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Xiaoxiang Jiao

Publications and source records attributed to Xiaoxiang Jiao.

9 recordsLinked to original sources
arXiv2026

Local Variational Properties of Non-degenerate Free Boundary Minimal Hypersurfaces

We establish local variational characterizations of non-degenerate free-boundary minimal hypersurfaces in compact Riemannian manifolds with boundary. For a two-sided strictly stable hypersurface, we show that it is the unique mass minimizer in its relative homology class, both in a small tubular and flat-neighborhood. As an application, for generic metrics in dimensions $3\le n+1\le 7$, we show that the first free-boundary width is realized by a multiplicity-one hypersurface of Morse index one. For an orientable non-degenerate free-boundary hypersurface of Morse index $k>0$, we construct a canonical local $k$-parameter family and obtain a local min--max characterization.

math.DG↗
arXiv2026

Barycentric Cuts at Maximal Depth and a Five-Cut Theorem

We determine the sharp number and the complete finite spectrum of barycentric hyperplanes through the Tukey median. We further prove that every convex body in $\mathbb R^n$, $n\ge3$, contains an interior point incident with at least five barycentric hyperplanes. In particular, this settles Grünbaum's original conjecture in dimension four.

math.MG↗
arXiv2026

Existence of multiple constant mean curvature hypersurfaces for varying Riemannian metrics

Given a closed Riemannian manifold $(M^{n+1},g)$,$3\leq n+1\leq7$.In this paper,we will prove that for any $c>0$,suppose the number of closed $c-CMC$ hypersurfaces is finite,then there exists a metric $h$ on $M$ such that the $c-CMC$ hypersurfaces in $(M,g)$ are also $c-CMC$ hypersurfaces in $(M,h)$ and the number of $c-CMC$ hypersurfaces in $(M,h)$ is strictly greater than the number of $c-CMC$ hypersurfaces in $(M,g)$.Moreover,we will give a precise upper bound for the $L^{\frac{n+1}{2}}$ norm of $(g-h)$,which depends on the metric $g$ and the number of $c-CMC$ hypersurfaces in $(M,g)$.

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↗
arXiv2020

Totally Real Flat Minimal Surface in Hyperquadric

In this paper, we study geometry of totally real minimal surfaces in the complex hyperquadric $Q_{N-2}$, and obtain some characterizations of the harmonic sequence generated by these minimal immersions. For totally real flat surfaces that are minimal in both $Q_{N-2}$ and $\mathbb{C}P^{N-1}$, we determine them for $N=4, 5, 6$, and give a classification theorem when they are Clifford solutions.

math.DG↗
arXiv2019

Rigidity of conformal minimal immersions of constant curvature from $S^2$ to $Q_4$

Geometry of conformal minimal two-spheres immersed in $G(2,6;\mathbb{R})$ is studied in this paper by harmonic maps. We construct a non-homogeneous constant curved minimal two-sphere in $G(2,6;\mathbb{R})$, and give a classification theorem of linearly full conformal minimal immersions of constant curvature from $S^2$ to $G(2,6;\mathbb{R})$, or equivalently, a complex hyperquadric $Q_{4}$, which illustrates minimal two-spheres of constant curvature in $Q_{4}$ are in general not congruent.

math.DG↗