Minimal Submanifolds of Spheres and Cones

Abstract

Intersections of cones of index zero with spheres are investigated. Fields of the corresponding minimal manifolds are found. In particular, the cone $\mathbb{K} = \left\{ {x_0^2 + x_1^2 + x_2^2 + x_3^2} \right\}$ is considered. Its intersection with the sphere $\mathbb{S}^{3}=\sum\nolimits_{i=0}^{3} x_{i}^{2}$ 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 ${\mathcal K} = \{ \sum\nolimits_{i = 0}^3x_0^2 = \sum _{i = 4}^7x_i^2\}$, often called the Simons cone because he proved that ${\mathcal K}$ specifies a single-valued nonsmooth globally defined minimal surface in ℝ⁸ 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.