Bernstein's problem

For Bernstein's problem in mathematical genetics, see Genetic algebra. For Bernstein's Degrees-of-Freedom problem in motor control, see Degrees of Freedom Problem (Motor Control). For its possible generalization in global differential geometry, see spherical 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

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

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, Bibcode:1969InMat...7..243B, 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

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