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{{MSC|53A10}}
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[[Category:Differential geometry]]
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[[Category:Partial differential equations]]
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{{TEX|done}}
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''on minimal surfaces''
 
''on minimal surfaces''
  
If a minimal surface is given by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015750/b0157501.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015750/b0157502.png" /> has continuous partial derivatives of the first and second orders for all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015750/b0157503.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015750/b0157504.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015750/b0157505.png" /> is a plane. A proof of this theorem, which is due to S.N. Bernstein [S.N. Bernshtein] [[#References|[1]]], is a consequence of a more general theorem on the behaviour of surfaces with non-positive curvature. Various generalizations of Bernstein's theorem have been proposed, most of them being of the three following kinds: 1) Quantitative improvements; e.g. obtaining a priori estimates of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015750/b0157506.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015750/b0157507.png" /> is the radius of the disc over which the minimal surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015750/b0157508.png" /> is defined and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015750/b0157509.png" /> is the Gaussian curvature of the surface at the centre of the disc. 2) The search for other a priori geometric conditions under which the minimal surface would be of a specific kind — a plane, a catenoid, etc.; for instance, if the spherical image of a complete minimal surface contains no open set on the sphere, then such a minimal surface is a plane. 3) The generalization of Bernstein's theorem to minimal surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015750/b01575010.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015750/b01575011.png" />, located in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015750/b01575012.png" />; for example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015750/b01575013.png" />, any minimal surface over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015750/b01575014.png" /> is uniquely determined if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015750/b01575015.png" />, and is a hyperplane, while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015750/b01575016.png" />, there exist non-planar minimal surfaces; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015750/b01575017.png" />, then already for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015750/b01575018.png" /> it is possible to find non-linear minimal surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015750/b01575019.png" />, defined over any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015750/b01575020.png" />.
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A "Liouville-type" theorem proved by S. N. Bernstein in {{Cite|Be}}, which states the following
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'''Theorem 1'''
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If $\Sigma$ is an entire minimal graph in $\mathbb R^3$, i.e. the graph of a function $f: \mathbb R^2 \to \mathbb R$ with mean curvature $0$, then $f$ is affine, i.e. $f(x) = a + b\cdot x$ for some constants $a\in \mathbb R, b\in \mathbb R^2$.
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The condition that the mean curvature vanishes is equivalent to $f$ being
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a solution of an elliptic nonlinear partial differential equation (see [[Bernstein problem in differential geometry|Bernstein problem]] for more details). Thus, the Bernstein theorem can regarded
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as an analog of the classical fact that harmonic functions with a polynomial growth must be polynomials.  
  
====References====
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Various generalizations of Bernstein's theorem have been developed subsequently, most of them being of the three following kinds:
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.N. [S.N. Bernshtein] Bernstein,  "Über ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom elliptischen Typus"  ''Math. Z.'' , '''26'''  (1927)  pp. 551–558  (Translated from French)  {{MR|1544873}}  {{ZBL|53.0670.01}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.C.C. Nitsche,  "Vorlesungen über Minimalflächen" , Springer  (1975)  {{MR|0448224}} {{ZBL|0319.53003}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Osserman,  "Minimal varieties"  ''Bull. Amer. Math. Soc.'' , '''75'''  (1969)  pp. 1092–1120  {{MR|0276875}} {{ZBL|0188.53801}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R. Osserman,  "A survey of minimal surfaces" , v. Nostrand  (1969)  {{MR|0256278}} {{ZBL|0209.52901}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.T. Fomenko,  "Plateau's problem" , Gordon &amp; Breach  (1987)  (Translated from Russian)  {{MR|}} {{ZBL|}} </TD></TR></table>
 
  
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1) Quantitative improvements; e.g. obtaining a priori estimates of the form $K (0) \leq C R^{-2}$,
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for a minimal graph of type $f: B_R (0)\to \mathbb R$ with $f(0)= 0$, where $K$ denotes the Gauss curvature of the graph (such estimates were first obtained by Heinz in {{Cite|He}}).
  
