发表机构
Bocconi University; UC San Diego（博科尼大学; 加州大学圣地亚哥分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在$\text{R}^9$中构造了一类极小超曲面，其奇异集含原点及收敛于原点的孤立奇点，关键步骤是构造了具有特定切锥性质的整体稳定超曲面。
AI 中文摘要
在$\boldsymbol{\text{R}}^9$的原点邻域中，我们构造了一个关于光滑共形欧氏度量的极小超曲面，其奇异集由原点和一列收敛于原点的孤立奇点组成。在每个孤立奇点处，该超曲面具有唯一的切锥$C_{4,3}$，即$S^4(\text{√}(4/7))\times S^3(\text{√}(3/7))$上的锥；而在原点处，其唯一切锥为锥$C_{3,3}\times\boldsymbol{\text{R}}$。该度量在每个孤立奇点附近为欧氏度量，且在原点处与欧氏度量无限阶一致。一个具有独立意义的关键步骤是：在欧氏$\boldsymbol{\text{R}}^9$中构造一个整体稳定超曲面，其原点处恰好有一个奇异点，该点处的唯一切锥为$C_{4,3}$，且其无穷远处的唯一切锥为$C_{3,3}\times\boldsymbol{\text{R}}$。
英文摘要
In a neighborhood of the origin in $\mathbb{R}^9$, we construct a minimal hypersurface with respect to a smooth conformally Euclidean metric, whose singular set consists of the origin and a sequence of isolated singular points converging to it. At each of the isolated singular points the hypersurface has the unique tangent cone $C_{4,3}$, the cone over $S^4(\sqrt{4/7})\times S^3(\sqrt{3/7})$, whereas at the origin its unique tangent cone is the cone $C_{3,3}\times\mathbb{R}$. The metric is Euclidean near each isolated singular point and agrees with the Euclidean metric to infinite order at the origin. A key step, of independent interest, is the construction of an entire stationary hypersurface in Euclidean $\mathbb{R}^9$ with exactly one singular point at the origin, at which its unique tangent cone is $C_{4,3}$, and whose unique tangent cone at infinity is $C_{3,3}\times\mathbb{R}$.
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