Homogeneous Hypersurfaces Research Articles

Orbit Spaces Arising from Isometric Actions on Hyperbolic Spaces

Let be a differentiable action of a Lie group on a differentiable manifold and consider the orbit space with the quotient topology. Dimension of is called the cohomogeneity of the action of on . If is a differentiable manifold of cohomogeneity one under the action of a compact and connected Lie group, then the orbit space is homeomorphic to one of the spaces , , or . In this paper we suppose that the hyperbolic space is of cohomogeneity two under the action of , a connected and closed subgroup of Then we prove that its orbit space is homeomorphic to or Also we prove that either all orbits are diffeomorphic to or there are nonnegative integers such that some orbits are diffeomorphic to , and the other orbits are diffeomorphic to , where may be a sphere, a homogeneous hypersurface of sphere or a helix in some Euclidean space. ../files/site1/files/0Abstract6.pdf

Fourier transforms of irregular mixed homogeneous hypersurface measures

AbstractWith the help of Van der Corput lemmas, decay estimates are proven for Fourier transforms of mixed homogeneous hypersurface measures with densities that can be quite irregular. The primary results are local in nature, but can be extended to global theorems in an appropriate sense. The estimates are sharp for a certain range of indices in the theorems.

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 $$(S^n,h)$$ .

Möbius homogeneous hypersurfaces with three distinct principal curvatures in S n+1

Let x: M n → S n+1 be an immersed hypersurface in the (n + 1)-dimensional sphere S n+1. If, for any points p, q ∈ M n , there exists a Mobius transformation ϕ: S n+1 → S n+1 such that ϕ ○ x(M n ) = x(M n ) and ϕ ○ x(p) = x(q), then the hypersurface is called a Mobius homogeneous hypersurface. In this paper, the Mobius homogeneous hypersurfaces with three distinct principal curvatures are classified completely up to a Mobius transformation.

Об аффинно-однородных вещественных гиперповерхностях общего положения в C3

Об аффинно-однородных вещественных гиперповерхностях общего положения в C3

Topologically equisingular deformations of homogeneous hypersurfaces with line singularities are equimultiple

We prove that if [Formula: see text] is a family of line singularities with constant Lê numbers and such that [Formula: see text] is a homogeneous polynomial, then [Formula: see text] is equimultiple. This extends to line singularities a well-known theorem of Gabrièlov and Kušnirenko concerning isolated singularities. As an application, we show that if [Formula: see text] is a topologically [Formula: see text]-equisingular family of line singularities, with [Formula: see text] homogeneous, then [Formula: see text] is equimultiple. This provides a new partial positive answer to the famous Zariski multiplicity conjecture for a special class of non-isolated hypersurface singularities.

Three-dimensional locally homogeneous nondegenerate centroaffine hypersurfaces with nondiagonalizable Tchebychev operator

Liu and Wang (Beitrage Algebra Geom 35:109–117, 1994) determined homogeneous nondegenerate centroaffine surfaces in the 3-dimensional affine space. However, n-dimensional homogeneous or locally homogeneous nondegenerate centroaffine hypersurfaces for \(n\ge 3\) have not been completely classified yet. In this paper, we determine 3-dimensional locally homogeneous nondegenerate centroaffine hypersurfaces with nondiagonalizable Tchebychev operator.

On homogeneous hypersurfaces in $${\mathbb {C}}^3$$ C 3

We consider a family \(M_t^n\), with \(n\geqslant 2\), \(t>1\), of real hypersurfaces in a complex affine n-dimensional quadric arising in connection with the classification of homogeneous compact simply connected real-analytic hypersurfaces in \({\mathbb {C}}^n\) due to Morimoto and Nagano. To finalize their classification, one needs to resolve the problem of the embeddability of \(M_t^n\) in \({\mathbb {C}}^n\) for \(n=3,7\). In our earlier article we showed that \(M_t^7\) is not embeddable in \({\mathbb {C}}^7\) for every t and that \(M_t^3\) is embeddable in \({\mathbb {C}}^3\) for all \(1<t<1+10^{-6}\). In the present paper, we improve on the latter result by showing that the embeddability of \(M_t^3\) in fact takes place for \(1<t<\sqrt{(2+\sqrt{2})/3}\). This is achieved by analyzing the explicit totally real embedding of the sphere \(S^3\) in \({\mathbb {C}}^3\) constructed by Ahern and Rudin. For \(t\geqslant {\sqrt{(2+\sqrt{2})/3}}\), the problem of the embeddability of \(M_t^3\) remains open.

