AI 中文总结
该论文将面积最小化超锥的定量不等式从正则情形推广到一般情形（允许非孤立奇点），通过使用极小叶状结构和正 Jacobi 场构造向量场，证明了周长差与对称差体积平方之间的下界。
AI 中文摘要
设 $\mathbf{C}=\partial E\subset\mathbb{R}^{n+1}$ 为一个面积最小化超锥。该锥可能具有非孤立奇点。我们证明存在常数 $c_{\mathbf{C}}>0$，使得对于每个 $R>0$ 以及每个局部有限周长且满足 $F\triangle E\Subset B_R$ 的集合 $F$，有 \\[ \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. \\] 这将对正则面积最小化超锥的无权重定量不等式推广到一般面积最小化超锥。我们遵循作者早期关于正则锥的工作中的校准论证。主要变化在于向量场的构造。正则情形中用于正 Jacobi 场的逐点渐近估计在此不直接适用。我们使用 Zhihan Wang 在锥两侧构造的极小叶状结构。在每个叶上，我们以正 Jacobi 场作为权重对投影核进行积分。核构造给出了所需的散度界。Wang 的弱 Harnack 不等式和增长估计给出了证明向量场线性增长所需的积分界。
英文摘要
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.
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