Non-compact Einstein manifolds with symmetry

Böhm, Christoph; Lafuente, Ramiro

doi:10.1090/jams/1022

Non-compact Einstein manifolds with symmetry
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by Christoph Böhm and Ramiro A. Lafuente;

J. Amer. Math. Soc. 36 (2023), 591-651

DOI: https://doi.org/10.1090/jams/1022

Published electronically: February 28, 2023

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Abstract:

For Einstein manifolds with negative scalar curvature admitting an isometric action of a Lie group $\mathsf {G}$ with compact, smooth orbit space, we show that the nilradical $\mathsf {N}$ of $\mathsf {G}$ acts polarly and that the $\mathsf {N}$-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.

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