Stability and Area-Minimizing Property of Higher-Dimensional Helicoids
Abstract
For each integer k≥ 1, we study the (k+1)-dimensional helicoid Hk⊂ℝ2k+1 parametrized by (u1,…,uk,s) longmapsto (u1eis,…,ukeis,s) ∈ ℂk×ℝ≅ ℝ2k+1. These helicoids form a basic and distinguished family of complete, properly embedded minimal submanifolds diffeomorphic to ℝk+1, and provide natural higher-dimensional analogues of the classical helicoid in ℝ3. We completely determine their stability: Hk is stable for k≥ 3 and unstable for k≤ 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 H3⊂ℝ7 is already stable. For k≥ 3, we also determine their area-minimizing property: Hk 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≥ 4, the (k+1)-dimensional helicoid Hk is an entire minimal graph in ℝ2k+1 that is area-minimizing.
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BibTeX
@misc{tsai2026stability,
title = {{Stability and Area-Minimizing Property of Higher-Dimensional Helicoids}},
author = {Chung-Jun Tsai and Mao-Pei Tsui and Jingbo Wan and Mu-Tao Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11481},
eprint = {2609.11481},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11481},
}
RIS
TY - UNPB
TI - Stability and Area-Minimizing Property of Higher-Dimensional Helicoids
AU - Chung-Jun Tsai
AU - Mao-Pei Tsui
AU - Jingbo Wan
AU - Mu-Tao Wang
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11481
UR - https://arxiv.org/abs/2609.11481
AB - For each integer k≥ 1, we study the (k+1)-dimensional helicoid H_k⊂ℝ^2k+1 parametrized by (u_1,…,u_k,s) longmapsto (u_1e^is,…,u_ke^is,s) ∈ ℂ^k×ℝ≅ ℝ^2k+1. These helicoids form a basic and distinguished family of complete, properly embedded minimal submanifolds diffeomorphic to ℝ^k+1, and provide natural higher-dimensional analogues of the classical helicoid in ℝ^3. We completely determine their stability: H_k is stable for k≥ 3 and unstable for k≤ 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⊂ℝ^7 is already stable. For k≥ 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≥ 4, the (k+1)-dimensional helicoid H_k is an entire minimal graph in ℝ^2k+1 that is area-minimizing.
ER -