Searcharxiv⌕ Search

arXiv subjects

Hongbin Cui

Publications and source records attributed to Hongbin Cui.

9 recordsLinked to original sources
arXiv2026

The Morse Index, Nullity and Jacobi Fields of Constant-Curvature 2-Spheres in Complex Projective Spaces

We determine the Morse index and normal nullity of minimal immersions $S^2\to \mathbb{C}P^N$ with constant Gauss curvature. For integers $n\ge1$ and $k\in\{0,\ldots,n\}$, the member $\phi_{n-2k,n}:S^2 \to \mathbb{C}P^n$ of the Veronese sequence satisfies \[ \operatorname{Ind}(\phi_{n-2k,n})=2k(n-k)(n+1),\qquad \operatorname{Nul}(\phi_{n-2k,n})=2(n-1)(n+3). \] We also obtain the corresponding formulas for its totally geodesic extensions to $\mathbb{C}P^N$, $N\ge n$, and identify the normal Jacobi kernel with infinitesimal deformations obtained by post-composing the rational normal directrix with projective linear embeddings into $\mathbb{C}P^N$. In particular, every normal Jacobi field is integrable through a family of minimal $2$-spheres.

math.DG↗
arXiv2026

A family of area-minimizing tensor varieties

By using Lawlor's curvature criterion, we prove that a class of real tensor varieties are regular area-minimizing cones except for one case, which gives a partial generalization of their minimality as in \cite{HKV23}. Such area-minimizing tensor varieties have not been found beyond matrix varieties. Moreover, we provide a new derivation for the important Lawlor ODE.

math.DG↗
arXiv2026

Bernstein-type theorem for stationary hypersurfaces of the Euler-Dierkes-Huisken functional

We say that a hypersurface $\Sigma \subset\mathbb{R}^{n+1}$ is $\alpha$-stationary if it is a critical point of the Euler-Dierkes-Huisken functional $\mathcal{E}_\alpha(\Sigma)=\int_\Sigma|X|^\alpha\, d\mathcal{H}^n$, introduced by Dierkes and Huisken in \cite{[DH-24]}. In this paper, we prove that every smooth, complete, connected, embedded $\alpha$-stationary hypersurface in $\mathbb{R}^{n+1}$ passing through the origin with $\alpha>0$ is a linear hyperplane.

math.DG↗
arXiv2025

On FKM isoparametric hypersurfaces in $\mathbb{S}^n \times \mathbb{S}^n$ and new area-minimizing cones

We present two generalizations for the celebrated works of Ferus-Karcher-M\"unzner \cite{FKM81} and Wang \cite{W94}. We first show that an isoparametric foliation on $\mathbb{S}^{2n+1}$ constructed by Ferus-Karcher-M\"unzner naturally yields an isoparametric foliation on its submanifold $\mathbb{S}^n \times \mathbb{S}^n$ with one same focal variety. The second part concerns area-minimizing cones; all known regular area-minimizing hypercones are realized as real algebraic varieties: isoparametric cones (cf. \cite{W94}). As a noteworthy application, we extend area-minimizing isoparametric hypercones in \cite{W94} to codimension-two cases, and obtain infinitely many families (each containing infinitely many members) of area-minimizing subcones of Simons cones.

