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 in dimensions n at most 8, but false in dimensions 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

\sum _{i=1}^{n-1}{\frac {\partial }{\partial x_{i}}}{\frac {\frac {\partial f}{\partial x_{i}}}{\sqrt {1+\sum _{j=1}^{n-1}({\frac {\partial f}{\partial x_{j}}})^{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 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 gave examples of locally stable cones in R⁸ and asked if they were globally area-minimizing.

Bombieri, De Giorgi & Giusti (1969) showed that Simons' cones are indeed globally minimizing, and showed 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 dimensions up to 8, and false in higher dimensions. A specific example is the surface $\{x\in \mathbb {R} ^{8}:x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}\}$ .

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: 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: 243–268, doi:10.1007/BF01404309, ISSN 0020-9910, MR 0250205
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: 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

External links

Encyclopaedia of Mathematics article on the Bernstein theorem