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2) A priori geometric conditions under which the minimal surface would be of a specific kind — a plane, a catenoid, etc.. For instance, Fujimoto's Theorem (see {{Cite|Fu}}) shows that any complete minimal surface in $\mathbb R^3$ whose Gauss map omits more than $4$ points is necessarily a $2$-dimensional plane. Fujimoto's theorem is optimal since the Gauss map of [[Scherk surface|Scherk's minimal surface]] omits precisely $4$ points and completes a body of work from several other mathematicians (see, for instance, {{Cite|Os}}).
  
====Comments====
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3) Generalizations to minimal graphs in higher dimensions. This issue has become famous as the Bernstein problem and was fully resolved in the sixties by the works of De Giorgi, Fleming, Almgren, Simons
As an important reference for the generalizations of Bernstein's theorem the paper of Bombieri–de Giorgi–Giusti [[#References|[a1]]] can be quoted. The original of [[#References|[1]]] is [[#References|[a2]]].
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(Theorem 1 holds for minimal graphs in $\mathbb R^n$ when $n\leq 8$) and Bombieri, De Giorgi and Giusti
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(there is an entire minimal graph in $\mathbb R^9$ which is not a hyperplane); see [[Bernstein problem in differential geometry|Bernstein problem]] for more details.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Bombieri,   E. de Giorgi,   E. Giusti,  "Minimal cones and the Bernstein theorem"  ''Inventiones Math.'' , '''7'''  (1969) pp. 243–269 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> S.N. Bernstein,   "Sur une théorème de géometrie et ses applications aux équations dérivées partielles du type elliptique"  ''Comm. Soc. Math. Kharkov'' , '''15''' (1915–1917) pp. 38–45  {{MR|}} {{ZBL|}} </TD></TR></table>
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{|
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|-
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|valign="top"|{{Ref|Be}}|| S.N. Bernstein, "Sur un théorème de géométrie et ses applications aux équations dérivées partielles du type elliptique"  ''Comm. Soc. Math.  Kharkov'' , '''15'''  (1915–1917)  pp. 38–45  {{MR|}} {{ZBL|}}
 +
|-
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|valign="top"|{{Ref|Be2}}|| S.N. Bernstein, "Über ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom elliptischen Typus"  ''Math. Z.'' , '''26'''  (1927)  pp. 551–558  (Translated from French)  {{MR|1544873}}  {{ZBL|53.0670.01}}
 +
|-
 +
|valign="top"|{{Ref|BDG}}|| E. Bombieri, E. De Giorgi, E. Giusti,  "Minimal cones and the Bernstein theorem"  ''Inventiones Math.'' , '''7'''  (1969) pp. 243–269 {{MR|0250205}}{{ZBL|0183.25901}} 
 +
|-
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|valign="top"|{{Ref|Fu}}|| H. Fujimoto, "On the number of exceptional values of the Gauss map of minimal surfaces" ''J. Math. Soc. Japan'' , '''40''' (1988) pp. 235-247
 +
|-
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|valign="top"|{{Ref|He}}|| E. Heinz, "Ueber die  Loesungen  der Minimalflaechengleichung" ''Nachr. Akad. Wiss. Goettingen Math. Phys.'' K1 ii, (1952) pp. 51-56
 +
|-
 +
|valign="top"|{{Ref|Ni}}|| J.C.C. Nitsche, "Vorlesungen über Minimalflächen" , Springer (1975) {{MR|0448224}} {{ZBL|0319.53003}}
 +
|-
 +
|valign="top"|{{Ref|Os}}|| R. Osserman, "A survey of minimal surfaces", v. Nostrand (1969) {{MR|0256278}}
 +
|-
 +
|valign="top"|{{Ref|Os2}}|| R. Osserman, "Minimal varieties" ''Bull. Amer. Math. Soc.'', '''75''' (1969) pp. 1092–1120 {{MR|0276875}} {{ZBL|0188.53801}}
 +
|-
 +
|valign="top"|{{Ref|Si}}|| J. Simons, "Minimal varieties in riemannian manifolds" ''Ann. of Math.'', '''88''' (1968) pp. 62-105 {{MR|233295}} {{ZBL|0181.49702}}
 +
|-
 +
|}

