Isoparametric Hypersurfaces Research Articles

Isoparametric foliations and critical sets of eigenfunctions

Jakobson and Nadirashvili \cite{JN} constructed a sequence of eigenfunctions on $T^2$ with a bounded number of critical points, answering in the negative the question raised by Yau \cite{Yau1} which asks that whether the number of the critical points of eigenfunctions for the Laplacian increases with the corresponding eigenvalues. The present paper finds three interesting eigenfunctions on the minimal isoparametric hypersurface $M^n$ in $S^{n+1}(1)$. The corresponding eigenvalues are $n$, $2n$ and $3n$, while their critical sets consist of $8$ points, a submanifold(infinite many points) and $8$ points, respectively. On one of its focal submanifolds, a similar phenomenon occurs.

On the equivalence of two curvature conditions for Lorentzian hypersurfaces

Let n≥3. We show that semi-symmetry and Ricci-semisymmetry conditions are equivalent for any n-dimensional Lorentzian hypersurface in a Lorentzian space form with nonzero curvature. We also show that these curvature conditions are equivalent for any n-dimensional Lorentzian isoparametric hypersurface in Minkowski space R1n+1, and we construct an example of a Ricci-semisymmetric 5-dimensional Lorentzian hypersurface in R16 which is not semi-symmetric.

CMC hypersurfaces with canonical principal direction in space forms

A hypersurface of the space form has a canonical principal direction (CPD) relative to the closed and conformal vector field Z of if the projection of Z to M is a principal direction of M. We show that CPD hypersurfaces with constant mean curvature are foliated by isoparametric hypersurfaces. In particular, we show that a CPD surface with constant mean curvature of space form is invariant by the flow of a Killing vector field whose action is polar on . As consequence we show that a compact CPD minimal surface of the sphere is a Clifford torus. Finally, we consider the case when a CPD Euclidean hypersurface has zero Gauss–Kronecker curvature.

Hamiltonian non-displaceability of Gauss images of isoparametric hypersurfaces

In this article we study the Hamiltonian non-displaceability of Gauss images of isoparametric hypersurfaces in the spheres as Lagrangian submanifolds embedded in complex hyperquadrics.

Conformal regular spacelike hypersurfaces in a conformal space $$\mathbb {Q}^{n+1}_1$$ Q 1 n + 1

Let \(x: M \rightarrow \mathbb {Q}^{n+1}_1\) be an \(n (n\ge 4)\)-dimensional conformal regular spacelike hypersurface in the conformal space \(\mathbb {Q}^{n+1}_1\) with vanishing conformal form. If x is a conformal Blaschke isoparametric spacelike hypersurface in \(\mathbb {Q}^{n+1}_1\) with three distinct conformal principal curvatures, one of which is simple or x is of parallel conformal Blaschke tensor, we obtain some classification of such conformal regular spacelike hypersurface x.

Errata of ``Isoparametric hypersurfaces with (g,m)=(6,2)"

We give a correction of the proof of the homogeneity of isoparametric hypersurfaces with (g;m) = (6; 2).

Isoparametric foliations, a problem of Eells–Lemaire and conjectures of Leung

In this paper, two sequences of minimal isoparametric hypersurfaces are constructed via representations of Clifford algebras. Based on these, we give estimates on eigenvalues of the Laplacian of the focal submanifolds of isoparametric hypersurfaces in unit spheres. This improves results of [TY13] and [TXY14]. Eells and Lemaire [EL83] posed a problem to characterize the compact Riemannian manifold M for which there is an eigenmap from M to S^n. As another application of our constructions, the focal maps give rise to many examples of eigenmaps from minimal isoparametric hypersurfaces to unit spheres. Most importantly, by investigating the second fundamental forms of focal submanifolds of isoparametric hypersurfaces in unit spheres, we provide infinitely many counterexamples to two conjectures of Leung [Le91] (posed in 1991) on minimal submanifolds in unit spheres. Notice that these conjectures of Leung have been proved in the case that the normal connection is flat [HV01].

