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.