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{{TEX|done}}
 
{{TEX|done}}
  
It is a well-known and elementary fact in complex analysis that a bounded and [[Holomorphic function|holomorphic function]] on the whole plane must be a constant (cf. [[Liouville theorems|Liouville theorems]]). In fact, more generally an holomorphic function with polynomial growth is necessarily a polynomial. The same property can be then inferred for (real-valued) harmonic functions.
+
It is a well-known and elementary fact in complex analysis that a bounded and [[Holomorphic function|holomorphic function]] on the whole plane (cf. also [[Entire function|entire function]]) must be a constant (cf. [[Liouville theorems|Liouville theorems]]). In fact, more generally an holomorphic function with polynomial growth is necessarily a polynomial. The same property can be then inferred for (real-valued) harmonic functions.
  
S.N. Bernstein proved an analogous result in [[Differential geometry|differential geometry]] (cf. {{Cite|Be}}), saying that,
+
S.N. Bernstein proved an analogous result in [[Differential geometry|differential geometry]] (cf. {{Cite|Be}}), saying that, if the graph of a $C^2$ function $f: \mathbb R^2\to \mathbb R$ is a [[Minimal surface|minimal surface]] (i.e. its mean curvature vanishes everywhere), then $f$ must be an affine function (see [[Bernstein theorem|Bernstein theorem]]). The question whether the corresponding statement holds or not in higher dimensions became famous in the last century as the Bernstein problem and was completely solved at the end of the sixties. More precisely, since the [[Mean curvature|mean curvature]] of a graph can be explicitely computed in terms of first and second derivatives of $f$, the Bernstein problem can be stated in the following terms.
if the graph of a $C^2$ function $f: \mathbb R^2\to \mathbb R$ is a [[Minimal surface|minimal surface]] (i.e. its mean curvature vanishes everywhere), then $f$ must be an affine function (see [[Bernstein theorem|Bernstein theorem]]). The question whether the corresponding statement holds or not in higher dimensions became famous in the last century as the Bernstein problem and was completely solved at the end of the sixties. More precisely, since the condition that the [[Mean curvature|mean curvature]] of a graph can be explicitely computed in terms of first and second derivatives of $f$, the Bernstein problem can be stated in the following terms.
 
  
 
'''Problem 1'''
 
'''Problem 1'''
Is it true that $C^2$ solutions $f: \mathbb R^n \to \mathbb R$ (cf. also [[Entire function|Entire function]]) of the minimal surface equation
+
Is it true that $C^2$ solutions $f: \mathbb R^n \to \mathbb R$ of the minimal surface equation
\begin{equation}\label{e:min_surf}
+
\[
{\rm div}\, \frac{\nabla f}{\sqrt{1+|\nabla f|^2}
+
{\rm div}\, \frac{\nabla f}{\sqrt{1+|\nabla f|^2}}
\end{equation}
+
\]
 
are necessarily affine functions?
 
are necessarily affine functions?
  
The answer to the above question is affirmative in the range <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b1103608.png" />; the proof of this is the result of the successive efforts of Bernstein [[#References|[a2]]], W.H. Fleming [[#References|[a7]]], [[#References|[a8]]], E. DeGiorgi [[#References|[a9]]], F. Almgren [[#References|[a1]]], and J. Simons [[#References|[a16]]]. On the other hand, a counterexample for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b1103609.png" />, which in turn renders a counterexample for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036010.png" /> by a standard construction, was obtained in 1969 by E. Bombieri, DeGiorgi and E. Giusti, cf. [[#References|[a3]]]. The complete solution of the classical Bernstein problem constitutes, indeed, an exciting chapter of global differential geometry, involving geometric measure theory and non-linear analysis (cf. also [[Plateau problem|Plateau problem]]).
+
The answer to the above question is affirmative in the range $n \leq 7$. The proof of this is the result of the successive efforts of W.H. Fleming {{Cite|Fl}}, {{Cite|Fl2}}, E. DeGiorgi {{Cite|DG}}, F. Almgren {{Cite|Al}}, and J. Simons {{Cite|Si}}. The answer is instead negative when $n\geq 8$, as was shown by E. Bombieri, De Giorgi and E. Giusti in {{Cite|BDG}}. In particular:
 +
* Fleming remarked that an affirmative answer in $\mathbb R^n$ can be concluded from the statement that area-minimizing hypercones in $\mathbb R^{n+1}$ are necessarily planar (thereby giving a new proof of Bernstein's original theorem).
 +
* De Giorgi relaxed Fleming's condition to the nonexistence of area-minimizing hypercones in $\mathbb R^n$, extending Bernstein's statement to $n=3$.
 +
* Almgren proved that all area-minimizing hypercones in $\mathbb R^4$ are planar, thereby solving Bernstein's problem for $n=4$.
 +
* Simons proved that area-minimizing hypercones in $\mathbb R^n$ are planar for any $n\leq 7$, hence generalizing Bernstein's statement to $n\leq 7$. Indeed Simons proved the stronger statement that any stable minimal hypercone in $\mathbb R^n$ must be planar for $n\leq 7$. His proof is based on an inequality for the norm of the second fundamental form of minimal surfaces, see [[Simons inequality]], which has several deep consequences in the theory of minimal hypersurfaces. Simons also pointed out the existence of a minimal nonplanar $7$-dimensional cone in $\mathbb R^8$ which is stable. Such a cone is nowadays known as the [[Simons cone]].
 +
* Finally, Bombieri, De Giorgi and Giusti showed that Simons' cone is indeed area minimizing and obtained as a consequence a non-affine entire solution $f: \mathbb R^8 \to \mathbb R$ of the minimal surface equation.
  
