Simons cone

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

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

S = { x ∈ 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 } ⊂ R 8 {\displaystyle 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.^[1][2]

The classical Bernstein theorem states that any minimal graph in R 3 {\displaystyle \mathbb {R} ^{3}} must be a plane. This was extended to R 4 {\displaystyle \mathbb {R} ^{4}} by Wendell Fleming in 1962 and Ennio De Giorgi in 1965, and to dimensions up to R 5 {\displaystyle \mathbb {R} ^{5}} by Frederick J. Almgren Jr. in 1966 and to R 8 {\displaystyle \mathbb {R} ^{8}} by Jim Simons in 1968. The existence of the Simons cone as a minimizing cone in R 8 {\displaystyle \mathbb {R} ^{8}} demonstrated that the Bernstein theorem could not be extended to R 9 {\displaystyle \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.^[1][2]

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

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