Bernstein Problem

In differential geometry, Bernstein's problem is as follows: if the graph of a function on R^''n''−1 is a minimal surface in R^''n'', does this imply that the function is linear?
This is true in dimensions ''n'' at most 8, but false in dimensions ''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 R^''n'', and the condition that this is a minimal surface is that ''f'' satisfies the minimal surface equation

:$\sum_^ \frac\frac = 0$

Bernstein's problem asks whether an ''entire'' function (a function defined throughout R^''n''−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 R^''n''−1 then the analogue of Bernstein's theorem is true in R^''n'', 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 gave examples of locally stable cones in R⁸ and asked if they were globally area-minimizing.

showed that Simons' cones are indeed globally minimizing, and showed that in R^''n'' 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 dimensions up to 8, and false in higher dimensions.
A specific example is the surface $\$.

References

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*{{eom, id=b/b110360, title=Bernstein problem in differential geometry, first=E. , last=Straume

External links

Encyclopaedia of Mathematics article on the Bernstein theorem

Differential geometry