Bombieri–De Giorgi–Giusti theorem

Remarks

Enrico Bombieri, Ennio De Giorgi and Enrico Giusti proved these results in 1969. For parts 1 and 2 they compared $E$ with a family of smooth minimal hypersurfaces that fill each side of the cone. Such a family forces any competitor to have at least as much perimeter. For part 3 they built a subsolution and a supersolution of the minimal surface equation, and found an entire solution between them. At infinity both behave like a homogeneous function that vanishes on the Simons cone. The function $u$ of part 3 gives, for every $m \ge 8$, a non-affine solution on $\R^m$ that does not depend on the last $m - 8$ variables. So the answer to Bernstein's problem is negative in all dimensions $m \ge 8$, while Simons had shown it is positive for $m \le 7$. Part 2 shows that the bound $n - 8$ on the dimension of the singular set of area-minimizing hypersurfaces in $\R^n$ is sharp.

Going deeper

For a student

The minimizing cone and the minimal graph are linked through a blow-down argument. If a non-affine entire minimal graph exists over $\R^m$, a blow-down argument of Fleming and De Giorgi produces a non-flat area-minimizing cone in $\R^m$. Simons showed that there is no such cone in $\R^m$ for $m \le 7$, which gives Bernstein's theorem for $m \le 7$. The theorem shows that this obstruction disappears in $\R^8$, and constructs the corresponding graph. It is a standard example of how the regularity theory of minimal surfaces and the Liouville-type theorems for the minimal surface equation depend on the dimension.