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.
Definition 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.
Applications
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.
See also Minimal surface Bernstein's problem Geometric measure theory
References
Original source J. Simons (1968), "Minimal varieties in riemannian manifolds" Annals of Mathematics, 88 pp. 62-105
