M.I. Zelikin, Yu.S. Osipov. Minimal submanifolds of spheres and cones ... P. 100-107

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

Received February 11, 2019

Revised March 11, 2019

Accepted March 18, 2019

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

Yury Sergeyevich Osipov, Dr. Phys.-Math. Sci., Prof., RAS Academician, Lomonosov Moscow State University, Moscow 119991, Russia, e-mail: yriyosipov@hotmail.com

Mikhail Ilyich Zelikin, Dr. Phys.-Math. Sci, Prof., RAS Corresponding Member, Lomonosov Moscow State University, Moscow 119991, Russia, e-mail: mzelikin@mtu-net.ru

REFERENCES

1.   Caratheodory C. $\ddot{\mathrm{U}}$ber die Variationsrechnung bei mehrfachen Integralen. Acta Szeged. 1929, vol. 4, pp. 193–216.

2.   Clifford W.K. On a surface of zero curvature and finite extent. Proc. London Math. Soc., 1873, vol. 4, pp. 381–395.

3.   Osipov Yu.S., Zelikin M.I. Multidimensional generalization of Jacobi envelope theorem. Russian J. Math. Phys., 2012, vol. 19, no. 1, pp. 101–106. doi: 10.1134/S1061920812010086 

4.   Postnikov M.M. Lectures in geometry: Lie groups and Lie algebras. Semester V. Moscow: Mir Publ., 1986, 437 p. ISBN (2nd ed.): 978-5-88417-024-6 . Original Russian text published in Postnikov M.M. Gruppy i algebry Li. Lektsii po geometrii, Semestr V. Moscow: Nauka Publ., 1982, 447 p.

5.   Simons J. Minimal varieties in Riemannian manifold. Ann. of Math., 1969, vol. 88, pp. 62–105. doi: 10.2307/1970556 

6.   Weyl H. Geodesic fields in the calculus of variations for multiple integrals. Ann. Math., 1935, vol. 36, pp. 607–629. doi: 10.2307/1968645 

7.   Zelikin M.I. Theory of fields of extremals for multiple integrals. Russian Math. Surveys, 2011, vol. 66, no. 4, pp. 733–765. doi: 10.1070/RM2011v066n04ABEH004754 

Cite this article as: M.I. Zelikin, Yu.S. Osipov. Minimal submanifolds of spheres and cones, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 3, pp. 100–107.