Derivations of the moduli algebras of weighted homogeneous hypersurface singularities

Let R=C[x1,x2,⋯,zn]/(f) where f is a weighted homogeneous polynomial defining an isolated singularity at the origin. Then R, and Der(R), the Lie algebra of derivations on R, are graded. It is well-known that Der(R) has no negatively graded component [10]. J. Wahl conjectured that the above fact is still true in higher codimensional case provided that R=C[x1,x2,⋯,xn]/(f1,f2,⋯,fm) is an isolated, normal and complete intersection singularity and f1,f2,⋯,fm are weighted homogeneous polynomials with the same weight type (w1,w2,⋯,wn). On the other hand the first author Yau conjectured that the moduli algebra A(V)=C[x1,x2,⋯,xn]/(∂f/∂x1,⋯,∂f/∂xn) has no negatively weighted derivations where f is a weighted homogeneous polynomial defining an isolated singularity at the origin. Assuming this conjecture has a positive answer, he gave a characterization of weighted homogeneous hypersurface singularities only using the Lie algebra Der(A(V)) of derivations on A(V). The conjecture of Yau can be thought as an Artinian analogue of J. Wahl's conjecture. For the low embedding dimension, the Yau conjecture has a positive answer. In this paper we prove this conjecture for any high-dimensional singularities under the condition that the lowest weight is bigger than or equal to half of the highest weight.

Biharmonic hypersurfaces in Riemannian symmetric spaces I

We classify biharmonic geodesic spheres in the Cayley projective plane. Our results completes the classification of all biharmonic homogeneous hypersurfaces in simply connected compact Riemannian symmetric spaces of rank 1. In addition we show that complex Grassmannian manifolds, and exceptional Lie groups $F_4$ and $G_2$ admit proper biharmonic real hypersurfaces.

A characterization of the homogeneous ruled real hypersurface in a complex hyperbolic space in terms of the first curvature of some integral curves

Due to the works of Berndt and others (J Reine Angew Math 395:132–141, 1989; Geom Dedicata 138:129–150, 2009; Trans Am Math Soc 359:3425–3438, 2007), we find that the class of all homogeneous real hypersurfaces M in \({\mathbb{C}H^n(c)}\) has just one example which is minimal in this space. Furthermore, in this ambient space, a homogeneous real hypersurface M is minimal if and only if it is ruled. This fact implies that it is interesting to study this minimal homogeneous real hypersurface from the viewpoint of the geometry of ruled real hypersurfaces. It is known that the shape operator A of every ruled real hypersurface M is given by the characteristic field \({\xi}\) and the vector field U. The purpose of this paper is to characterize this minimal homogeneous real hypersurface in \({\mathbb{C}H^n(c)}\) in the class of all ruled real hypersurfaces M by investigating the first curvature of all integral curves of the vector fields \({\xi}\) and U. Note that there exist minimal non-homogeneous ruled real hypersurfaces in \({\mathbb{C}H^n(c)}\) (see Geom Dedicata 79:267–286, 1999; Hokkaido Math J 43:1–14, 2014).

Biharmonic homogeneous hypersurfaces in compact symmetric spaces

In this paper, we study biharmonic hypersurfaces in Einstein manifolds. Then, we determine all the biharmonic hypersurfaces in irreducible symmetric spaces of compact type which are regular orbits of commutative Hermann actions of cohomogeneity one.