Isoparametric hypersurfaces in Minkowski spaces

In this paper, we introduce isoparametric functions and isoparametric hypersurfaces in Finsler manifolds and give the necessary and sufficient conditions for a transnormal function to be isoparametric. We then prove that hyperplanes, Minkowski hyperspheres and F⁎-Minkowski cylinders in a Minkowski space with BH-volume (resp. HT-volume) form are all isoparametric hypersurfaces with one and two distinct constant principal curvatures respectively. Moreover, we give a complete classification of isoparametric hypersurfaces in Randers–Minkowski spaces.

Singular critical points for variational free boundary problem from isoparametric hypersurfaces

We construct three families of singular critical points for a variational free boundary problem. These critical points are homogeneous solutions of degree one to some overdetermined boundary value problem. The intersections of the level sets of these solutions with the unit sphere are isoparametric hypersurfaces and their focal submanifolds.

On submanifolds whose tubular hypersurfaces have constant higher order mean curvatures

Motivated by the theory of isoparametric hypersurfaces, we study submanifolds whose tubular hypersurfaces have some constant higher order mean curvatures. Here a k-th order mean curvature Qkν (k ≥ 1) of a submanifold Mn is defined as the k-th power sum of the principal curvatures, or equivalently, of the shape operator with respect to the unit normal vector ν. We show that if all nearby tubular hypersurfaces of M have some constant higher order mean curvatures, then the submanifold M itself has some constant higher order mean curvatures Qkν independent of the choice of ν. Many identities involving higher order mean curvatures and Jacobi operators on such submanifolds are also obtained. In particular, we generalize several classical results in isoparametric theory given by E. Cartan, K. Nomizu, H. F. M¨unzner, Q. M. Wang, et al. As an application, we finally get a geometrical filtration for the focal submanifolds of isoparametric functions on a complete Riemannian manifold.

Heat flow, heat content and the isoparametric property

Let M be a Riemannian manifold and $$\Omega $$ a compact domain of M with smooth boundary. We study the solution of the heat equation on $$\Omega $$ having constant unit initial conditions and Dirichlet boundary conditions. The purpose of this paper is to study the geometry of domains for which, at any fixed value of time, the normal derivative of the solution (heat flow) is a constant function on the boundary. We express this fact by saying that such domains have the constant flow property. In constant curvature spaces known examples of such domains are given by geodesic balls and, more generally, by domains whose boundary is connected and isoparametric. The question is: are they all like that? This problem is the analogous (for the heat equation) of the classical Serrin’s problem for harmonic domains. In this paper we give an affirmative answer to the above question: in fact we prove more generally that, if a domain in an analytic Riemannian manifold has the constant flow property, then every component of its boundary is an isoparametric hypersurface. For space forms, we also relate the order of vanishing of the heat content with fixed boundary data with the constancy of the r-mean curvatures of the boundary and with the isoparametric property. Finally, we discuss the constant flow property in relation to other well-known overdetermined problems involving the Laplace operator, like the Serrin problem or the Schiffer problem.

Rigidity of linear Weingarten hypersurfaces in locally symmetric manifolds

Our purpose in this paper is to study the rigidity of complete linear Weingarten hypersurfaces immersed in a locally symmetric manifold obeying some standard curvature conditions (in particular, in a Riemannian space with constant sectional curvature). Under appropriated constrains on the scalar curvature function, we prove that such a hypersurface must be either totally umbilical or isometric to an isoparametric hypersurface with two distinct principal curvatures, one of them being simple. Furthermore, we also deal with the parabolicity of these hypersurfaces with respect to a suitable Cheng–Yau modified operator.

A Class of Special Hypersurfaces in Real Space Forms

We define the generalized golden- and product-shaped hypersurfaces in real space forms. A hypersurfaceMin real space formsRn+1,Sn+1, andHn+1is isoparametric if it has constant principal curvatures. Based on the classification of isoparametric hypersurfaces, we obtain the whole families of the generalized golden- and product-shaped hypersurfaces in real space forms.

