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Niang Chen

Publications and source records attributed to Niang Chen.

7 recordsLinked to original sources
arXiv2026

Radial pinching and topological rigidity for free boundary Gaussian $f$-minimal submanifolds

Let $M^k\subset \overline{B}_R^N$ be a smooth compact connected orientable free boundary $f_c$-minimal submanifold of the closed Euclidean ball, where $f_c(x)=c|x|^2/2$ and $c\ge 0$. Assume that $cR^2\le k$ and $|A_{x^\perp}|^2\le 1+\frac{1}{k-1}(1-c|x^\perp|^2)^2$, where $A_{x^\perp}(X,Y)=\langle x^\perp,A(X,Y)\rangle$. We prove that $M$ is diffeomorphic either to $D^k$ or to $S^1\times D^{k-1}$; strict pinching yields the disk. The proof uses Hessian convexity of the squared-distance function, a nullity estimate along its minimum set, and a sublevel-set argument. In dimension two and codimension one, the non-disk branch is rotationally symmetric. We also construct a local family of embedded rotational examples for small $c\ge 0$, with the $c=0$ member equal to the critical catenoid.

math.DG↗
arXiv2026

Cohomology Vanishing for Free Boundary $f$-Minimal Submanifolds in Gaussian-Weighted Euclidean Balls

Let $M^n\subset \overline{\B_R^{n+k}}\subset \R^{n+k}$ be a compact orientable free boundary $f$-minimal submanifold of the Gaussian-weighted Euclidean ball $\left(\overline{\B_R^{n+k}},g_{\rm can},e^{-f}\dd V\right), f(x)=\frac c2 |x|^2,c\ge 0.$ We prove a cohomology vanishing theorem under the pointwise pinching condition $ |A|^2\le \frac{n-p}{R^2},1\le p<n.$ More precisely, the space of tangential $f$-harmonic $p$-forms vanishes, and hence$H^p(M;\R)=0.$ The proof is based on three elementary ingredients in the Gaussian-weighted ball: a weighted Hardy inequality obtained from the identity $\divf(x^T)=n-c|x|^2$, a cancellation in the weighted Weitzenb\"ock curvature operator, and a boundary reduction showing that tangential $f$-harmonic forms satisfy the same local absolute-boundary algebra as in the unweighted case. The constant pinching threshold is independent of the Gaussian parameter $c$, and the argument also includes the unweighted case $c=0$; the strict interior positivity comes from the full Hardy--Weitzenb\"ock coefficient rather than from the sign of $c$ alone.

math.DG↗
arXiv2026

Rigidity and Gap Phenomena in the Sphere--Ball Correspondence

This survey reviews a collection of parallel phenomena between free boundary submanifolds in the Euclidean unit ball and closed submanifolds in the sphere, with particular emphasis on rigidity mechanisms, pinching thresholds, and canonical models. We do not regard the two theories as a unified system in one-to-one correspondence. Rather, we emphasize that in several typical settings -- including low topology, strong pinching, spectral extremality, and symmetry reduction -- the free boundary condition often forces stronger rigidity in the unit ball than in the closed setting. The exposition is organized around six interconnected themes. We first contrast the failure of the spherical Bernstein problem in high dimensions with the dimension-independent rigidity of free boundary minimal disks in the unit ball. We then discuss the parallel roles played by the Clifford torus and the critical catenoid in uniqueness, Morse index, and eigenvalue characterizations. Next, we review the transition from the Lawson--Simons stable currents method to the Bochner--Hardy techniques developed for free boundary problems, summarize pinching and gap theorems driven by the second fundamental form and its traceless part, and outline the linear comparison framework between Morse index and topology in the minimal, constant mean curvature, and weighted settings. Finally, we survey existence results obtained from group actions, isoparametric foliations, and recent equivariant eigenvalue optimization, thereby illustrating both the striking analogies and the essential boundary-driven differences between the closed spherical theory and the free boundary theory in the ball.

math.DG↗
arXiv2026

ACS Condition on Minimal Isoparametric Hypersurfaces of Positive Ricci Curvature in Unit Spheres

We study the Ambrozio--Carlotto--Sharp (ACS) criterion on minimal isoparametric hypersurfaces $N^{n+1}\subset S^{n+2}$ with positive Ricci curvature, motivated by the Schoen--Marques--Neves conjecture on Morse index.For $g=4$ distinct principal curvatures with multiplicities $m_1,m_2$, we prove that the pointwise ACS inequality holds if and only if $\min\{m_1,m_2\}\ge 4$. Sufficiency is obtained via a moment-relaxation technique yielding the sharp bound $4a^2$ on the quadratic part of the integrand; necessity follows from an explicit extremal configuration in the top eigenspace of the shape operator. We also verify the ACS condition for $g=3$ with $m_1=m_2\in\{4,8\}$.As a consequence, for any closed embedded minimal hypersurface $M^n$ in such an ambient manifold, $\operatorname{index}(M)\ge \tfrac{2}{d(d-1)}\, b_1(M)$ with $d=n+3$.

math.DG↗
arXiv2022

Cohomology vanishing theorems for free boundary submanifolds

In this paper, via a new Hardy type inequality, we establish some cohomology vanishing theorems for free boundary compact submanifolds $M^n$ with $n\geq2$ immersed in the Euclidean unit ball $\mathbb{B}^{n+k}$ under one of the pinching conditions $|Φ|^2\leq C$, $|A|^2\leq \widetilde{C}$, or $|Φ|\leq R(p,|H|)$, where $A$ $(Φ)$ is the (traceless) second fundamental form, $H$ is the mean curvature, $C,\widetilde{C}$ are positive constants and $R(p,|H|)$ is a positive function. In particular, we remove the condition on the flatness of the normal bundle, solving the first question, and partially answer the second question on optimal pinching constants proposed by Cavalcante, Mendes and Vitório.

math.DG↗
arXiv2022

An isoperimetric inequality of minimal hypersurfaces in spheres

Let $ M^n$ be a closed immersed minimal hypersurface in the unit sphere $\mathbb{S}^{n+1}$. We establish a special isoperimetric inequality of $M^n$. As an application, if the scalar curvature of $ M^n$ is constant, then we get a uniform lower bound independent of $M^n$ for the isoperimetric inequality. In addition, we obtain an inequality between Cheeger's isoperimetric constant and the volume of the nodal set of the height function.

math.DG↗