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differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, Bernstein's problem is as follows: if the graph of a function on R''n''−1 is a
minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that ...
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 Sergei Natanovich Bernstein (russian: Серге́й Ната́нович Бернште́йн, sometimes Romanized as ; 5 March 1880 – 26 October 1968) was a Ukrainian and Russian mathematician of Jewish origin known for contributions to parti ...
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

* * German translation in * * * * * *{{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