The paper classifies isoparametric hypersurfaces in conic Finsler spaces.
problem Identifying new isoparametric hypersurfaces in conic Finsler spaces.
method Introduced isoparametric functions and hypersurfaces in conic Finsler spaces, classified them in specific spaces.
result Found additional isoparametric hypersurfaces in conic Minkowski spaces, such as helicoids.
The paper classifies isoparametric hypersurfaces in Randers space forms.
problem Classifying isoparametric hypersurfaces in Randers space forms.
method Anisotropic submanifolds and isoparametric hypersurfaces in Randers space forms (N,F) with (h,W) were studied.
result The isoparametric hypersurfaces in a Randers space form (N,F) are the same as in Riemannian space, despite different isoparametric functions.
The paper defines and studies equiaffine isoparametric hypersurfaces and functions.
problem Understanding and characterizing equiaffine isoparametric hypersurfaces and functions.
method Introduced equiaffine parallel hypersurfaces, defined equiaffine isoparametric hypersurfaces and functions, and proved their relationship.
result Equiaffine isoparametric hypersurfaces are regular level sets of equiaffine isoparametric functions.
Navigation in Lorentz Finsler geometry induces isoparametric hypersurfaces.
problem Defining and analyzing isoparametric hypersurfaces in Lorentz Finsler geometry.
method Using a navigation process with a Finsler metric and a tangent vector field, isoparametric functions and hypersurfaces are defined and analyzed.
result Local correspondences between isoparametric functions and hypersurfaces are established.
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.
problem Characterizing and constructing hypersurfaces with specific curvature properties in Finsler geometry.
method Analyzing isoparametric and constant-curvature hypersurfaces on Randers manifolds.
result Construction of a conformally flat Randers manifold with nonisoparametric hyperplanes of constant curvatures.
Classifies isoparametric hypersurfaces in 3D manifolds.
problem Identifying isoparametric hypersurfaces in specific 3D manifolds.
method By proving constant angle and principal curvatures.
result Established classification of hypersurfaces.
Study isoparametric hypersurfaces in Finsler space forms, proving anisotropic-minimal focal submanifolds.
problem Investigate isoparametric hypersurfaces in Finsler space forms.
method Investigate focal points, tubes, and parallel hypersurfaces of submanifolds; prove anisotropic-minimal focal submanifolds; derive Cartan-type formula.
result Prove isoparametric hypersurfaces in Finsler space forms have anisotropic-minimal focal submanifolds.
Study isoparametric hypersurfaces in a Randers sphere with constant flag curvature.
problem Characterize isoparametric hypersurfaces in a Randers sphere of constant flag curvature.
method Analyze isoparametric hypersurfaces for the standard round sphere and show their behavior under navigation.
result Provide a classification of special isoparametric hypersurfaces and their ambient metrics.
Minimal hypersurfaces in spheres generated by isoparametric foliations are found.
problem Existence of minimal hypersurfaces in spheres generated by isoparametric foliations.
method Generalized rotational ansatz formed by the union of homothetic copies of isoparametric leaves, reducing the minimal surface equation to an ordinary differential equation.
result Closed embedded minimal hypersurfaces of topological type S1imesM are found for any isoparametric hypersurface M⊂Sn. The study classifies isoparametric and homogeneous hypersurfaces in product spaces.
problem Classifying hypersurfaces in product spaces.
method Exploiting rigidity of constant angle functions and constant principal curvatures.
result Complete classification of isoparametric and homogeneous hypersurfaces in SnimesRm and HnimesRm. The paper classifies and explores isoparametric hypersurfaces in pseudo-Riemannian space forms.
problem Investigating isoparametric hypersurfaces in pseudo-Riemannian space forms.
method Petrov's classification theorem and shape operator analysis.
result No isoparametric hypersurfaces of index 2 with complex principal curvatures exist.
Survey of Cartan's work on isoparametric hypersurfaces in spheres.
problem Understanding isoparametric hypersurfaces and their focal submanifolds.
method Review of Cartan's original papers from 1938-1940.
result Detailed description of isoparametric hypersurfaces and their focal submanifolds.
New isoparametric hypersurfaces found in Damek-Ricci spaces.
problem Characterizing new isoparametric hypersurfaces in Damek-Ricci spaces.
method Defining and studying 'sphere-like' hypersurfaces formed by extending horospheres.
result Found a new family of isoparametric hypersurfaces connecting geodesic spheres to previously known ones.
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…
Surveying isoparametric hypersurfaces in spheres, focusing on Élie Cartan's techniques.
problem Classifying isoparametric hypersurfaces in spheres.
method Historical survey and focus on Élie Cartan's techniques.
result Attention to Élie Cartan's techniques in the case of four principal curvatures.
Survey of Cartan and Münzner's work on isoparametric hypersurfaces.
problem Understanding isoparametric hypersurfaces in real space forms.
method Review of four papers by Cartan and two papers by Münzner.
result Complete classification of isoparametric hypersurfaces in spheres.
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…
The paper classifies isoparametric hypersurfaces in product spaces with different curvatures.
problem Characterizing isoparametric hypersurfaces in product spaces with varying curvatures.
method Analyzing hypersurfaces in product spaces with constant sectional curvatures.
result Classification of isoparametric hypersurfaces in product spaces with different curvatures.
Mean curvature flow by parallel hypersurfaces is solved for isoparametric hypersurfaces.
problem Finding solutions to mean curvature flow by parallel hypersurfaces.
method Solving an ordinary differential equation for isoparametric hypersurfaces.
result Explicit solutions and exact collapsing time for isoparametric hypersurfaces of the sphere.
