Einstein hypersurfaces in irreducible symmetric spaces

TL;DR

This paper classifies Einstein hypersurfaces in irreducible symmetric spaces of rank greater than one, revealing specific geometric structures and foliations, extending previous rank-one results to higher ranks.

Contribution

It provides a classification of Einstein hypersurfaces in higher rank irreducible symmetric spaces, identifying their geometric types and foliations, which was previously known only for rank-one cases.

Findings

In noncompact type, hypersurfaces are Einstein solvmanifolds.

In special cases, hypersurfaces are foliated by totally geodesic spheres or hyperbolic planes.

The space of leaves relates to Legendrian surfaces and affine spheres.

Abstract

We show that if $M$ is an Einstein hypersurface in an irreducible Riemannian symmetric space $\overline{M}$ of rank greater than $1$ (the classification in the rank-one case was previously known), then either $\overline{M}$ is of noncompact type and $M$ is a codimension one Einstein solvmanifold, or $\overline{M}=\mathrm{SU}(3)/\mathrm{SO}(3)$ (respectively, $\overline{M}=\mathrm{SL}(3)/\mathrm{SO}(3)$) and $M$ is foliated by totally geodesic spheres (respectively, hyperbolic planes) of $\overline{M}$, with the space of leaves parametrised by a special Legendrian surface in $S^5$ (respectively, by a proper affine sphere in $\mathbb{R}^3$).