arXiv research

The study characterizes Clifford hypersurfaces in terms of curvature constants.

The paper studies inhomogeneous isoparametric hypersurfaces in pseudo-spheres.

We characterize the first min-max width of real projective spaces of any dimension. The width is the minimum area over the Clifford hypersurfaces. We also compute the Morse index of the Clifford hypersurfaces in the complex and quaternionic projective spaces.

The paper explores CMC hypersurfaces in spheres, verifying Yau's conjecture.

Study on minimal hypersurfaces in a unit sphere, proving specific isometries.

Sharp focal radius estimate for hypersurfaces in manifolds with positive curvature.

Four constructions of constant mean curvature (CMC) hypersurfaces in the (n+1)-sphere are given, which should be considered analogues of `classical' constructions that are possible for CMC hypersurfaces in Euclidean space. First, Delaunay-like hypersurfaces, consisting roughly of a chain of hyperspheres winding multipl…

The paper proves conditions for closed minimally immersed hypersurfaces in a 5-sphere.

In this paper, we show that the catenoids and the Clifford minimal hypersurfaces are the only complete minimal hypersurfaces satisfying the Simons' equation (3.9) in the space forms.

Paper finds conditions for minimal hypersurfaces in S^6 with constant scalar curvature.

Using representations of Clifford algebras we construct indecomposable singular Riemannian foliations on round spheres, most of which are non-homogeneous. This generalizes the construction of non-homogeneous isoparametric hypersurfaces due to by Ferus, Karcher and Munzner.

Paper characterizes a special hypersurface in 5D sphere.

The (n+1)-sphere contains a simple family of constant mean curvature (CMC) hypersurfaces which are products of lower-dimensional spheres called the generalized Clifford hypersurfaces. This paper demonstrates that new, topologically non-trivial CMC hypersurfaces resembling a pair of neighbouring generalized Clifford tor…

Study on free-boundary CMC hypersurfaces in upper hemisphere, proving Morse index and eigenvalue bounds.

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.

Study proves mean curvature flows on spheres in higher dimensions.

Study on Gehring link problem and width of bands in curved manifolds.

The study characterizes embedded minimal hypersurfaces in $S^{n+1}$ with symmetries.

Paper proves minimal hypersurface index for specific cases.

Classifies hypersurfaces with constant isotropic curvature in space forms.

We study conformally flat hypersurfaces $f\colon M^{3} \to \Q^{4}(c)$ with three distinct principal curvatures and constant mean curvature $H$ in a space form with constant sectional curvature $c$. First we extend a theorem due to Defever when $c=0$ and show that there is no such hypersurface if $H\neq 0$. Our main res…

Constructs CMC hypersurfaces in S^4 from piecewise-smooth unions of spheres.

Let $φ:M\to\mathbb{S}^{n+1}\subset\mathbb{R}^{n+2}$ be an immersion of a complete $n$-dimensional oriented manifold. For any $v\in\mathbb{R}^{n+2}$, let us denote by $\ell_v:M\to\mathbb{R}$ the function given by $\ell_v(x)=φ(x),v$ and by $f_v:M\to\mathbb{R}$, the function given by $f_v(x)=ν(x),v$, where $ν:M\to\mathbb{…

Using a new estimate for the Peng-Terng invariant and the multiple-parameter method, we verify a rigidity theorem on the stronger version of Chern Conjecture for minimal hypersurfaces in spheres. More precisely, we prove that if $M$ is a compact minimal hypersurface in $\mathbb{S}^{n+1}$ whose squared length of the sec…

The paper constructs metrics with non-negative curvature and harmonic maps.

We construct biharmonic real hypersurfaces and Lagrangian submanifolds of Clifford torus type in $CP^n$ via the Hopf fibration; and get new examples of biharmonic submanifolds in $S^{2n+1}$ as byproducts .

Proves rigidity in product spaces using index theory.

