Isoparametric Hypersurfaces In Spheres Research Articles

Isoparametric hypersurfaces with four principal curvatures, IV

We prove that an isoparametric hypersurface with four principal curvatures and multiplicity pair $(7,8)$ is either the one constructed by Ozeki and Takeuchi, or one of the two constructed by Ferus, Karcher, and M\"{u}nzner. This completes the classification of isoparametric hypersurfaces in spheres that \'{E}. Cartan initiated in the late 1930s.

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.

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 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.

On Focal Submanifolds of Isoparametric Hypersurfaces and Simons Formula

The focal submanifolds of isoparametric hypersurfaces in spheres are all minimal Willmore submanifolds, mostly being $${\mathcal{A}}$$ -manifolds in the sense of A.Gray but rarely Ricci-parallel, see Li et al. (Sci China Math 58, 2015), Qian et al. (Ann Glob Anal Geom 43:47–62, 2013), Tang and Yan (Isoparametric foliation and a problem of Besse on generalizations of Einstein condition arXiv:1307.3807 , 2013). In this paper we study the geometry of the focal submanifolds via Simons formula. We show that all the focal submanifolds with g ≥ 3 are not normally flat by estimating the normal scalar curvatures. Moreover, we give a complete classification of the semiparallel submanifolds among the focal submanifolds.

MOMENT MAPS AND ISOPARAMETRIC HYPERSURFACES IN SPHERES — HERMITIAN CASES

We are studying a relationship between isoparametric hypersurfaces in spheres with four distinct principal curvatures and the moment maps of certain Hamiltonian actions. In this paper, we consider the isoparametric hypersurfaces obtained from the isotropy representations of compact irreducible Hermitian symmetric spaces of rank two. We prove that the Cartan-M\"unzner polynomials of these hypersurfaces can be written as squared-norms of the moment maps for some Hamiltonian actions. The proof is based on the structure theory of symmetric spaces.

Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces, I

The image of the Gauss map of any oriented isoparametric hypersurface in the standard unit sphere $S^{n+1}(1)$ is a minimal Lagrangian submanifold in the complex hyperquadric $Q_n(\mathrm{C})$. In this paper we show that the Gauss image of a compact oriented isoparametric hypersurface with $g$ distinct constant principal curvatures in $S^{n+1}(1)$ is a compact monotone and cyclic embedded Lagrangian submanifold with minimal Maslov number $2n / g$. We obtain the Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces of classical type with $g = 4$. Combining with our results in On Lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres and Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces II, we completely determine the Hamiltonian stability of the Gauss images of all homogeneous isoparametric hypersurfaces.

Homogeneous isoparametric hypersurfaces in spheres with four distinct principal curvatures and moment maps

We study relations between moment maps of Hamiltonian actions and isoparametric hypersurfaces in spheres with four distinct principal curvatures. In this paper, we deal with the isoparametric hypersurfaces given by the isotropy representations of compact irreducible Hermitian symmetric spaces of classical type and of rank two. We show that such isoparametric hypersurfaces can be obtained by moment maps. More precisely, certain squared-norms of moment maps coincide with Cartan-Münzner polynomials, which are defining-equations, of above isoparametric hypersurfaces.

On Lagrangian submanifolds in complex hyperquadrics and isoparametric hypersurfaces in spheres

The n-dimensional complex hyperquadric is a compact complex algebraic hypersurface defined by the quadratic equation in the (n+1)-dimensional complex projective space, which is isometric to the real Grassmann manifold of oriented 2-planes and is a compact Hermitian symmetric space of rank 2. In this paper, we study geometry of compact Lagrangian submanifolds in complex hyperquadrics from the viewpoint of the theory of isoparametric hypersurfaces in spheres. From this viewpoint we provide a classification theorem of compact homogeneous Lagrangian submanifolds in complex hyperquadrics by using the moment map technique. Moreover we determine the Hamiltonian stability of compact minimal Lagrangian submanifolds embedded in complex hyperquadrics which are obtained as Gauss images of isoparametric hypersurfaces in spheres with g(= 1, 2, 3) distinct principal curvatures.

