正则面积极小超锥的严格稳定性与严格极小性：定量刻画
Strict Stability and Strict Minimality of Regular Area-Minimizing Hypercones: A Quantitative Characterization
浏览论文内容
中文总结 AI 辅助
该论文针对正则面积极小超锥，通过定量不等式刻画其严格稳定性与严格极小性，推广了Lawson锥的相关不等式，并确定了稳定正则极小超锥的最优Dirichlet谱常数。
中文摘要 AI 辅助
设$\boldsymbol{C}=\boldsymbol{\nabla}E\boldsymbol{\nabla}\boldsymbol{R}^{n+1}$为正则面积极小超锥。我们首先证明：$\boldsymbol{C}$同时为严格稳定且严格极小当且仅当存在$c_{\boldsymbol{C}}>0$，使得对任意$R>0$及满足$F\boldsymbol{\triangle}E\boldsymbol{\boxminus}B_R$的有限周长集$F$，有$\text{Per}(F;B_R)-\text{Per}(E;B_R)\boldsymbol{\text{geq}}c_{\boldsymbol{C}}\boldsymbol{\text{int}}_{F\boldsymbol{\triangle}E}\frac{\text{dist}(x,\boldsymbol{C})}{|x|^2}\text{d}x$，该内蕴距离加权不等式精确刻画了稳定性与极小性的同时严格性。其次，在无任何严格性假设下，每个正则面积极小超锥均满足尺度不变二次不等式：$\frac{\text{Per}(F;B_R)-\text{Per}(E;B_R)}{R^n}\boldsymbol{\text{geq}}c_{\boldsymbol{C}}\boldsymbol{\text{left(}}\frac{|F\boldsymbol{\triangle}E|}{R^{n+1}}\boldsymbol{\text{right)}}^2$，该结果推广了此前针对面积极小Lawson锥建立的不等式。最后，面积极小假设对谱结果并非必要：对每个稳定正则极小超锥，$\boldsymbol{C}\boldsymbol{\boxminus}B_R$上Jacobi算子的第一Dirichlet特征值$\boldsymbol{\text{lambda}}_{\boldsymbol{C}}^D(R)$精确为$\boldsymbol{\text{lambda}}_{\boldsymbol{C}}^D(R)=\frac{j_{b_1,1}^2}{R^2}$，其中$b_1^2=\frac{(n-2)^2}{4}+\boldsymbol{\text{mu}}_1$，$\boldsymbol{\text{mu}}_1$为联络Jacobi算子的第一特征值，$j_{b_1,1}$为Bessel函数$J_{b_1}$的第一个正零点，该结果明确了每个稳定正则极小超锥的最优Dirichlet谱常数。
英文摘要
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.
