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.

Statement

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

Template:Harvtxt 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.

Template:Harvtxt 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.

Template:Harvtxt 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.

Template:Harvtxt showed there are no non-planar minimizing cones in R⁴, thus extending Bernstein's theorem to R⁵.

Template:Harvtxt 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

$\{x\in {ℝ}^8:x_1^2+x_2^2+x_3^2+x_4^2=x_5^2+x_6^2+x_7^2+x_8^2\}$

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

Template:Harvtxt 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.

References

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External links

• Encyclopaedia of Mathematics article on the Bernstein theorem