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.

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 a function satisfying this equation is necessarily linear.

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 minimal cone in R³.

De Giorgi (1965) showed that if there is no minimal 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 minimal cones in R⁴, thus extending Bernstein's theorem to R⁵.

Simons (1968) showed there are no minimal 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 global minima for area.

Bombieri, De Giorgi & Giusti (1969) showed that Simon's cones are indeed globally of minimal area, 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.

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, ISSN 0003-486X, MR0200816
• 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{{citation}}: CS1 maint: date format (link) 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, 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, MR0250205
• De Giorgi, Ennio (1965), "Una estensione del teorema di Bernstein", Ann. Scuola Norm. Sup. Pisa (3), 19: 79–85, MR0178385
• 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, MR0157263
• 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, ISSN 0003-486X, MR0233295
• Straume, E. (2001) [1994], "Bernstein problem in differential geometry", Encyclopedia of Mathematics, EMS Press