature is diffeomorphic to a Euclidean space.