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 Trudy Inst. Mat. i Mekh. UrO RAN, 2019, Volume 25, Number 3, Pages 100–107 (Mi timm1650)

Minimal submanifolds of spheres and cones

M. I. Zelikin, Yu. S. Osipov

Lomonosov Moscow State University

Abstract: 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.

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

 Funding Agency Grant Number Russian Foundation for Basic Research 17-01-00805 This work was supported by the Russian Foundation for Basic Research (project no. 17-01-00805).

DOI: https://doi.org/10.21538/0134-4889-2019-25-3-100-107

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Bibliographic databases:

UDC: 523.46/.481
MSC: 49Q05, 11R52
Revised: 11.03.2019
Accepted:18.03.2019

Citation: M. I. Zelikin, Yu. S. Osipov, “Minimal submanifolds of spheres and cones”, Trudy Inst. Mat. i Mekh. UrO RAN, 25, no. 3, 2019, 100–107

Citation in format AMSBIB
\Bibitem{ZelOsi19} \by M.~I.~Zelikin, Yu.~S.~Osipov \paper Minimal submanifolds of spheres and cones \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2019 \vol 25 \issue 3 \pages 100--107 \mathnet{http://mi.mathnet.ru/timm1650} \crossref{https://doi.org/10.21538/0134-4889-2019-25-3-100-107} \elib{http://elibrary.ru/item.asp?id=39323540}