发表机构
National Taiwan University; National Center for Theoretical Sciences; Laboratoire Jacques-Louis Lions de Sorbonne Université; Columbia University（台湾大学; 国家理论科学中心; 索邦大学雅克-路易·利翁斯实验室; 哥伦比亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究完全确定了高维螺旋面的稳定性与面积最小化性质：当k≥3时稳定，k≤2时不稳定；当k为偶数时面积最小化，k为奇数时非面积最小化，并通过显式校准和竞争者证明。
AI 中文摘要
对于每个整数$k\geq 1$，我们研究由\\[(u_1,\ldots,u_k,s) \longmapsto \bigl(u_1e^{is},\ldots,u_ke^{is},s\bigr) \in \mathbb{C}^k\times\mathbb{R}\cong \mathbb{R}^{2k+1}\\]参数化的$(k+1)$维螺旋面$H_k\subset\mathbb{R}^{2k+1}$。这些螺旋面构成了一类基本且特殊的光滑、恰当嵌入的极小子流形族，它们与$\mathbb{R}^{k+1}$微分同胚，并为$\mathbb{R}^3$中的经典螺旋面提供了自然的高维类比。我们完全确定了它们的稳定性：$H_k$在$k\geq 3$时稳定，在$k\leq 2$时不稳定。在$k=3$处的尖锐转变尤为引人注目：虽然经典螺旋面（$k=1$）及其第一个高维类比（$k=2$）不稳定，但四维螺旋面$H_3\subset\mathbb{R}^7$已经稳定。对于$k\geq 3$，我们还确定了它们的面积最小化性质：$H_k$在$k$为偶数时面积最小化，在$k$为奇数时非面积最小化。面积最小化的结果通过构造显式校准来证明，而非面积最小化的结果则来自显式的竞争者。特别地，对于每个偶数$k\geq 4$，$(k+1)$维螺旋面$H_k$是$\mathbb{R}^{2k+1}$中面积最小化的整体极小图。
英文摘要
For each integer $k\geq 1$, we study the $(k+1)$-dimensional helicoid $H_k\subset\mathbb{R}^{2k+1}$ parametrized by \[ (u_1,\ldots,u_k,s) \longmapsto \bigl(u_1e^{is},\ldots,u_ke^{is},s\bigr) \in \mathbb{C}^k\times\mathbb{R}\cong \mathbb{R}^{2k+1}. \] These helicoids form a basic and distinguished family of complete, properly embedded minimal submanifolds diffeomorphic to $\mathbb{R}^{k+1}$, and provide natural higher-dimensional analogues of the classical helicoid in $\mathbb{R}^3$. We completely determine their stability: $H_k$ is stable for $k\geq 3$ and unstable for $k\leq 2$. The sharp transition at $k=3$ is particularly striking: while the classical helicoid $(k=1)$ and its first higher-dimensional analogue $(k=2)$ are unstable, the four-dimensional helicoid $H_3\subset\mathbb{R}^7$ is already stable. For $k\geq 3$, we also determine their area-minimizing property: $H_k$ is area-minimizing when $k$ is even and not area-minimizing when $k$ is odd. The area-minimizing result is proved by constructing an explicit calibration, while the non-area-minimizing result follows from an explicit competitor. In particular, for every even $k\geq 4$, the $(k+1)$-dimensional helicoid $H_k$ is an entire minimal graph in $\mathbb{R}^{2k+1}$ that is area-minimizing.
Comments32 pages, 2 figures
