We provide an explicit classification of the following four families of surfaces in any homogeneous 3-manifold with 4-dimensional isometry group: isoparametric surfaces, surfaces with constant principal curvatures, homogeneous surfaces, and surfaces with constant mean curvature and vanishing Abresch-Rosenberg different…
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48 results for “isoparametric surfaces”
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…
The paper proves conditions for a 4D minimal surface to be isoparametric.
problem Conditions for a 4D minimal surface to be isoparametric.
method Analyzes the properties of a closed immersed minimal hypersurface in with specific curvature conditions.
result If conditions on curvature are met, the surface is isoparametric.
Associated with isoparametric foliations of unit spheres, there are two classes of minimal surfaces minimal isoparametric hypersurfaces and focal submanifolds. By virtue of their rich structures, we find new series of minimizing cones. They are cones over focal submanifolds and cones over suitable products among th…
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 are found for any isoparametric hypersurface .
Minimal hypersurfaces in real projective spaces have at least n+2 Morse index.
problem Understanding the Morse index of minimal hypersurfaces in real projective spaces.
method Analyzing unstable one-sided and two-sided minimal hypersurfaces in real projective spaces.
result The Morse index of minimal hypersurfaces is at least n+2, with specific examples provided.
Study classifies triharmonic surfaces in 3D homogeneous spaces.
problem Classifying triharmonic surfaces in 3D homogeneous spaces.
method Classification through isoparametric and CMC surfaces.
result Complete classification of CMC r-harmonic Hopf cylinders in BCV-spaces.
A hypersurface is said to be totally biharmonic if all its geodesics are biharmonic curves in the ambient space. We prove that a totally biharmonic hypersurface into a space form is an isoparametric biharmonic hypersurface, which allows us to give the full classification of totally biharmonic hypersurfaces in these spa…
The study classifies hypersurfaces with constant principal curvatures in and .
problem Classifying hypersurfaces with constant principal curvatures in specific product spaces.
method Analyzing isoparametric surfaces and using isoparametric properties to classify hypersurfaces.
result Hypersurfaces with constant principal curvatures are cylinders over isoparametric surfaces in .
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…
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.
Very recently Ben Andrews and Haizhong Li showed that every embedded cmc torus in the three dimensional sphere is axially symmetric. There is a two-parametric family of axially symmetric cmc surfaces; more precisely, for every real number H and every C > 2 (H+\sqrt{1+H^2}) there is an axially symmetry surface Σ_{H,C} w…
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.
Compact Dupin Hypersurfacesmath.DG
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.
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 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 . Motivated by the case on the Euclidean space , we first introduce the concept of equiaffine parallel hypersurfaces in , 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 -Minkowski cylinders in a Minkowski space with -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 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…
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…
Isoparametric hypersurfaces and hypersurfaces with constant principal curvatures in Finsler spacesmath.DG
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 and .
Extends isoparametric foliations and area-minimizing cones in product manifolds.
problem Generalizing isoparametric foliations and area-minimizing cones in .
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.
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.
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.
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 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 and are built.
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 and .
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.
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.
A submanifold in space forms is isoparametric if the normal bundle is flat and principal curvatures along any parallel normal fields are constant. We study the mean curvature flow with initial data an isoparametric submanifold in Euclidean space and sphere. We show that the mean curvature flow preserves the isoparametr…
Isoparametric hypersurfaces and their application to special geometries