Simons cone

In geometry and geometric measure theory, the Simons cone refers to a specific minimal hypersurface in ${\mathbb{ℝ}}^8$ that plays a crucial role in resolving Bernstein's problem in higher dimensions. It is named after American mathematician Jim Simons.

Definition

The Simons cone is defined as the hypersurface given by the equation

$S=\{x\in {\mathbb{ℝ}}^8|x_1^2+x_2^2+x_3^2+x_4^2=x_5^2+x_6^2+x_7^2+x_8^2\}\subset {\mathbb{ℝ}}^8$.

This 7-dimensional cone has the distinctive property that its mean curvature vanishes at every point except at the origin, where the cone has a singularity.

Applications

See also: Bernstein's problem

The classical Bernstein theorem states that any minimal graph in ${\mathbb{ℝ}}^3$ must be a plane. This was extended to ${\mathbb{ℝ}}^4$ by Wendell Fleming in 1962 and Ennio De Giorgi in 1965, and to dimensions up to ${\mathbb{ℝ}}^5$ by Frederick J. Almgren Jr. in 1966 and to ${\mathbb{ℝ}}^8$ by Jim Simons in 1968. The existence of the Simons cone as a minimizing cone in ${\mathbb{ℝ}}^8$ demonstrated that the Bernstein theorem could not be extended to ${\mathbb{ℝ}}^9$ and higher dimensions. Bombieri, De Giorgi, and Enrico Giusti proved in 1969 that the Simons cone is indeed area-minimizing, thus providing a negative answer to the Bernstein problem in higher dimensions.

See also

• Minimal surface
• Bernstein's problem
• Geometric measure theory

Original source

• J. Simons (1968), "Minimal varieties in riemannian manifolds" Annals of Mathematics, 88 pp. 62-105