A hypersurface in a Riemannian manifold is called isoparametric if itself as well as its nearby equidistant hypersurfaces have constant mean curvature (CMC), equivalently, if it is a regular level set of an isoparametric function on the ambient manifold. Isoparametric hypersurfaces and even isoparametric foliations of high codimensions in real space forms have been classified via many contributions (cf. [Q.-S. Chi, in: Proceedings of the international consortium of Chinese mathematicians, 2018. Second meeting, Taipei, Taiwan, December 2018. Somerville, MA: International Press. 197–260 (2020; Zbl 1465.53016)]). Also, isoparametric hypersurfaces and isoparametric foliations in symmetric spaces of rank one have been almost classified by J. C. Díaz-Ramos and the authors [J. Reine Angew. Math. 779, 189–222 (2021; Zbl 1483.53073); São Paulo J. Math. Sci. 15, No. 1, 75–110 (2021; Zbl 1480.53068)] as well as their collaborators in several other papers (see also J. Ge et al. [J. Math. Soc. Japan 67, No. 3, 1179–1212 (2015; Zbl 1331.53086)]). Isoparametric hypersurfaces in symmetric spaces of rank two were also studied and classified in some cases (such as in [F. Urbano, Commun. Anal. Geom. 27, No. 6, 1381–1416 (2019; Zbl 1429.53078)]).
In the paper under review, the authors treat symmetric spaces of non-compact type and rank \(\geq3\). Via a new general extension method for submanifolds from Euclidean spaces to symmetric spaces of non-compact type (which fits for those of rank \(\geq3\)), originally by M. Domínguez-Vázquez [Int. Math. Res. Not. 2015, No. 22, 12114–12125 (2015; Zbl 1331.53075)], they are able to construct minimal hypersurfaces, inhomogeneous, isoparametric, hyperpolar, singular Riemannian foliations (even of codimension higher than one) in a non-compact symmetric space of rank \(\geq3\). A surprising and remarkable case is that the construction leads to inhomogeneous isoparametric hypersurfaces in the product \(\mathbb{H}^2\times\mathbb{H}^2\times\mathbb{H}^2\) of three real hyperbolic planes, despite the fact that the isoparametric hypersurfaces of the irreducible factors are all homogeneous. The paper includes three main Theorems A, B and C, where A follows from B and B from C, the latter being exactly the general extension construction. We only state Theorem A as the most representative result.
Theorem A. Each symmetric space \(M\) of non-compact type and rank at least three admits inhomogeneous isoparametric families of hypersurfaces with non-austere focal submanifolds. If the rank is greater than or equal to four, there exist uncountably many such examples, up to congruence.
Focal submanifolds of isoparametric hypersurfaces in Riemannian manifolds were shown to be minimal in general, and even austere submanifolds for those isoparametric hypersurfaces with constant principal curvatures (by J. Ge and Z. Tang [Asian J. Math. 18, No. 1, 117–126 (2014; Zbl 1292.53040)]). The austerity of focal submanifolds has played a crucial role in the classification of isoparametric hypersurfaces in unit spheres and complex hyperbolic spaces. The fundamental construction theorem of C. Qian and Z. Tang [Adv. Math. 272, 611–629 (2015; Zbl 1312.53059)] states that a closed manifold with a Morse-Bott function \(f\) with critical sets of codimension \(\geq2\) (or a DDBD structure, i.e., a manifold with two (“double”) disk bundle decompositions) admits a Riemannian metric such that \(f\) is isoparametric and the focal submanifolds are totally geodesic. Non-austere focal submanifolds may only occur when the isoparametric hypersurfaces are inhomogeneous and have non-constant principal curvatures. Theorem A provides the first example of non-austere focal submanifolds.
The paper is very well-written and certainly an important reference in isoparametric theory especially on symmetric spaces of non-compact type.