Simons cone

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

:$S\; =\; \backslash \{x\; \backslash in\; \backslash mathbb\; R^8\; |\; x\_1^2\; +\; x\_2^2\; +\; x\_3^2\; +\; x\_4^2\; =\; x\_5^2\; +\; x\_6^2\; +\; x\_7^2\; +\; x\_8^2\; \backslash \}\; \backslash subset\; \backslash mathbb\; R^8$.

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.Bombieri, E., De Giorgi, E., and Giusti, E. (1969). "Minimal cones and the Bernstein problem". Inventiones Mathematicae, 7: 243-268.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

{{See also|Bernstein's problem}}

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

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

== References ==

{{reflist}}

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

Category:Measure theory

Category:Surfaces

Category:Geometry

Category:Minimal surfaces