arXiv subjects

Gongping Niu

Publications and source records attributed to Gongping Niu.

7 recordsLinked to original sources

arXiv2026

A quantitative inequality for general area-minimizing hypercones

Let $\mathbf{C}=\partial E\subset\mathbb{R}^{n+1}$ be an area-minimizing hypercone. The cone may have nonisolated singularities. We prove that there is a constant $c_{\mathbf{C}}>0$ such that \[ \frac{\operatorname{Per}(F;B_R)-\operatorname{Per}(E;B_R)}{R^n} \geq c_{\mathbf{C}} \left(\frac{|F\triangle E|}{R^{n+1}}\right)^2 \] for every $R>0$ and every set $F$ of locally finite perimeter with $F\triangle E\Subset B_R$. This extends the unweighted quantitative inequality from regular area-minimizing hypercones to general area-minimizing hypercones. We follow the calibration argument in the author's earlier work on regular cones. The main change is the construction of the vector fields. The pointwise asymptotic estimates for positive Jacobi fields used in the regular case do not directly apply here. We use the minimal foliations constructed by Zhihan Wang on the two sides of the cone. On each leaf, we integrate projection kernels with a positive Jacobi field as the weight. The kernel construction gives the required divergence bound. Wang's weak Harnack inequality and growth estimates give the integral bounds needed to prove linear growth of the vector fields.

math.DG↗

arXiv2026

Generic Uniqueness of Isoperimetric Regions in Arbitrary Dimension

Let $M^{n+1}$ be a closed, connected smooth manifold, where $n\geq 1$. For each prescribed volume fraction $s\in(0,1)\setminus\{\frac12\}$, we prove that the isoperimetric region of volume $s \operatorname{Vol}_g(M)$ is unique for a generic set of smooth Riemannian metrics $g$. At half volume, a generic metric has exactly two minimizers, a region $E$ and its complement $E^c$. As consequences, for every $m>0$, uniqueness holds for a generic set of metrics $g$ satisfying $m<\operatorname{Vol}_g(M)$, and it also holds for a generic set of pairs $(g,m)$ with $0<m<\operatorname{Vol}_g(M)$. The proof is variational and requires neither boundary regularity nor nondegeneracy of the constrained Jacobi operator. In particular, the results apply even when isoperimetric boundaries are singular.

math.DG↗

arXiv2026

Isoperimetry by stretching

We construct isoperimetric regions from separating hypersurfaces in closed manifolds. This yields isoperimetric boundaries exhibiting a wide variety of topological types and singular sets.

math.DG↗

arXiv2026

Strict Stability and Strict Minimality of Regular Area-Minimizing Hypercones: A Quantitative Characterization

Let $\mathbf{C}=\partial E\subset \mathbb{R}^{n+1}$ be a regular area-minimizing hypercone. We first prove that $\mathbf{C}$ is simultaneously strictly stable and strictly minimizing if and only if there exists $c_\mathbf{C}>0$ such that \[ \operatorname{Per}(F;B_R)-\operatorname{Per}(E;B_R) \geq c_\mathbf{C} \int_{F\mathbin\triangle E} \frac{\operatorname{dist}(x,\mathbf{C})}{|x|^2}\,dx \] for every $R>0$ and finite-perimeter set $F$ with $F\mathbin\triangle E\Subset B_R$. Thus this intrinsic distance-weighted inequality characterizes exactly the simultaneous strictness of the stability and minimizing properties. Second, without either strictness assumption, every regular area-minimizing hypercone satisfies the scale-invariant quadratic inequality \[ \frac{\operatorname{Per}(F;B_R)-\operatorname{Per}(E;B_R)}{R^n} \geq c_\mathbf{C} \left( \frac{|F\mathbin\triangle E|}{R^{n+1}} \right)^2. \] This extends the inequality previously established for area-minimizing Lawson cones. Finally, the area-minimizing assumption is unnecessary for our spectral result: for every stable regular minimal hypercone, the first Dirichlet eigenvalue $λ_\mathbf{C}^D(R)$ of the Jacobi operator on $\mathbf{C}\cap B_R$ is given exactly by \[ λ_\mathbf{C}^D(R) = \frac{j_{b_1,1}^2}{R^2}, \qquad b_1^2 = \frac{(n-2)^2}{4}+μ_1, \] where $μ_1$ is the first eigenvalue of the link Jacobi operator and $j_{b_1,1}$ is the first positive zero of the Bessel function $J_{b_1}$. In particular, this identifies the optimal Dirichlet spectral constant for every stable regular minimal hypercone.

math.DG↗

arXiv2025

Generic regularity of isoperimetric regions in dimension eight

We establish generic regularity results for isoperimetric regions in closed Riemannian manifolds of dimension eight. In particular, we show that every isoperimetric region has a smooth nondegenerate boundary for a generic choice of smooth metric and enclosed volume, or for a fixed enclosed volume and a generic choice of smooth metric.

math.DG↗

arXiv2023

Existence of singular isoperimetric regions

It is well known that isoperimetric regions in a smooth compact $(n+1)$-manifold are smooth, up to a closed set of codimension at most $6$. In this note, we first construct an $8$-dimensional compact smooth manifold whose unique isoperimetric region with half volume that of the manifold exhibits two isolated singularities. And then, for $n\geq 7$, using Smale's construction of singular homological area minimizers for higher dimensions, we construct a Riemannian manifold such that the unique isoperimetric region of half volume, with singular set the submanifold $§^{n-7}$.

math.DG↗

arXiv2017

A Study of finitely generated Free Groups via the Fundamental Groups

Free groups have many applications in Algebraic Topology. In this paper I specifically study the finitely generated free groups by using the covering spaces and fundamental groups. By the Van Kampen's theorem, we have a famous fact that the fundamental group of a wedge sum of circles is a free group. Therefore, to study free groups, we could try to figure out the covering spaces of the wedge sum of circles. And in the appendix B, I prove the Nielsen-Schreier theorem which I will use this to study finitely index subgroups of a finitely generated free group.

math.AT↗