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.
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.
Historyedit
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.
Referencesedit
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), Springer Berlin / Heidelberg, 26: 551–558, doi:10.1007/BF01475472, ISSN 0025-5874
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
Encyclopaedia of Mathematics article on the Bernstein theorem
February 15, 2024
bernstein, problem, mathematical, genetics, genetic, algebra, bernstein, degrees, freedom, problem, motor, control, degrees, freedom, problem, motor, control, possible, generalization, global, differential, geometry, spherical, differential, geometry, follows,. 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 Contents 1 Statement 2 History 3 References 4 External linksStatement editSuppose 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 i 1 n 1 x i f x i 1 j 1 n 1 f x j 2 0 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 nbsp 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 editBernstein 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 x 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 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 nbsp is a locally stable cone in R8 and asked if it is globally area minimizing Bombieri De Giorgi amp 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 editAlmgren 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 theoreme de geometrie et ses applications aux equations derivees partielles du type elliptique Comm Soc Math Kharkov 15 38 45 German translation in Bernstein Serge 1927 Uber ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom elliptischen Typus Mathematische Zeitschrift in German 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 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 PressExternal links editEncyclopaedia of Mathematics article on the Bernstein theorem Retrieved from https en wikipedia org w index php title Bernstein 27s problem amp oldid 1192063299, wikipedia, wiki, book, books, library,
