Bernstein's problem

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

:$\backslash sum\_\{i=1\}^\{n-1\}\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\_i\}\backslash frac\{\backslash frac\{\backslash partial\; f\}\{\backslash partial\; x\_i\}\}\{\backslash sqrt\{1+\backslash sum\_\{j=1\}^\{n-1\}\backslash left(\backslash frac\{\backslash partial\; f\}\{\backslash partial\; x\_j\}\backslash right)^2\}\}\; =\; 0$

Bernstein's problem asks whether an entire function (a function defined throughout Rⁿ⁻¹ ) that solves this equation is necessarily a degree-1 polynomial.

{{harvtxt|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.

{{harvtxt|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.

{{harvtxt|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.

{{harvtxt|Almgren|1966}} showed there are no non-planar minimizing cones in R⁴, thus extending Bernstein's theorem to R⁵.

{{harvtxt|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

:$\backslash \{\; x\; \backslash in\; \backslash mathbb\{R\}^8\; :\; x\_1^2+x\_2^2+x\_3^2+x\_4^2=x\_5^2+x\_6^2+x\_7^2+x\_8^2\; \backslash \}$

is a locally stable cone in R⁸, and asked if it is globally area-minimizing.

{{harvtxt|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.

• Simons cone

• {{Citation | last1=Almgren | first1=F. J. | author1-link=Frederick J. Almgren Jr. | title=Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem | jstor=1970520 | mr=0200816 | year=1966 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=84 | issue=2 | pages=277–292 | doi=10.2307/1970520}}
• {{Citation | last1=Bernstein | first1=S. N. | author1-link=Sergei Natanovich Bernstein | title=Sur une théorème de géometrie et ses applications aux équations dérivées partielles du type elliptique | year=1915–1917 | journal=Comm. Soc. Math. Kharkov | volume=15 | pages=38–45}} German translation in {{Citation | last1=Bernstein | first1=Serge | title=Über ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom elliptischen Typus | doi=10.1007/BF01475472 | publisher=Springer Berlin / Heidelberg | language=German | year=1927 | journal=Mathematische Zeitschrift | issn=0025-5874 | volume=26 | pages=551–558}}
• {{Citation | last1=Bombieri | first1=Enrico | author1-link=Enrico Bombieri | last2=De Giorgi | first2=Ennio | author2-link=Ennio De Giorgi | last3=Giusti | first3=E. | author3-link=Enrico Giusti | title=Minimal cones and the Bernstein problem | doi=10.1007/BF01404309 | mr=0250205 | year=1969 | journal=Inventiones Mathematicae | issn=0020-9910 | volume=7 | issue=3 | pages=243–268| bibcode=1969InMat...7..243B | s2cid=59816096 }}
• {{Citation | last1=De Giorgi | first1=Ennio | author1-link=Ennio De Giorgi | title=Una estensione del teorema di Bernstein | url=http://www.numdam.org/item?id=ASNSP_1965_3_19_1_79_0 | mr=0178385 | year=1965 | journal=Ann. Scuola Norm. Sup. Pisa (3) | volume=19 | pages=79–85}}
• {{Citation | last1=Fleming | first1=Wendell H. | author1-link=Wendell Fleming | title=On the oriented Plateau problem | doi=10.1007/BF02849427 | mr=0157263 | year=1962 | journal=Rendiconti del Circolo Matematico di Palermo. Serie II | issn=0009-725X | volume=11 | pages=69–90}}
• {{eom|id=b/b015750|title=Bernstein theorem|first=I. Kh. |last=Sabitov}}
• {{Citation | last1=Simons | first1=James | author1-link=James Harris Simons | title=Minimal varieties in riemannian manifolds | jstor=1970556 | mr=0233295 | year=1968 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=88 | issue=1 | pages=62–105 | doi=10.2307/1970556| url=https://www.jstor.org/stable/1970556 }}
• {{eom|id=b/b110360|title=Bernstein problem in differential geometry|first=E. |last=Straume}}

• [http://www.encyclopediaofmath.org/index.php/Bernstein_theorem Encyclopaedia of Mathematics article on the Bernstein theorem]

Category:Minimal surfaces

Category:Functions and mappings

Category:Dimension

Category:Theorems in differential geometry