Geometry of isoparametric hypersurfaces in Riemannian manifolds

Abstract

In our previous work, we studied isoparametric functions on Riemannian manifolds, especially on exotic spheres. One result there says that, in the family of isoparametric hypersurfaces of a closed Riemannian manifold, there exists at least one minimal isoparametric hypersurface. In this paper, we show such a minimal isoparametric hypersurface is also unique in the family if the ambient manifold has positive Ricci curvature. Moreover, we give a proof of Theorem D claimed by Q.M.Wang (without proof) which asserts that the focal submanifolds of an isoparametric function on a complete Riemannian manifold are minimal. Further, we study isoparametric hypersurfaces with constant principal curvatures in general Riemannian manifolds. It turns out that in this case the focal submanifolds have the same properties as those in the standard sphere, i.e., the shape operator with respect to any normal direction has common constant principal curvatures. Some necessary conditions involving Ricci curvature and scalar curvature are also derived.2010 Mathematics Subject Classification. 53C20.

Lie Groups and Geometric Aspects of Isometric Actions

“…Remark 6.44. Some of the above submanifold geometry properties of orbits of a cohomogeneity one action were proved to remain true for general isoparametric foliations by Ge and Tang [95,96], see Remark 5.52.…”

“…Interesting aspects of submanifold geometry of the level sets of isoparametric functions on compact manifolds were studied by Ni [174], Wang [224], and Ge and Tang [95,96]. For example, it is proved that focal submanifolds of an isoparametric function are minimal, and that there exists at least one minimal hypersurface level set.…”

Isoparametric functions and nodal solutions of the Yamabe equation

“…Ge and Tang proved in [20] that if the co-dimension of each components of the focal submanifolds of f is at least 2 (which is actually equivalent to the focal set of any regular level set of f is exactly the focal submanifold M − ∪ M + ), then all the level sets are connected and there exists at least one isoparametric hypersurface induced by f that is a minimal hypersurface. Moreover, if the Ricci curvature is positive this minimal isoparametric hypersurface is unique, see (Corollary 2.1, [21]). The isoparametric functions with focal submanifolds of co-dimension greater than 1 are called proper isoparametric functions.…”

“…The isoparametric functions with focal submanifolds of co-dimension greater than 1 are called proper isoparametric functions. It was proved in [21] that the focal submanifolds M − and M + of an proper isoparametric function are minimal submanifolds. Actually this result was proved by Miyaoka in [38] for transnormal functions.…”

Geometric rigidity of constant heat flow

“…In the next subsection we will discuss the main class of examples of isoparametric tubes, namely, spherical domains bounded by isoparametric hypersurfaces. We finish this section by pointing out the following fact, which is proved in [16] (and first proved in [25] when the ambient manifold is the sphere). We discuss it in more detail in the last part of the Appendix.…”