arXiv research

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

New method constructs explicit $p$-harmonic functions on Lie groups.

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…

The paper studies geodesics and isoparametric functions on Finsler spheres.

This note constructs complex structures on specific isoparametric hypersurfaces.

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

Classifies foliations of complex and quaternionic projective spaces.

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…

We classify isoparametric hypersurfaces in complex hyperbolic spaces.

Navigation in Lorentz Finsler geometry induces isoparametric hypersurfaces.

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

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…

The paper classifies shapes of translating solitons from isoparametric graphs.

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 classifies isoparametric hypersurfaces in conic Finsler spaces.

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…

The study classifies rational isoparametric functions on Damek-Ricci spaces.

Irreducible isoparametric foliations of arbitrary codimension q on complex projective spaces CP^n are classified, except if n=15 and q=1. Remarkably, there are noncongruent examples that pull back under the Hopf map to congruent foliations on the sphere. Moreover, there exist many inhomogeneous isoparametric foliations…

The abstract discusses the equivalence of transnormal and isoparametric functions on compact manifolds.

The study classifies isoparametric and homogeneous hypersurfaces in product spaces.

The study classifies isoparametric hypersurfaces in specific product spaces.

The first part of the paper is to improve the fundamental theory of isoparametric functions on general Riemannian manifolds. Next we focus our attention on exotic spheres, especially on "exotic" 4-spheres (if exist) and the Gromoll-Meyer sphere. In particular, as one of main results we prove: there exists no properly t…

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

Study on gradient Ricci solitons with isoparametric potential functions.

Study on 4-manifolds restricts negatively curved metrics and fundamental groups.

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 classify the isoparametric functions on $\mathbb{R}^n\times\mathbb{M}^m$, $n, m\geq2$, with compact level sets, where $\mathbb{M}^m$ is a connected, closed Riemannian manifold of dimension $m$. Also, we classify the isoparametric hypersurfaces in $\mathbb{S}^2\times\mathbb{R}^2$ with constant principal curvatures.

We prove existence results for nodal solutions of the Yamabe equation that are constant along the level sets of an isoparametric function.

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…

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

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 provide some examples of harmonic unit vector fields as normalized gradients of isoparametric functions from a K-contact geometry setting.

New classification of Kähler-Ricci solitons linked to isoparametric functions and contact geometry.

Classifies isoparametric hypersurfaces in 3D manifolds.

The paper proves solutions for Yamabe equations on manifolds with boundary.

The paper classifies shapes of translating solitons for a specific flow.

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…

The $n$-dimensional complex hyperquadric is a compact complex algebraic hypersurface defined by the quadratic equation in the $(n+1)$-dimensional complex projective space, which is isometric to the real Grassmann manifold of oriented 2- planes and is a compact Hermitian symmetric space of rank 2. In this paper we study…

This paper introduces the notion of $k$-isoparametric hypersurface in an $(n+1)$-dimensional Riemannian manifold for $k=0,1,...,n$. Many fundamental and interesting results (towards the classification of homogeneous hypersurfaces among other things) are given in complex projective spaces, complex hyperbolic spaces, and…

A hypersurface $M^{n-1}$ in a real space-form ${\bf R}^n$, $S^n$ or $H^n$ is isoparametric if it has constant principal curvatures. For ${\bf R}^n$ and $H^n$, the classification of isoparametric hypersurfaces is complete and relatively simple, but as Elie Cartan showed in a series of four papers in 1938-1940, the subje…

Complete classification of isoparametric hypersurfaces in product spaces of space forms.

This paper extends widely the work in \cite{GT13}. Existence and non-existence results of isoparametric functions on exotic spheres and Eells-Kuiper projective planes are established. In particular, every homotopy $n$-sphere ($n>4$) carries an isoparametric function (with certain metric) with 2 points as the focal set,…

This paper is a continuation of [TY12] and [QTY13]. We show that both focal submanifolds of each isoparametric hypersurface in the sphere with six distinct principal curvatures are Willmore.

The paper examines how isoparametric foliations affect the Pompeiu property in compact Riemannian manifolds.

In this paper, we provide conceptional explanations for the geodesic and Jacobi field correspondences for homothetic navigation, and then let them guide us to the shortcuts to some well known flag curvature and S-curvature formulas. They also help us directly see the local correspondence between isoparametric functions…

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…

We study curvature-adapted submanifolds of general symmetric spaces. We generalize Cartan's theorem for isoparametric hypersurfaces of spheres and Wang's classification of isoparametric Hopf hypersurfaces in complex projective spaces to any compact symmetric space. Our second objective is to investigate such hypersurfa…

Study examines how wind affects shortest paths on Finsler manifolds.