arXiv research

The paper classifies isoparametric hypersurfaces in conic Finsler spaces.

Navigation in Lorentz Finsler geometry induces isoparametric hypersurfaces.

In this note, we give a classification of complete anisotropic isoparametric hypersurfaces, i.e., hypersurfaces with constant anisotropic principal curvatures, in Euclidean spaces, which is in analogue with the classical case for isoparametric hypersurfaces in Euclidean spaces. On the other hand, by an example of local…

This paper gives a survey of recent progress in isoparametric functions and isoparametric hypersurfaces, mainly in two directions. (1) Isoparametric functions on Riemannian manifolds, including exotic spheres. The existences and non-existences will be considered. (2) The Yau conjecture on the first eigenvalues of the e…

Study on isoparametric and constant-curvature hypersurfaces in Finsler spaces.

Classifies isoparametric hypersurfaces in 3D manifolds.

In this paper, we introduce and study the locally strongly convex equiaffine isoparametric hypersurfaces and equiaffine isoparametric functions on the affine space $A^{n+1}$. Motivated by the case on the Euclidean space $E^{n+1}$, we first introduce the concept of equiaffine parallel hypersurfaces in $A^{n+1}$, obtaini…

In this paper, we discuss anisotropic submanifolds and isoparametric hypersurfaces in a Randers space form (N,F) with the navigation datum (h,W). We find that (N, F) with respect to the BH-volume and (N,h) have the same isoparametric hypersurfaces although, in general, their isoparametric functions are different. This …

Minimal hypersurfaces in spheres generated by isoparametric foliations are found.

The study classifies isoparametric and homogeneous hypersurfaces in product spaces.

The paper classifies and explores isoparametric hypersurfaces in pseudo-Riemannian space forms.

Survey of Cartan's work on isoparametric hypersurfaces in spheres.

New isoparametric hypersurfaces found in Damek-Ricci spaces.

Surveying isoparametric hypersurfaces in spheres, focusing on Élie Cartan's techniques.

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 $BH$-vo…

Survey of Cartan and Münzner's work on isoparametric hypersurfaces.

The paper classifies isoparametric hypersurfaces in product spaces with different curvatures.

In this note, we study properties of the gradient map of the isoparametric polynomial. For a given isoparametric hypersurface in sphere, we calculate explicitly the gradient map of its isoparametric polynomial which turns out many interesting phenomenons and applications. We find that it should map not only the focal s…

This note constructs complex structures on specific isoparametric hypersurfaces.

The study classifies isoparametric hypersurfaces in specific product spaces.

The paper proves isoparametric functions on Finsler space forms under specific conditions.

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 proc…

Revisits and applies a formula for hypersurface Euler characteristics.

We classify isoparametric hypersurfaces in complex hyperbolic spaces.

The study classifies isoparametric hypersurfaces in product spaces with constant angle function.

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 L…

Isoparametric hypersurfaces and their application to special geometries

In this paper, we obtain a Cartan type identity for curvature-adapted isoparametric hypersurfaces in symmetric spaces of compact type or non-compact type. This identity is a generalization of Cartan-D'Atri's identity for curvature-adapted(=amenable) isoparametric hypersurfaces in rank one symmetric spaces. Furthermore,…

The paper constructs complex structures on specific manifolds using isoparametric theory.

In a 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 exist at least one minimal isoparametric hypersurface. In this note, we show such minimal isoparam…

We construct examples of inhomogeneous isoparametric real hypersurfaces in complex hyperbolic spaces.

Study shows how certain hypersurfaces evolve under mean curvature flow.

In this paper, we study isoparametric hypersurfaces in Finsler space forms by investigating focal points, tubes and parallel hypersurfaces of submanifolds. We prove that the focal submanifolds of isoparametric hypersurfaces are anisotropic-minimal and obtain the Cartan-type formula in a Finsler space form with vanishin…

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 Münzner. This completes the classification of isoparametric hypersurfaces in spheres that É. Cartan initiated…

Paper proves isoparametric property for certain hypersurfaces.

The paper studies inhomogeneous isoparametric hypersurfaces in pseudo-spheres.

Complete classification of isoparametric hypersurfaces in product spaces of 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 hypersurfaces in Lorentzian space forms and provide a local characterization of such hypersu…

New examples of hypersurfaces found in quaternionic hyperbolic spaces.

We classify isoparametric hypersurfaces in spheres with $(g,m)=(6,1)$ and thereby reprove a result of Dorfmeister and Neher.

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 space…

The study of Laguerre isotropic hypersurfaces with rigidity and isoparametric properties.

The image of the Gauss map of any oriented isoparametric hypersurface of the unit standard sphere $S^{n+1}(1)$ is a minimal Lagrangian submanifold in the complex hyperquadric $Q_n({\mathbf C})$. In this paper we show that the Gauss image of a compact oriented isoparametric hypersurface with $g$ distinct constant princi…

Survey on Dupin hypersurfaces and isoparametric hypersurfaces in spheres.

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.

We construct uncountably many isoparametric families of hypersurfaces in Damek-Ricci spaces. We characterize those of them that have constant principal curvatures by means of the new concept of generalized Kahler angle. It follows that, in general, these examples are inhomogeneous and have nonconstant principal curvatu…

In this note we will review the most important results and questions related to Chern conjecture and isoparametric hypersurfaces, as well as their interactions and applications to various aspects in mathematics.

A new proof of the homogeneity of isoparametric hypersurfaces with six simple principal curvatures (Dorfmeister-Neher's theorem) is given in a method applicable to the multiplicity two case.