# 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

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

${\displaystyle \sum _{i=1}^{n-1}{\frac {\partial }{\partial x_{i}}}{\frac {\frac {\partial f}{\partial x_{i}}}{\sqrt {1+\sum _{j=1}^{n-1}\left({\frac {\partial f}{\partial x_{j}}}\right)^{2}}}}=0}$

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

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

${\displaystyle \{x\in \mathbb {R} ^{8}:x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}\}}$

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.

## References

• Almgren, F. J. (1966), "Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem", Annals of Mathematics, Second Series, 84 (2): 277–292, doi:10.2307/1970520, ISSN 0003-486X, JSTOR 1970520, MR 0200816
• Bernstein, S. N. (1915–1917), "Sur une théorème de géometrie et ses applications aux équations dérivées partielles du type elliptique", Comm. Soc. Math. Kharkov, 15: 38–45 German translation in Bernstein, Serge (1927), "Über ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom elliptischen Typus", Mathematische Zeitschrift (in German), 26, Springer Berlin / Heidelberg: 551–558, doi:10.1007/BF01475472, ISSN 0025-5874
• Bombieri, Enrico; De Giorgi, Ennio; Giusti, E. (1969), "Minimal cones and the Bernstein problem", Inventiones Mathematicae, 7 (3): 243–268, doi:10.1007/BF01404309, ISSN 0020-9910, MR 0250205, S2CID 59816096
• De Giorgi, Ennio (1965), "Una estensione del teorema di Bernstein", Ann. Scuola Norm. Sup. Pisa (3), 19: 79–85, MR 0178385
• Fleming, Wendell H. (1962), "On the oriented Plateau problem", Rendiconti del Circolo Matematico di Palermo. Serie II, 11: 69–90, doi:10.1007/BF02849427, ISSN 0009-725X, MR 0157263
• Sabitov, I. Kh. (2001) [1994], "Bernstein theorem", Encyclopedia of Mathematics, EMS Press
• Simons, James (1968), "Minimal varieties in riemannian manifolds", Annals of Mathematics, Second Series, 88 (1): 62–105, doi:10.2307/1970556, ISSN 0003-486X, JSTOR 1970556, MR 0233295
• Straume, E. (2001) [1994], "Bernstein problem in differential geometry", Encyclopedia of Mathematics, EMS Press