math.DG↗
arXiv2024

On Euler-Dierkes-Huisken variational problem

In this paper, we study the stability and minimizing properties of higher codimensional surfaces in Euclidean space associated with the $f$-weighted area-functional $$\mathcal{E}_f(M)=\int_M f(x)\; d \mathcal{H}_k$$ with the density function $f(x)=g(|x|)$ and $g(t)$ is non-negative, which develop the recent works by U. Dierkes and G. Huisken (Math. Ann., 20 October 2023) on hypersurfaces with the density function $|x|^\alpha$. Under suitable assumptions on $g(t)$, we prove that minimal cones with globally flat normal bundles are $f$-stable, and we also prove that the regular minimal cones satisfying Lawlor curvature criterion, the highly singular determinantal varieties and Pfaffian varieties without some exceptional cases are $f$-minimizing. As an application, we show that $k$-dimensional minimal cones over product of spheres are $|x|^\alpha$-stable for $\alpha\geq-k+2\sqrt{2(k-1)}$, the oriented stable minimal hypercones are $|x|^\alpha$-stable for $\alpha\geq 0$, and we also show that the minimal cones over product of spheres $\mathcal{C}=C \left(S^{k_1} \times \cdots \times S^{k_{m}}\right)$ are $|x|^\alpha$-minimizing for $\dim \mathcal{C} \geq 7$, $k_i>1$ and $\alpha \geq 0$, the Simons cones $C(S^{p} \times S^{p})(p\geq 1)$ are $|x|^\alpha$-minimizing for any $\alpha \geq 1$, which relaxes the assumption $1\leq\alpha \leq 2p$ in \cite{DH23}.

math.DG↗
arXiv2023

On area-minimizing Pfaffian varieties

There are two significant families of minimal real matrix varieties: determinantal varieties and skew-symmetric determinantal varieties, the later ones are also known as Pfaffian varieties. In 1999, Kerckhove and Lawlor [Duke Math.J. 96(2),401--424,1999] proved that determinantal varieties are area-minimizing except for two families. In this paper we prove that all Pfaffian varieties are area-minimizing with the exception of Pfaffian hypersurfaces.

math.AG↗
arXiv2021

Area-minimizing Cones over Products of Grassmannian Manifolds

This paper is the continuation of the previous one \cite{Cui2021}, where we re-proved the area-minimization of cones over Grassmannians of $n$-planes $G(n,m;\mathbb{F})(\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H})$, Cayley plane $\mathbb{O}P^2$ from the point view of Hermitian orthogonal projectors, and gave area-minimizing cones associated to oriented real Grassmannians $\widetilde{G}(n,m;\mathbb{R})$. In this paper, we make a further step on showing that the cones, of dimension no less than $\mathbf{8}$, over minimal products of $G(n,m;\mathbb{F})$ are area-minimizing. Moreover, those cones are very similar to the classical cones over products of spheres, and for the critical situation -- the cones of dimension $\mathbf{7}$ \cite{lawlor1991sufficient}, we gain more area-minimizing cones by carefully computing the Jacobian $inf_{v}det(I-tH^{v}_{ij})$. Certain minimizing cones among them had been found from the perspective of $R$-spaces\cite{Ohno2021area}, or isoparametric theory\cite{tang2020minimizing}, and others are completely new. We also prove that the cones over minimal product of $\widetilde{G}(n,m;\mathbb{R})$ are area-minimizing.

math.DG↗
arXiv2021

Area-minimizing Cones over Grassmannian Manifolds

It is a well-known fact that there exists a standard minimal embedding map for the Grassmannians of $n$-planes $G(n,m;\mathbb{F})(\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H})$ and Cayley plane $\mathbb{O}P^2$ into Euclidean spheres, then an natural question is that if the cones over these embedded Grassmannians are area-minimizing? In this paper, detailed descriptions for this embedding map are given from the point view of Hermitian orthogonal projectors which can be seen as an direct generalization of Gary R. Lawlor's(\cite{lawlor1991sufficient}) original considerations for the case of real projective spaces, then we re-prove the area-minimization of those cones which was gradually obtained in \cite{kerckhove1994isolated}, \cite{kanno2002area} and \cite{ohno2015area} from the perspectives of isolated orbits of adjoint actions or canonical embedding of symmetric $R$-spaces, all based on the method of Gary R. Lawlor's Curvature Criterion. Additionally, area-minimizing cones over almost all common Grassmannians has been given by Takahiro Kanno, except those cones over oriented real Grassmannians $\widetilde{G}(n,m;\mathbb{R})$ which are not Grassmannians of oriented $2$-planes. The second part of this paper is devoted to complement this result, a natural and key observation is that the oriented real Grassmannians can be considered as unit simple vectors in the exterior vector spaces, we prove that all their cones are area-minimizing except $\widetilde{G}(2,4;\mathbb{R})$.

math.DG↗