Distinguished dimensions for special Riemannian geometries

The paper is based on relations between a ternary symmetric form defining the SO ( 3 ) geometry in dimension five and Cartan’s works on isoparametric hypersurfaces in spheres. As observed by Bryant such a ternary form exists only in dimensions n k = 3 k + 2 , where k = 1 , 2 , 4 , 8 . In these dimensions it reduces the orthogonal group to the subgroups H k ⊂ SO ( n k ) , with H 1 = SO ( 3 ) , H 2 = SU ( 3 ) , H 4 = Sp ( 3 ) and H 8 = F 4 . This enables studies of special Riemannian geometries with structure groups H k in dimensions n k . The necessary and sufficient conditions for the H k geometries to admit the characteristic connection are given. As an illustration nontrivial examples of SU ( 3 ) geometries in dimension 8 admitting characteristic connection are provided. Among them are the examples having nonvanishing torsion and satisfying Einstein equations with respect to either the Levi-Civita or the characteristic connections. The torsionless models for the H k geometries have the respective symmetry groups G 1 = SU ( 3 ) , G 2 = SU ( 3 ) × SU ( 3 ) , G 3 = SU ( 6 ) and G 4 = E 6 . The groups H k and G k constitute a part of the ‘magic square’ for Lie groups. The ‘magic square’ Lie groups suggest studies of ten other classes of special Riemannian geometries. Apart from the two exceptional cases, they have the structure groups U ( 3 ) , S ( U ( 3 ) × U ( 3 ) ) , U ( 6 ) , E 6 × SO ( 2 ) , Sp ( 3 ) × SU ( 2 ) , SU ( 6 ) × SU ( 2 ) , SO ( 12 ) × SU ( 2 ) and E 7 × SU ( 2 ) and should be considered in respective dimensions 12, 18, 30, 54, 28, 40, 64 and 112. The two ‘exceptional’ cases are: SU ( 2 ) × SU ( 2 ) geometries in dimension 8 and SO ( 10 ) × SO ( 2 ) geometries in dimension 32. The case of SU ( 2 ) × SU ( 2 ) geometry in dimension 8 is examined closer. We determine the tensor that reduces SO ( 8 ) to SU ( 2 ) × SU ( 2 ) leaving the more detailed studies of the geometries based on the magic square ideas to the forthcoming paper.

Tight polyhedral models of isoparametric families, and PL-taut submanifolds

Abstract We present polyhedral models for isoparametric families in the sphere with at most three principal curvatures. Each member of the family (including the analogues of the focal sets) is tight in the boundary complex of an ambient convex polytope. In particular, the tube around the real (or complex) Veronese surface is represented as a tight polyhedron in 5-space (or 8-space). The examples are based on a certain Bier sphere triangulation of S 4 or S 7, respectively. In the 4-dimensional case there are simplicial branched coverings of these triangulations in the complex projective plane and in S 2 × S 2 which are branched precisely along the polyhedral analogues of the Veronese surface. Moreover, we introduce a notion of PL-tautness and discuss its relationship with tightness of polyhedra. In particular, each member of our polyhedral isoparametric family is PL-taut. For an extended abstract see [W. Kühnel, Discrete models of isoparametric hypersurfaces in spheres. Oberwolfach Reports 12, 665–667 (2006).].

Isoparametric hypersurfaces in spheres, integrable nondiagonalizable systems of hydrodynamic type, and N-wave systems

Isoparametric hypersurface M n ⊂ S n + 1 can be defined as an intersection of the unit sphere r 2 = ( u 1) 2 + … + ( u n + 2 ) 2 = 1 with the level set F( u) = const of a homogeneous polynomial F of degree g, satisfying Cartan-Munzner equations (▽ F) 2 = g 2 r 2 g − 2 , Δ F = cr g − 2 c = const. We introduce a hamiltonian system of hydrodynamic type u i t = 1 g δ ij d dx ∂F ∂u j , with the hamiltonian operator δ ij d dx and the hamiltonian density F (u) g . Under the additional assumption of the homogeneity of the hypersurface M n , the restriction of this system to M n proves to be nondiagonalizable, but integrable and can be transformed to an appropriate integrable reduction of the N-wave system. Possible generalizations to isoparametric submanifolds (finite or infinite dimensional) are also briefly indicated.

Projective planes and isoparametric hypersurfaces

The isoparametric hypersurfaces in spheres with three distinct principal curvatures were classified by Cartan in 1939. We give a new proof for this result by showing that every such hypersurface can be naturally identified with the flag space of a compact connected Moufang plane. Our approach also leads to a uniform and explicit description of these hypersurfaces.

Tits distance of Hadamard manifolds and isoparametric hypersurfaces

We prove that the Tits distance of nonhomogeneous isoparametric hypersurfaces in spheres cannot occur as Tits distance (in the sense of Gromov) at the boundary of a Hadamard manifold.

An algebraic approach to isoparametric hypersurfaces in spheres, II

An algebraic approach to isoparametric hypersurfaces in spheres, II

An algebraic approach to isoparametric hypersurfaces in spheres, I

An algebraic approach to isoparametric hypersurfaces in spheres, I

Isoparametric triple systems of algebra type

Introduction. In this paper we continue our study of isoparametric triple systems. These triple systems have been introduced in [3] and are studied in [3], [4] and [5]. They are in one-to-one correspondence with isoparametric hypersurfaces in spheres which have four distinct principal curvatures. The classes of isoparametric hypersurfaces which have been considered up to now are the homogeneous ones ([10], [11]), the surfaces of FKM-type ([5], [6]), the surfaces satisfying condition (A) and (B) ([9], [10]) and the surfaces where the multiplicity of one of the principal curvatures is < 2 ([12],

On some types of isoparametric hypersurfaces in spheres, II

On some types of isoparametric hypersurfaces in spheres, II

On some types of isoparametric hypersurfaces in spheres, I

On some types of isoparametric hypersurfaces in spheres, I