Minimal submanifolds of spheres and cones

M. I. Zelikin ; Yu. S. Osipov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 3, pp. 100-107

Résumé

Intersections of cones of index zero with spheres are investigated. Fields of the corresponding minimal manifolds are found. In particular, we consider the cone $\mathbb{K} =\{x_0^2+x_1^2=x_2^2+x_3^2\}$. Its intersection with the sphere $\mathbb{S}^3=\sum_{i=0}^3x_i^2$ is often called the Clifford torus $\mathbb{T}$, because Clifford was the first to notice that the metric of this torus as a submanifold of $\mathbb{S}^3$ with the metric induced from $\mathbb{S}^3$ is Euclidian. In addition, the torus $\mathbb{T}$ considered as a submanifold of $\mathbb{S}^3$ is a minimal surface. Similarly, it is possible to consider the cone $\mathcal{K} =\{\sum_{i=0}^3x_i^2=\sum_{i=4}^7x_i^2\}$, often called the Simons cone because he proved that $\mathcal{K}$ specifies a single-valued nonsmooth globally defined minimal surface in $\mathbb{R}^8$ which is not a plane. It appears that the intersection of $\mathcal{K}$ with the sphere $\mathbb{S}^7$, like the Clifford torus, is a minimal submanifold of $\mathbb{S}^7$. These facts are proved by using the technique of quaternions and the Cayley algebra.

Détail

Keywords: minimal surface, gaussian curvature, octonions (Cayley numbers), field of extremals, Weierstrass function.
Mots-clés : quaternions