Isoparametric Hypersurfaces Research Articles

On the linear Weingarten spacelike submanifolds immersed in a locally symmetric semi-Riemannian space

Let $$M^{n}$$ be an n-dimensional complete linear Weingarten spacelike submanifold immersed with parallel normalized mean curvature vector field and flat normal bundle in a locally symmetric semi-Riemannian space $$L_{p}^{n+p}$$ of index p, which obeys standard curvature constraints (such an ambient space can be regarded as an extension of a semi-Riemannian space form). In this setting, our purpose is to establish sufficient conditions guaranteeing that such a spacelike submanifold $$M^{n}$$ be either totally umbilical or isometric to an isoparametric hypersurface of a totally geodesic submanifold $$L_{1}^{n+1}\hookrightarrow L_{p}^{n+p}$$, with two distinct principal curvatures, one of which is simple. Our approach is based on a suitable Simons type formula jointly with a version of the Omori–Yau’s generalized maximum principle for a Cheng–Yau’s modified operator.

Geodesic Chord Property and Hypersurfaces of Space Forms

In the Euclidean space E n , hyperplanes, hyperspheres and hypercylinders are the only isoparametric hypersurfaces. These hypersurfaces are also the only ones with chord property, that is, the chord connecting two points on them meets the hypersurfaces at the same angle at the two points. In this paper, we investigate hypersurfaces in nonflat space forms with the so-called geodesic chord property and classify such hypersurfaces completely.

Equiaffine Isoparametric Functions and Their Regular Level Hypersurfaces

In this paper, we aim to introduce and study the (locally strongly convex) equiaffine isoparametric functions on the affine space $$A^{n+1}$$, making the emphasis on their relation with the affine isoparametric hypersurfaces. Motivated by the case in the Euclidean space $$E^{n+1}$$, we first introduce the concept of equiaffine parallel hypersurfaces in $$A^{n+1}$$, and then equivalently re-define the equiaffine isoparametric hypersurfaces to be ones that are among families of equiaffine parallel hypersurfaces in $$A^{n+1}$$ of constant affine mean curvature. As the main result, we prove that an equiaffine isoparametric hypersurface is nothing but exactly a regular level set of some equiaffine isoparametric function.

Robust index bounds for minimal hypersurfaces of isoparametric submanifolds and symmetric spaces

We find many examples of compact Riemannian manifolds (M, g) whose closed minimal hypersurfaces satisfy a lower bound on their index that is linear in their first Betti number. Moreover, we show that these bounds remain valid when the metric g is replaced with $$g'$$ in a neighbourhood of g. Our examples (M, g) consist of certain minimal isoparametric hypersurfaces of spheres, their focal manifolds, the Lie groups $${\text {SU}}(n)$$ for $$n\le 17$$ and $${\text {Sp}}(n)$$ for all n, and all quaternionic Grassmannians.

Submanifold geometry in symmetric spaces of noncompact type

In this survey article we provide an introduction to submanifold geometry in symmetric spaces of noncompact type. We focus on the construction of examples and the classification problems of homogeneous and isoparametric hypersurfaces, polar and hyperpolar actions, and homogeneous CPC submanifolds.

Null Screen Isoparametric Hypersurfaces in Lorentzian Space Forms

In this paper, we develop the notion of screen isoparametric hypersurface for null hypersurfaces of Robertson–Walker spacetimes. Using this formalism we derive Cartan identities for the screen principal curvatures of null screen isoparametric hypersurfaces in Lorentzian space forms and provide a local characterization of such hypersurfaces.

The mean curvature flow by parallel hypersurfaces

It is shown that a hypersurface of a space form is the initial data for a solution to the mean curvature flow by parallel hypersurfaces if, and only if, it is isoparametric. By solving an ordinary differential equation, explicit solutions are given for all isoparametric hypersurfaces of space forms. In particular, for such hypersurfaces of the sphere, the exact collapsing time into a focal submanifold is given in terms of its dimension, the principal curvatures and their multiplicities.

Isoparametric functions on [formula omitted

We classify the isoparametric functions on Rn×Mm, n,m≥2, with compact level sets, where Mm is a connected, closed Riemannian manifold of dimension m. Also, we classify the isoparametric hypersurfaces in S2×R2 with constant principal curvatures.

Lorentzian isoparametric hypersurfaces in the Lorentzian sphere $\boldsymbol{S_1^{n+1}}$

There are many research papers for isoparametric hypersurfaces in Riemannian space forms and almost perfect results have already been got till now. In the late 20th century, the Lorentzian isoparametric hypersurfaces in the Lorentzian space $\bbr_1^{n+1}$ and the Lorentzian hyperbolic space $H_1^{n+1}$ have been completely classified. Continuing with our previous work, we investigate Lorentzian isoparametric hypersurfaces in the Lorentzian sphere $S_1^{n+1}$ in this paper.Then complete classification and analytic expression are given for all kinds of Lorentzian isoparametric hypersurfaces in $S_1^{n+1}$.

Spacelike Dupin hypersurfaces in Lorentzian space forms

Similar to the definition in Riemannian space forms, we define the spacelike Dupin hypersurface in Lorentzian space forms. As conformal invariant objects, spacelike Dupin hypersurfaces are studied in this paper using the framework of the conformal geometry of spacelike hypersurfaces. Further we classify the spacelike Dupin hypersurfaces with constant Mobius curvatures, which are also called conformal isoparametric hypersurface.

