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.12
Applications
See also: Bernstein's problem
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.34
See also
References
Bombieri, E., De Giorgi, E., and Giusti, E. (1969). "Minimal cones and the Bernstein problem". Inventiones Mathematicae, 7: 243-268. /wiki/Inventiones_Mathematicae ↩
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 ↩
Bombieri, E., De Giorgi, E., and Giusti, E. (1969). "Minimal cones and the Bernstein problem". Inventiones Mathematicae, 7: 243-268. /wiki/Inventiones_Mathematicae ↩
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 ↩