A filtration for isoparametric hypersurfaces in Riemannian manifolds

Motivated by the well known Cartan-Munzner equations and the Chern conjecture, we introduce the notion of k-isoparametric hypersurface in an (n + 1)- dimensional Riemannian manifold for k = 0,1,...,n. Many fundamental and inter- esting results ( towards the classification of homogeneous hypersurfaces among other things ) are given in complex projective spaces, complex hyperbolic spaces, and even in locally rank one symmetric spaces. Further investigations in this direction are expected.

On complete list of affine homogeneous surfaces of (ɛ, 0)-types in the space ℂ3

The problem of description of affine homogeneous real hypersurfaces in complex spaces is an important part of the problem of holomorphic classification of homogeneous manifolds, which has no complete solution till now, even in the 3-dimensional case. The method developed by the authors, based on affine canonical equations and techniques of matrix Lie algebras, allowed them earlier to obtain complete descriptions of two natural classes of affine homogeneous real hypersurfaces in the 3-dimensional complex space. In this paper, we present the complete description of one more class. The description includes examples already known and (obtained with the use of symbolic computations) Lie algebras corresponding to the other homogeneous manifolds belonging to the types under consideration. The main result of the paper is obtained by means of integration of these algebras.

Affine-homogeneous surfaces of type (0, 0) in the space ℂ3

The problem of describing affine homogeneous real hypersurfaces of the complex space C3 was posed in [1]. Such an affine setting is closely related to the problem of classifying holomorphically homogeneous submanifolds of multidimensional complex spaces (see [2]–[4]). A scheme for the study of the affine homogeneity property based on the use of the canonical equations of the manifolds under consideration was proposed in [5]–[7]. The scheme envisages a separate study of several (for example, seven) types of strictly pseudoconvex hypersurfaces depending on the second-order Taylor coefficients of their canonical equations. This scheme was used in [6]–[8] to give a complete description of two of the seven cases mentioned above. Below we describe the third possible type.

Complete characterization of isolated homogeneous hypersurface singularities

Complete characterization of isolated homogeneous hypersurface singularities

Classification of 2-dimensional graded normal hypersurfaces with a(R) ≤ 6

We classify 2-dimensional normal weighted homogeneous hypersurface R = k[X, Y, Z]/(f) with given a-invariant a(R) ≤ 6. We show that for a(R) > 0, the number of “types” are finite.

The Generalized Cayley Hypersurfaces and Their Geometrical Characterization

In this paper, we study a whole family of n-dimensional equiaffine homogeneous hypersurfaces with a parameter α, constructed by Eastwood and Ezhov (Proc Steklov Inst Math 253:221–224, 2006), called the generalized Cayley hypersurfaces. By introducing a new parametrization we find that the generalized Cayley hypersurfaces are improper affine hypersphere with flat affine metric and vanishing Pick invariant, whose difference tensor K satisfies \({\nabla^{(\alpha)}K=0 \,\,{\rm and}\,\, K^{n-1} \neq 0}\), where the affine \({\alpha{\rm -connection}\,\, \nabla^{(\alpha)}}\) of information geometry is first introduced on affine hypersurface for each \({\alpha\in\mathbb{R}}\). As main result, we establish a characterization of the generalized Cayley hypersurfaces by the last two properties for some nonzero constant α.

On two methods for reconstructing homogeneous hypersurface singularities from their Milnor algebras

On two methods for reconstructing homogeneous hypersurface singularities from their Milnor algebras

On a family of real hypersurfaces in a complex quadric

We discuss a family Mtn, with n⩾2, t>1, of real hypersurfaces in a complex affine n-dimensional quadric arising in connection with the classification of homogeneous compact simply-connected real-analytic hypersurfaces in Cn due to Morimoto and Nagano. To finalize their classification, one needs to resolve the problem of the embeddability of Mtn in Cn for n=3,7. We show that Mt7 does not embed in C7 for every t and observe that Mt3 embeds in C3 for all t sufficiently close to 1. As a consequence of analyzing a map constructed by Ahern and Rudin, we also conjecture that Mt3 embeds in C3 for all 1<t<(2+2)/3.