The study classifies isoparametric hypersurfaces in specific product spaces.
problem Classifying isoparametric hypersurfaces in product spaces.
method Analyzing the angle function and constant principal curvatures.
result A complete classification of isoparametric and homogeneous hypersurfaces in SnimesSm and SnimesHm. The paper studies special null hypersurfaces in spacetimes.
problem Characterizing null screen isoparametric hypersurfaces in Lorentzian space forms.
method Developed screen isoparametric hypersurface concept for null hypersurfaces of Robertson-Walker spacetimes, derived Cartan identities, and provided local characterizations.
result Derived Cartan identities for the screen principal curvatures of null screen hypersurfaces in Lorentzian space forms and provided a local characterization.
This note constructs complex structures on specific isoparametric hypersurfaces.
problem Building complex structures on isoparametric hypersurfaces.
method Constructing almost or complex structures on isoparametric hypersurfaces in unit spheres.
result Complex structures on S1imesS7imesS6 and S1imesS3imesS2 are built. The paper proves isoparametric functions on Finsler space forms under specific conditions.
problem Understanding isoparametric functions in Finsler space forms.
method Proving transnormal functions as isoparametric functions and constructing global and local isoparametric functions using the distance function.
result Generalization of Theorem B to Finsler space forms.
Revisits and applies a formula for hypersurface Euler characteristics.
problem Understanding isoparametric hypersurfaces in space forms.
method Re-proves Allendoerfer-Weil's formula and applies it to isoparametric hypersurfaces.
result New insights into isoparametric hypersurfaces.
We classify isoparametric hypersurfaces in complex hyperbolic spaces.
The study classifies isoparametric hypersurfaces in product spaces with constant angle function.
problem Classifying isoparametric hypersurfaces in product spaces with specific curvature conditions.
method Proving constant angle function and using it to classify hypersurfaces.
result Classification of isoparametric and homogeneous hypersurfaces in product spaces.
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
The paper constructs complex structures on specific manifolds using isoparametric theory.
problem Constructing explicit complex structures on closed manifolds.
method Isoparametric theory applied to specific manifolds.
result The construction of complex structures on isoparametric hypersurfaces and focal submanifolds.
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,…
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…
Study disproves a conjecture about constant curvature hypersurfaces.
problem Whether every hypersurface with constant principal curvatures is isoparametric.
method Constructed an explicit example of a conformally flat Riemannian manifold.
result Disproved the conjecture by providing a counterexample.
Study shows how certain hypersurfaces evolve under mean curvature flow.
problem Evolution of isoparametric hypersurfaces under mean curvature flow.
method Reparametrization of the parallel family in short time.
result Evolution given by a reparametrization of the parallel family.
We construct examples of inhomogeneous isoparametric real hypersurfaces in complex hyperbolic spaces.
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…
This paper determines bounds on normal scalar curvature of isoparametric hypersurface focal submanifolds.
problem Classifying points with specific conditions on isoparametric hypersurface focal submanifolds.
method Analyzing the second fundamental form and scalar curvature of focal submanifolds.
result Points with Condition A achieve an upper bound of normal scalar curvature.
Paper proves isoparametric property for certain hypersurfaces.
problem Chern conjecture for isoparametric hypersurfaces.
method Analyzes hypersurfaces with constant mean curvature and scalar curvature, showing isoparametric property under specific conditions.
result Generalizes Chern's conjecture to any dimension, proving isoparametric property.
The paper studies inhomogeneous isoparametric hypersurfaces in pseudo-spheres.
problem Investigating inhomogeneous isoparametric hypersurfaces in pseudo-sphere.
method Construction of Clifford systems and analysis of isoparametric hypersurfaces.
result Connected isoparametric hypersurfaces of OT-FKM-type in pseudo-spheres are inhomogeneous under specific conditions.
Complete classification of isoparametric hypersurfaces in product spaces of space forms.
problem Classifying isoparametric hypersurfaces in product spaces of space forms.
method Proving constant product angle function and removing constant principal curvatures condition.
result Complete classification of isoparametric hypersurfaces in product spaces of space forms.
Classifies isoparametric functions and hypersurfaces in specific spaces.
problem Classifying isoparametric functions and hypersurfaces in complex spaces.
method Analyzing functions and hypersurfaces on RnimesMm and S2imesR2. result Identified isoparametric functions and hypersurfaces with compact level sets.
New examples of hypersurfaces found in quaternionic hyperbolic spaces.
problem Classifying actions on symmetric spaces of rank one.
method Cohomogeneity one actions and orbit equivalence.
result Uncountably many inhomogeneous isoparametric families of hypersurfaces.
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.
problem Characterizing and understanding Laguerre isotropic hypersurfaces.
method Analyzing hypersurfaces with zero Laguerre form and constant eigenvalues of the Laguerre tensor.
result For L-isotropic hypersurfaces, if they are also L-isoparametric, the constant λ must be zero. The image of the Gauss map of any oriented isoparametric hypersurface of the unit standard sphere Sn+1(1) is a minimal Lagrangian submanifold in the complex hyperquadric Qn(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.
problem Understanding the relationship between compact proper Dupin hypersurfaces and isoparametric hypersurfaces in spheres.
method Survey of existing results and related developments.
result Progress on Cecil and Ryan's conjecture on compact proper Dupin hypersurfaces.