Latest revision as of 05:52, 17 July 2024

2020 Mathematics Subject Classification: Primary: 53A10 [MSN][ZBL]

on minimal surfaces

A "Liouville-type" theorem proved by S. N. Bernstein in [Be], which states the following

Theorem 1 If $\Sigma$ is an entire minimal graph in $\mathbb R^3$, i.e. the graph of a function $f: \mathbb R^2 \to \mathbb R$ with mean curvature $0$, then $f$ is affine, i.e. $f(x) = a + b\cdot x$ for some constants $a\in \mathbb R, b\in \mathbb R^2$.

The condition that the mean curvature vanishes is equivalent to $f$ being a solution of an elliptic nonlinear partial differential equation (see Bernstein problem for more details). Thus, the Bernstein theorem can regarded as an analog of the classical fact that harmonic functions with a polynomial growth must be polynomials.

Various generalizations of Bernstein's theorem have been developed subsequently, most of them being of the three following kinds:

1) Quantitative improvements; e.g. obtaining a priori estimates of the form $K (0) \leq C R^{-2}$, for a minimal graph of type $f: B_R (0)\to \mathbb R$ with $f(0)= 0$, where $K$ denotes the Gauss curvature of the graph (such estimates were first obtained by Heinz in [He]).

2) A priori geometric conditions under which the minimal surface would be of a specific kind — a plane, a catenoid, etc.. For instance, Fujimoto's Theorem (see [Fu]) shows that any complete minimal surface in $\mathbb R^3$ whose Gauss map omits more than $4$ points is necessarily a $2$-dimensional plane. Fujimoto's theorem is optimal since the Gauss map of Scherk's minimal surface omits precisely $4$ points and completes a body of work from several other mathematicians (see, for instance, [Os]).

3) Generalizations to minimal graphs in higher dimensions. This issue has become famous as the Bernstein problem and was fully resolved in the sixties by the works of De Giorgi, Fleming, Almgren, Simons (Theorem 1 holds for minimal graphs in $\mathbb R^n$ when $n\leq 8$) and Bombieri, De Giorgi and Giusti (there is an entire minimal graph in $\mathbb R^9$ which is not a hyperplane); see Bernstein problem for more details.

References

[Be] S.N. Bernstein, "Sur un théorème de géométrie et ses applications aux équations dérivées partielles du type elliptique" Comm. Soc. Math. Kharkov , 15 (1915–1917) pp. 38–45
[Be2] S.N. Bernstein, "Über ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom elliptischen Typus" Math. Z. , 26 (1927) pp. 551–558 (Translated from French) MR1544873 Zbl 53.0670.01
[BDG] E. Bombieri, E. De Giorgi, E. Giusti, "Minimal cones and the Bernstein theorem" Inventiones Math. , 7 (1969) pp. 243–269 MR0250205Zbl 0183.25901
[Fu] H. Fujimoto, "On the number of exceptional values of the Gauss map of minimal surfaces" J. Math. Soc. Japan , 40 (1988) pp. 235-247
[He] E. Heinz, "Ueber die Loesungen der Minimalflaechengleichung" Nachr. Akad. Wiss. Goettingen Math. Phys. K1 ii, (1952) pp. 51-56
[Ni] J.C.C. Nitsche, "Vorlesungen über Minimalflächen" , Springer (1975) MR0448224 Zbl 0319.53003
[Os] R. Osserman, "A survey of minimal surfaces", v. Nostrand (1969) MR0256278
[Os2] R. Osserman, "Minimal varieties" Bull. Amer. Math. Soc., 75 (1969) pp. 1092–1120 MR0276875 Zbl 0188.53801
[Si] J. Simons, "Minimal varieties in riemannian manifolds" Ann. of Math., 88 (1968) pp. 62-105 MR233295 Zbl 0181.49702
How to Cite This Entry:
Bernstein theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bernstein_theorem&oldid=28155
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article