Bernstein's problem

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 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

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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

{\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.

History

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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

\{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}\}

{\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.