The study classifies rational isoparametric functions on Damek-Ricci spaces.
problem Classifying isoparametric functions on Damek-Ricci spaces.
method Using polynomial functions divided by t for classification. result New isoparametric functions discovered and their properties studied.
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.
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 consta…
Analyzes constant heat flow properties on Riemannian manifolds.
problem Classify manifolds with constant heat flow property.
method Analyzes overdetermined parabolic problems and uses isoparametric tubes.
result Analytic manifolds with constant flow property are isoparametric tubes.
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 studies geodesics and isoparametric functions on Finsler spheres.
problem Analyzing geodesics and isoparametric functions on Finsler spheres.
method Global expressions of geodesics and isoparametric functions derived using navigation and Cartan-Münzner polynomials.
result Construction of isoparametric families and focal submanifolds.
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.
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.
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…
The abstract discusses open problems in isoparametric theory.
problem Open problems in isoparametric theory.
method Survey and propose open problems.
result Discussion of open problems in isoparametric theory.
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.
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.
Survey on isoparametric theory applications and results.
problem Yau's conjecture and problems in isoparametric theory.
method Various examples and counterexamples in isoparametric theory.
result Affirmative answer to Yau's conjecture on the first eigenvalue of Laplacian.
Surveying isoparametric foliations and related geometric constructions.
problem Constructing isoparametric foliations using Clifford algebra.
method Representation theory of Clifford algebra and mean curvature flow.
result Investigation of geometric constructions related to isoparametric hypersurfaces.
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.
Direct proof for all conformally flat isoparametric submanifolds in Euclidean space.
problem Classifying conformally flat isoparametric submanifolds in Euclidean space.
method Direct proof approach.
result Complete classification of conformally flat isoparametric submanifolds of Euclidean space.
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…
Study finds nodal solutions of Yamabe equation constant along isoparametric levels.
problem Existence of nodal solutions for the Yamabe equation.
method Constant along isoparametric levels of a function.
result Existence of nodal solutions proven.
We show that an isoparametric submanifold of a complex hyperbolic plane, according to the definition of Heintze, Liu and Olmos', is an open part of a principal orbit of a polar action. We also show that there exists a non-isoparametric submanifold of the complex hyperbolic plane that is isoparametric according to the d…
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…
Weyl's tube formula holds for various cross-sections under symmetry conditions.
problem Can the volume of tubes around submanifolds be calculated for non-round cross-sections?
method Investigated the volume of tubes with general cross-sections D under symmetry conditions.
result The volume of tubes around submanifolds can be calculated for general cross-sections under symmetry conditions.
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.
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…
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.
New examples of isoparametric foliations on SnimesSn are provided.
problem Isoparametric foliations on SnimesSn of codimension 1. method Construction of new examples and classification of bi-homogeneous polynomials.
result Isoparametric foliations can be realized as level sets of specific bi-homogeneous polynomials.
Paper solves a central question about nonnegative polynomials related to isoparametric polynomials.
problem Whether a given nonnegative polynomial is a sum of squares of polynomials.
method Solves the problem completely for nonnegative polynomials associated with isoparametric polynomials.
result The paper provides a complete solution for the specific case of isoparametric polynomials.
Computes tube formulas for valuations in complex space forms.
problem Computing values of valuations on complex space forms.
method Develops tube formulas for valuations in complex space forms and generalizes classical formulas.
result Generalizes classical formulas of Weyl, Gray and others.
The paper explores the topology and curvature of isoparametric families in spheres.
problem Investigating the topology and curvature of isoparametric families in spheres.
method The paper investigates the topology and curvature of isoparametric families in spheres using homotopy, homeomorphism, diffeomorphism types, parallelizability, and Lusternik-Schnirelmann category.
result The paper determines conditions for non-negative sectional curvatures and positive Ricci curvatures in isoparametric families.
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. 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.
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. Extends isoparametric foliations and area-minimizing cones in product manifolds.
problem Generalizing isoparametric foliations and area-minimizing cones in SnimesSn. method Analyzes isoparametric foliations and area-minimizing cones, extending known results.
result Extends known area-minimizing cones to codimension-two cases, yielding infinitely many families of area-minimizing subcones.
The paper defines marginal tubes and proves their null nature.
problem Understanding the geometry of spacelike surfaces in spacetimes.
method Introducing marginal tubes and studying spacelike surfaces with double null coordinates.
result If every spacelike section of a marginal tube is a marginal surface, then the marginal tube is null.
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.
The paper classifies shapes of translating solitons from isoparametric graphs.
problem Understanding shapes of translating solitons from isoparametric graphs.
method Analyzing ordinary differential equations and isoparametric functions.
result Classification of shapes of translating solitons.
Classifies foliations of complex and quaternionic projective spaces.
problem Classifying isoparametric foliations of complex and quaternionic projective spaces.
method Investigating projections of inhomogeneous isoparametric foliations of the 31-sphere under Hopf fibrations.
result Solved the last remaining open cases in the classification.
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…
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.
It was conjectured, twenty years ago, the following result that would generalize the so-called rank rigidity theorem for homogeneous Euclidean submanifolds: let M^n, n>=2, be a full and irreducible homogeneous submanifold of the sphere SN−1⊂RN and such that the normal holonomy group is not transitive (on t…
New examples of isoparametric families on non-compact symmetric spaces.
problem Constructing isoparametric families with non-austere focal sets.
method New extension method from Euclidean spaces to symmetric spaces of non-compact type.
result First examples of isoparametric families on non-compact symmetric spaces.
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 proves certain Zoll manifolds with entire Grauert tubes are isometric.
problem Characterizing Zoll manifolds with entire Grauert tubes.
method Computed algebraic indices of tangent bundles; proved isometry.
result Any Zoll manifold of type HP^2 with entire Grauert tube is isometric to canonical HP^2.
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.
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.
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 study limits the number of specific foliations with bounded geometry.
problem Bounding the number of isoparametric foliations with bounded geometry.
method Proving finitely many foliations with specific properties and constructing infinite families of non-diffeomorphic foliations.
result There are only finitely many isoparametrically foliated closed connected Riemannian manifolds with bounded geometry, up to foliated diffeomorphism.
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. Zero entropy found in entire Grauert tubes of certain manifolds.
problem Analyzing the geodesic flow on Grauert tubes with zero entropy.
method Examined entire Grauert tubes on real analytic Riemannian manifolds.
result Geodesic flow on entire Grauert tubes has zero topological entropy.