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
- [math]\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 }[/math]
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
- [math]\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 \} }[/math]
is a locally stable cone in R8, and asked if it is globally area-minimizing.
(Bombieri De Giorgi) 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
- 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" (in German), Mathematische Zeitschrift (Springer Berlin / Heidelberg) 26: 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
- De Giorgi, Ennio (1965), "Una estensione del teorema di Bernstein", Ann. Scuola Norm. Sup. Pisa (3) 19: 79–85, http://www.numdam.org/item?id=ASNSP_1965_3_19_1_79_0
- 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
- Hazewinkel, Michiel, ed. (2001), "Bernstein theorem", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=b/b015750
- Simons, James (1968), "Minimal varieties in riemannian manifolds", Annals of Mathematics, Second Series 88 (1): 62–105, doi:10.2307/1970556, ISSN 0003-486X, https://www.jstor.org/stable/1970556
- Hazewinkel, Michiel, ed. (2001), "Bernstein problem in differential geometry", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=b/b110360
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