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Bernstein's problem

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Problem in differential geometry For Bernstein's problem in mathematical genetics, see Genetic algebra. For Bernstein's Degrees-of-Freedom problem in motor control, see Degrees of Freedom Problem (Motor Control). For its possible generalization in global differential geometry, see spherical Bernstein's problem.

In differential geometry, Bernstein's problem is as follows: if the graph of a function on R is a minimal surface in R, 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 R, and the condition that this is a minimal surface is that f satisfies the minimal surface equation

∑ i = 1 n − 1 ∂ ∂ x i ∂ f ∂ x i 1 + ∑ j = 1 n − 1 ( ∂ f ∂ x j ) 2 = 0 {\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 R ) that solves this equation is necessarily a degree-1 polynomial.

History

Bernstein (1915–1917) proved Bernstein's theorem that a graph of a real function on R that is also a minimal surface in R 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 R.

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

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

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

{ x ∈ 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 } {\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 R, and asked if it is globally area-minimizing.

Bombieri, De Giorgi & Giusti (1969) showed that Simons' cone is indeed globally minimizing, and that in R 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 R for n≤8, and false in higher dimensions.

References

External links

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