Bernstein's problem

In differential geometry, Bernstein's problem is as follows: if the graph of a function on Rⁿ⁻¹ is a minimal surface in Rⁿ, does this imply that the function is linear? This is true for n at most 8, but false for n at least 9. The problem is named for Sergei Natanovich Bernstein who solved the case n = 3 in 1914.

Suppose that f is a function of n − 1 real variables. The graph of f is a surface in Rⁿ, and the condition that this is a minimal surface is that f satisfies the minimal surface equation

${\displaystyle \sum _{i=1}^{n-1}{\frac {\partial }{\partial x_{i}}}{\frac {\frac {\partial f}{\partial x_{i}}}{\sqrt {1+\sum _{j=1}^{n-1}\left({\frac {\partial f}{\partial x_{j}}}\right)^{2}}}}=0}$

Bernstein's problem asks whether an entire function (a function defined throughout Rⁿ⁻¹ ) that solves this equation is necessarily a degree-1 polynomial.

Bernstein (1915–1917) proved Bernstein's theorem that a graph of a real function on R² that is also a minimal surface in R³ must be a plane.

Fleming (1962) gave a new proof of Bernstein's theorem by deducing it from the fact that there is no non-planar area-minimizing cone in R³.

De Giorgi (1965) showed that if there is no non-planar area-minimizing cone in Rⁿ⁻¹ then the analogue of Bernstein's theorem is true for graphs in Rⁿ, which in particular implies that it is true in R⁴.

Almgren (1966) showed there are no non-planar minimizing cones in R⁴, thus extending Bernstein's theorem to R⁵.

Simons (1968) showed there are no non-planar minimizing cones in R⁷, thus extending Bernstein's theorem to R⁸. He also showed that the surface defined by

${\displaystyle \{x\in \mathbb {R} ^{8}:x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}\}}$

is a locally stable cone in R⁸, and asked if it is globally area-minimizing.

Bombieri, De Giorgi & Giusti (1969) showed that Simons' cone is indeed globally minimizing, and that in Rⁿ for n≥9 there are graphs that are minimal, but not hyperplanes. Combined with the result of Simons, this shows that the analogue of Bernstein's theorem is true in Rⁿ for n≤8, and false in higher dimensions.