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.
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
| n-1 | |
| \sum | |
| i=1 |
| \partial | |
| \partialxi |
| ||||||||||||
|
Bernstein's problem asks whether an entire function (a function defined throughout Rn-1) that solves this equation is necessarily a degree-1 polynomial.
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 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.
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 showed that the surface defined by
\{x\inR8:
| 2 | |
| x | |
| 8 |
\}
is a locally stable cone in R8, and asked if it is globally area-minimizing.
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.