Stability and Area-Minimizing Property of Higher-Dimensional Helicoids

For each integer $k\ge 1$, we study the $(k+1)$-dimensional helicoid $H_k\subset {\mathbb{R}}^{2k+1}$ parametrized by $$(u_1,\dots ,u_k,s)⟼(u_1{e}^{is},\dots ,u_k{e}^{is},s)\in {\mathbb{C}}^k\times \mathbb{R}\cong {\mathbb{R}}^{2k+1}\mathrm{.}$$ 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 $k\ge 3$ and unstable for $k\le 2$. 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 $k\ge 3$, 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 $k\ge 4$, the $(k+1)$-dimensional helicoid $H_k$ is an entire minimal graph in ${\mathbb{R}}^{2k+1}$ that is area-minimizing.