Simons cone

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

Definition

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

${\displaystyle S=\{x\in \mathbb {R} ^{8}|x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}\}\subset \mathbb {R} ^{8}}$ .

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]

Applications

See also: Bernstein's problem

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

See also

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

References

1. Bombieri, E., De Giorgi, E., and Giusti, E. (1969). "Minimal cones and the Bernstein problem". Inventiones Mathematicae, 7: 243-268.
2. G. De Philippis, E. Paolini (2009). "A short proof of the minimality of Simons cone". Rendiconti del Seminario Matematico della Università di Padova, 121. pp. 233-241

Original source

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