For a compact minimal hypersurface $M$ in $S^{n+1}$ with the squared length of the second fundamental form $S$ we confirm that there exists a positive constant $\de(n)$ depending only on $n,$ such that if $n\leq S\leq n +δ(n)$, then $S\equiv n$, i.e., $M$ is a Clifford minimal hypersurface, in particular, when $n\ge 6,…

Constructs minimal hypersurfaces in S^4(1) by doubling equatorial S^3.

We provide explicit spinor representations for Clifford algebras.

We obtain a complete classification of proper biharmonic hypersurfaces with at most three distinct principal curvatures in sphere spaces with arbitrary dimension. Precisely, together with known results of Balmuş-Montaldo-Oniciuc, we prove that compact orientable proper biharmonic hypersurfaces with at most three distin…

Let $M\subset S^{n+1}\subset\mathbb{R}^{n+2}$ be a compact minimal hypersurface of the $n$-dimensional Euclidean unit sphere. Let us denote by $|A|^2$ the square of the norm of the second fundamental form and $J(f)=-Δf-nf-|A|^2f$ the stability operator. It is known that the index (the number of negative eigenvalues of …

The study examines stability of triharmonic hypersurfaces in space forms.

A well-known conjecture of Yau states that the area of one of Clifford minimal hypersurfaces $S^k\big{(}\sqrt{\frac{k}{n}}\, \big{)}\times S^{n-k}\big{(}\sqrt{\frac{n-k}{n}}\, \big{)}$ gives the lowest value of area among all non-totally geodesic compact minimal hypersurfaces in the unit sphere $S^{n+1}(1)$. The presen…

The paper proves properties of specific hypersurfaces in a 5-sphere.

1. Translated by Thomas E. Cecil, Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, MA 01610, USA; E-mail address: cecil@mathcs.holycross.edu 2. Typed by Wenjiao Yan, School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing 10…

The study characterizes hypersurfaces in spheres with constant scalar curvature.

In this paper we study Clifford and harmonic analysis on some conformal flat spin manifolds. In particular we treat manifolds that can be parametrized by $U / Γ$ where $U$ is a simply connected subdomain of either $S^{n}$ or $R^{n}$ and $Γ$ is a Kleinian group acting discontinuously on $U$. Examples of such manifolds t…

In this work we characterize certain immersed closed hypersurfaces of some ambient manifolds via the second eigenvalue of the Jacobi operator. First, we characterize the Clifford torus as the surface which maximizes the second eigenvalue of the Jacobi operator among all closed immersed orientable surfaces of $\mathbb S…

Estimates the first eigenvalue of a Schrödinger operator on minimal submanifolds.

In this paper, two sequences of minimal isoparametric hypersurfaces are constructed via representations of Clifford algebras. Based on these, we give estimates on eigenvalues of the Laplacian of the focal submanifolds of isoparametric hypersurfaces in unit spheres. This improves results of [TY13] and [TXY14]. Eells and…

Let $M$ be an isoparametric hypersurface in the sphere $S^n$ with four distinct principal curvatures. Münzner showed that the four principal curvatures can have at most two distinct multiplicities $m_1, m_2$, and Stolz showed that the pair $(m_1,m_2)$ must either be $(2,2)$, $(4,5)$, or be equal to the multiplicities o…

The study of conformal biharmonic maps and hypersurfaces in various spaces.

The paper proves the regularity of cohomogeneity two problems and constructs minimal hypersurfaces on spheres.

The paper studies inverse mean curvature flow on hypersurfaces in space forms.

We verify that if $M$ is a compact minimal hypersurface in $\mathbb{S}^{n+1}$ whose squared length of the second fundamental form satisfying $0\leq |A|^2-n\leq\frac{n}{22}$, then $|A|^2\equiv n$ and $M$ is a Clifford torus. Moreover, we prove that if $M$ is a complete self-shrinker with polynomial volume growth in $\ma…

Motivated by the celebrated Schoen-Yau-Gromov-Lawson surgery theory on metrics of positive scalar curvature, we construct a double manifold associated with a minimal isoparametric hypersurface in the unit sphere. The resulting double manifold carries a metric of positive scalar curvature and an isoparametric foliation …

Sharp pinching theorem for submanifolds in spheres.