On complete linear Weingarten hypersurfaces in locally symmetric Riemannian manifolds

Our aim is to apply suitable generalized maximum principles in order to obtain characterization results concerning complete linear Weingarten hypersurfaces immersed in a locally symmetric Riemannian manifold, whose sectional curvature is supposed to obey standard constraints. In this setting, we establish sufficient conditions to guarantee that such a hypersurface must be either totally umbilical or an isoparametric hypersurface with two distinct principal curvatures one of which is simple.

Complete hypersurfaces with two distinct principal curvatures in a locally symmetric Riemannian manifold

We deal with complete hypersurfaces with two distinct principal curvatures in a locally symmetric Riemannian manifold, which is supposed to obey some appropriated curvature constraints. Initially, considering the case that such a hypersurface has constant mean curvature, we apply a Simons type formula jointly with the well known generalized maximum principle of Omori–Yau to show that it must be isometric to an isoparametric hypersurface of the ambient space. Afterwards, we use a Cheng–Yau modified operator in order to obtain a sort of extension of this previously mentioned result for the context of linear Weingarten hypersurfaces, that is, hypersurfaces whose mean and scalar curvatures are linearly related.

Willmore submanifolds in the unit sphere via isoparametric functions

In this paper, we show that both focal submanifolds of each isoparametric hypersurface in the sphere with six distinct principal curvatures are Willmore, hence all focal submanifolds of isoparametric hypersurfaces in the sphere are Willmore.

A holonomy invariant anisotropic surface energy in a Riemannian manifold

In this paper, we investigate a holonomy invariant elliptic anisotropic surface energy for hypersurfaces in a complete Riemannian manifold, where “holonomy invariant” means that the elliptic parametric Lagrangian (i.e., a Finsler metric) of the Riemannian manifold used to define the anisotropic surface energy is constant along each holonomy subbundle of the tangent bundle of the Riemannian manifold. First we obtain the first variational formula for this anisotropic surface energy. Next we shall introduce the notions of an anisotropic convex hypersurface, an anisotropic equifocal hypersurface and an anisotropic isoparametric hypersurface for this anisotropic surface energy. Also, we shall introduce the notion of an anisotropic tube for this anisotropic surface energy. We prove that anisotropic tubes over a one-point set in a symmetric space are anisotropic convex hypersurfaces and that anisotropic tubes over a certain kind of reflective submanifold in a symmetric space are anisotropic isoparametric and anisotropic equifocal hypersurfaces.

Blaschke isoparametric hypersurfaces in the conformal space ℚ 1 n+1 , I

Let x: M → Q1n+1 be a regular hypersurface in the conformal space Q1n+1. We classify all the space-like Blaschke isoparametric hypersurfaces with two distinct Blaschke eigenvalues in the conformal space up to the conformal equivalence.

Isoparametric foliation and a problem of Besse on generalizations of Einstein condition

The focal sets of isoparametric hypersurfaces in spheres with g=4 are all Willmore submanifolds, being minimal but mostly non-Einstein [25,22]. Inspired by A. Gray's view, the present paper shows that these focal sets are all A-manifolds but rarely Ricci parallel, except possibly for the only unclassified case. As a byproduct, it gives infinitely many simply-connected examples to the problem 16.56(i) of Besse concerning generalizations of the Einstein condition.

Clifford systems, Cartan hypersurfaces and Riemannian submersions

Cartan hypersurfaces are minimal isoparametric hypersurfaces with 3 distinct constant principal curvatures in unit spheres. In this article, we firstly build a relationship between the focal submanifolds of Cartan hypersurfaces and the Hopf fiberations and give a new proof of the classification result on Cartan hypersurfaces. Nextly, we show that there exists a Riemannian submersion with totally geodesic fibers from each Cartan hypersurface M3m to the projective planes \({{\mathbb{F}}P^2}\) (\({{\mathbb{F}}={\mathbb{R}},{\mathbb{C}},{\mathbb{H}},{\mathbb{O}}}\) for m = 1, 2, 4, 8, respectively) endowed with the canonical metrics. As an application, we give several interesting examples of Riemannian submersions satisfying a basic equality due to Chen (Proc Jpn Acad Ser A Math Sci 81:162–167, 2005).