Bernstein's problem

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 :\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. ==History==

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. 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. 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. showed there are no non-planar minimizing cones in R4, thus extending Bernstein's theorem to R5. 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 :\{ 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. 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. == See also ==