Normal Scalar Curvature Inequality on the Focal Submanifolds of Isoparametric Hypersurfaces

An isoparametric hypersurface in unit spheres has two focal submanifolds. Condition A plays a crucial role in the classification theory of isoparametric hypersurfaces in [CCJ07], [Chi16] and [Miy13]. This paper determines $C_A$, the set of points with Condition A in focal submanifolds. It turns out that the points in $C_A$ reach an upper bound of the normal scalar curvature $\rho^{\bot}$ (sharper than that in DDVV inequality [GT08], [Lu11]). We also determine the sets $C_P$ (points with parallel second fundamental form) and $C_E$ (points with Einstein condition), which achieve two lower bounds of $\rho^{\bot}$.

On CMC hypersurfaces in 𝕊 n+1 with constant Gauß–Kronecker curvature

Abstract After nearly 50 years of research the Chern conjecture for isoparametric hypersurfaces in spheres is still an unsolved and important problem. Here we give a partial result for CMC hypersurfaces with constant Gauß–Kronecker curvature, mainly using a result given in [3] by Otsuki.

Biharmonic hypersurfaces in 5-dimensional non-flat space forms

Abstract We prove that every biharmonic hypersurface having constant higher order mean curvature Hr for r > 2 in a space form M 5(c) is of constant mean curvature. In particular, every such biharmonic hypersurface in 𝕊5(1) has constant mean curvature. There exist no such compact proper biharmonic isoparametric hypersurfaces M in 𝕊5(1) with four distinct principal curvatures. Moreover, there exist no proper biharmonic hypersurfaces in hyperbolic space ℍ5 or in E 5 having constant higher order mean curvature Hr for r > 2.

Isoparametric hypersurfaces in a Randers sphere of constant flag curvature

In this paper, I study the isoparametric hypersurfaces in a Randers sphere $$(S^n,F)$$ of constant flag curvature, with the navigation datum (h, W). I prove that an isoparametric hypersurface M for the standard round sphere $$(S^n,h)$$ which is tangent to W remains isoparametric for $$(S^n,F)$$ after the navigation process. This observation provides a special class of isoparametric hypersurfaces in $$(S^n,F)$$ , which can be equivalently described as the regular level sets of isoparametric functions f satisfying $$-f$$ is transnormal. I provide a classification for these special isoparametric hypersurfaces M, together with their ambient metric F on $$S^n$$ , except the case that M is of the OT-FKM type with the multiplicities $$(m_1,m_2)=(8,7)$$ . I also give a complete classification for all homogeneous hypersurfaces in $$(S^n,F)$$ . They all belong to these special isoparametric hypersurfaces. Because of the extra W, the number of distinct principal curvature can only be 1, 2 or 4, i.e. there are less homogeneous hypersurfaces for $$(S^n,F)$$ than those for $$(S^n,h)$$ .

A new structural approach to isoparametric hypersurfaces in spheres

In this paper we show that the long-standing problem of classifying all isoparametric hypersurfaces in spheres with six different principal curvatures is still not complete. Moreover, we develop a structural approach that may be helpful for such a classification. Instead of working with the isoparametric hypersurface family in the sphere, we consider the associated Lagrangian submanifold of the real Grassmannian of oriented 2-planes in {mathbb {R}}^{n+2}. We obtain new geometric insights into classical invariants and identities in terms of the geometry of the Lagrangian submanifold.

Isoparametric hypersurfaces in complex hyperbolic spaces

We classify isoparametric hypersurfaces in complex hyperbolic spaces.

Isoparametric hypersurfaces in Funk spaces

Funk metrics are a kind of important Finsler metrics with constant negative flag curvature. In this paper, it is proved that any isoparametric hypersurface in Funk spaces has at most two distinct principal curvatures. Moreover, a complete classification of isoparametric families in a Funk space is given.

Ricci curvature of double manifolds via isoparametric foliations

Given a closed manifold M and a vector bundle ξ of rank n over M, by gluing two copies of the disc bundle of ξ, we can obtain a closed manifold D(ξ,M), the so-called double manifold.In this paper, we firstly prove that each sphere bundle Sr(ξ) of radius r>0 is an isoparametric hypersurface in the total space of ξ equipped with a connection metric, and for r>0 small enough, the induced metric of Sr(ξ) has positive Ricci curvature under the additional assumptions that M has a metric with positive Ricci curvature and n≥3.As an application, if M admits a metric with positive Ricci curvature and n≥2, then we construct a metric with positive Ricci curvature on D(ξ,M). Moreover, under the same metric, D(ξ,M) admits a natural isoparametric foliation.For a compact minimal isoparametric hypersurface Yn in Sn+1(1), which separates Sn+1(1) into S+n+1 and S−n+1, one can get double manifolds D(S+n+1) and D(S−n+1). Inspired by Tang, Xie and Yan's work on scalar curvature of such manifolds with isoparametric foliations (cf. [25]), we study Ricci curvature of them with isoparametric foliations in the last part.

Möbius curvature, Laguerre curvature and Dupin hypersurface

In this paper we show that a Dupin hypersurface with constant Möbius curvatures is Möbius equivalent to either an isoparametric hypersurface in the sphere or a cone over an isoparametric hypersurface in a sphere. We also show that a Dupin hypersurface with constant Laguerre curvatures is Laguerre equivalent to a flat Laguerre isoparametric hypersurface. These results solve the major issues related to the conjectures of Cecil et al. on the classification of Dupin hypersurfaces.

Linear Weingarten hypersurfaces in locally symmetric manifolds

In this paper, we discuss about the complete linear Weingarten hypersurfaces in locally symmetric manifold and obtain a rigidity theorem. More precisely, under a suitable restriction on the square norm of the second fundamental form, we prove that such a hypersurface must be either totally umbilical or an isoparametric hypersurface with two distinct principal curvatures, one of which is simple.