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.
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…
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.
Hilbert's 17th problem asks that whether every nonnegative polynomial can be a sum of squares of rational functions. It has been answered affirmatively by Artin. However, the question as to whether a given nonnegative polynomial is a sum of squares of polynomials is still a central question in real algebraic geometry. …
The paper studies SDP feasibility and sos ranks for specific polynomials.
problem Characterizing sos representations of nonnegative polynomials.
method Explicit SDP formulation based on Clifford systems.
result Quantitative rank bounds for sos representations, with rigidity.
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.
In this paper, we prove that a quartic polynomial solution of the eikonal equation ∣∇xf∣2=16x6 in Rn is either an isoparametric polynomial or congruent to a polynomial f=(∑i=1nxi2)2−8(∑i=1kxi2)(∑i=k+1nxi2), k=0,1,.˙.,[2n].
We give a generalization of the well-known result of E. Cartan on isoparametric cubics by showing that a homogeneous cubic polynomial solution of the eiconal equation ∣∇f∣2=9∣x∣4 must be rotationally equivalent to either xn3−3xn(x12+...+xn−12), or to one of four exceptional Cartan cubic polynomials…
In contrast to an infinite family of explicit examples of two-dimensional p-harmonic functions obtained by G.Aronsson in the late 80s, there is very little known about the higher-dimensional case. In this paper, we show how to use isoparametric polynomials to produce diverse examples of p-harmonic and biharmonic fu…
We show that any homogeneous polynomial solution of |\nabla F(x)|^2=m^2|x|^(2m-2), m>1, is either a radially symmetric polynomial F(x)=\pm |x|^m (for even m's) or it is a composition of a Chebychev polynomial and a Cartan-Münzner polynomial.
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 classification of isoparametric hypersurfaces with four principal curvatures in spheres in [2] hinges on a crucial characterization, in terms of four sets of equations of the 2nd fundamental form tensors of a focal submanifold, of an isoparametric hypersurface of the type constructed by Ferus, Karcher and Münzner. …
Study degenerate solutions on product of spheres using bifurcation theory.
problem Existence of degenerate solutions on product of spheres.
method Bifurcation theory, isoparametric functions, Gegenbauer polynomials.
result Existence of degenerate solutions that depend on both factors.
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.
Investigates polynomial solutions to minimal surface equation, proving constraints and structure theorems.
problem Finding polynomial solutions to the minimal surface equation.
method Proves structure theorem, analyzes polynomial constraints, and uses eigenvalue estimates.
result Polynomial solutions must contain terms of both high and low degree, and have specific factorization properties.
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 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 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 …
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.
In this paper, we introduce and study the locally strongly convex equiaffine isoparametric hypersurfaces and equiaffine isoparametric functions on the affine space An+1. Motivated by the case on the Euclidean space En+1, we first introduce the concept of equiaffine parallel hypersurfaces in An+1, obtaini…
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…
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 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.
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.
This is a survey on the recent progress in several applications of isoparametric theory, including an affirmative answer to Yau's conjecture on the first eigenvalue of Laplacian in the isoparametric case, a negative answer to Yau's 76th problem in his Problem Section, new examples of Willmore submanifolds in spheres, a…
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.
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.
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.
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. Ancient solutions found for mean curvature flows in hyperbolic spaces.
problem Mean curvature flows of isoparametric submanifolds in hyperbolic spaces.
method Study mean curvature flows of isoparametric submanifolds in hyperbolic spaces.
result Ancient solutions always exist for these flows.
We classify isoparametric hypersurfaces in complex hyperbolic spaces.
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.
Based on representation theory of Clifford algebra, Ferus, Karcher and Münzner constructed a series of isoparametric foliations. In this paper, we will survey recent studies on isoparametric hypersurfaces of OT-FKM type and investigate related geometric constructions with mean curvature flow.