Simons cone

2020 Mathematics Subject Classification: Primary: 53A10 [MSN][ZBL]

In the theory of minimal surfaces, the term is used for the $7$-dimensional cone \[ \{x_1^2 + x_2^2 + x_3^2 + x_4^2 = x_5^2 + x_6^2 + x_7^2 + x_8^2\}\subset \mathbb R^8\, . \] The mean curvature of the Simons' cone vanishes at every point outside the origin: therefore the first variation of the area vanishes along any deformation induced by compactly supported smooth vector fields. It was pointed out by Simons in his fundamental work on minimal varieties [Si] that this cone is also stable, meaning that the second variation of the area along smooth deformations is nonnegative. Eventually Bombieri, De Giorgi and Giusti showed in [BDG] that the Simons cone is in fact a minimizer of the area, i.e. that any hypersurface which coincides with the cone outside a compact set $K$ must have larger area in $K$. Given the singularity of the cone, such minimizing problem must be phrased in the appropriate setting, where non-smooth competitors are allowed (cf. Geometric measure theory); the setting of Bombieri, De Giorgi and Giusti is that of sets of finite perimeter (cf. Function of bounded variation), which however includes all smooth competitor surfaces.

The paper of Bombieri, De Giorgi and Giusti gave a negative answer to the Bernstein problem in high dimensions.

For a recent, elegant and simple proof of the minimizing property of the Simons' cone, see [DP].

References

[BDG]	E. Bombieri, E. De Giorgi, E. Giusti, "Minimal cones and the Bernstein theorem" Inventiones Math. , 7 (1969) pp. 243–269 MR0250205Zbl 0183.25901
[DP]	G. De Philippis, E. Paolini, "A short proof of the minimality of Simons cone" Rend. Sem. Mat. Univ. Padova, 121 (2009) pp. 233-241
[Si]	J. Simons, "Minimal varieties in riemannian manifolds" Ann. of Math., 88 (1968) pp. 62-105 MR233295 Zbl 0181.49702