Among the various generalizations of the above problem, the so-called spherical Bernstein problem is a natural and challenging one in the realm of global differential geometry. Let the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036011.png" />-sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036012.png" /> be imbedded as a minimal hypersurface of the Euclidean n-sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036013.png" />. Is it necessarily an equator?
+
The complete solution of the classical Bernstein problem constitutes, indeed, an exciting chapter of global differential geometry, involving geometric measure theory and nonlinear analysis (cf. also [[Plateau problem|Plateau problem]]).
  
This problem is due to S.S. Chern (cf. [[#References|[a5]]]; he also proposed it at the International Congress of Mathematicians at Nice (1970), cf. [[#References|[a6]]]). The problem has, for example, a direct bearing on the possible local structures of isolated singularities of minimal hypersurfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036014.png" /> in a general [[Riemannian manifold|Riemannian manifold]]. Indeed, the (regular) tangent cone of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036015.png" /> at the singularity is a minimal cone in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036016.png" />, whose intersection with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036017.png" /> is a minimal hypersurface.
+
====Spehrical Bernstein problem====
 +
Among the various generalizations of Problem 1, the so-called spherical Bernstein problem is a natural and challenging one in the realm of global differential geometry. It is due to S.S. Chern (cf. {{Cite|Ch}}; he also proposed it at the International Congress of Mathematicians at Nice (1970), cf. {{Cite|Ch2}}).
  
By the Almgren–Calabi theorem [[#References|[a1]]], [[#References|[a4]]], an immersion (cf. [[Immersion of a manifold|Immersion of a manifold]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036018.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036019.png" /> must, in fact, be an equator. Thus, at least the beginning case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036020.png" /> of the spherical Bernstein problem was known to have a positive answer. However, no further progress was made until 1983, when Wu-yi Hsiang, in the framework of equivariant [[Differential geometry|differential geometry]], constructed infinitely many mutually non-congruent minimal imbeddings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036021.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036022.png" />, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036023.png" /> (cf. [[#References|[a11]]], [[#References|[a12]]]). The basic ideas of this approach, which was initiated by Hsiang and H.B. Lawson [[#References|[a10]]], is to choose an orthogonal representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036024.png" /> with codimension-two principal orbits, and to search for solutions of the geometric problem which are invariant under the induced orthogonal transformation group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036025.png" />. Thus, the original partial differential equation associated with the minimal hypersurface condition is reduced to an ordinary second-order differential equation on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036026.png" />-dimensional orbit space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036027.png" />, which is geometrically a spherical lune. Here, the "closed" (in a certain sense) solution curves represent minimal hypersurfaces, and they will have the correct topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036028.png" /> thanks to the trivial summand added to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036029.png" />.
+
'''Problem 2'''
 +
Are the equators in $\mathbb S^{n+1}$ the only smooth embedded minimal hypersurfaces which are topological
 +
$n$-dimensional spheres?
  
During the 1980s, Hsiang and his collaborators obtained further results related to the spherical Bernstein problem, by investigating equivariant systems of the above type, as well as additional isoparametric foliations on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036030.png" />. For instance, many new examples of imbedded as well as immersed minimal hyperspheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036031.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036032.png" /> have been constructed, and moreover, the stability of the singularity at the origin of the corresponding minimal cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036033.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036034.png" /> has been investigated, see, e.g., [[#References|[a13]]]. P. Tomter [[#References|[a15]]] has shown the existence of a minimal and imbedded hypersphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036035.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036036.png" /> which is not an equator, for each even <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036037.png" />. On the other hand, although the methods give infinitely many non-congruent minimal immersions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036038.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036039.png" />, the problem of finding a non-equatorial imbedded sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036040.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036041.png" /> for odd <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036042.png" /> remains open.
+
As it happens for the classical Bernstein problem, the question has a direct bearing on the possible local structures of isolated singularities of minimal hypersurfaces $\Sigma$ in a general [[Riemannian manifold|Riemannian manifold]] $N$ of dimension $n+1$. Indeed, the tangent cone of $\Sigma$ at an isolated singularity is a minimal cone in the Euclidean space
 +
$\mathbb R^{n+1}$, whose intersection with $\mathbb S^n$ is a minimal hypersurface.
  
Finally, as a generalization of the spherical Bernstein problem, it is also natural to replace the ambient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110360/b11036043.png" /> by a simply-connected, compact symmetric space. See [[#References|[a14]]] for some recent results in this direction.
+
By the Almgren–Calabi theorem ({{Cite|Al}} and {{Cite|Ca}}), any immersion (cf. [[Immersion of a manifold]]) of $\mathbb S^2$ into $\mathbb S^3$ must, in fact, be an equator. Thus, at least the beginning case $n=2$ of the spherical Bernstein problem was known to have a positive answer. However, no further progress was made until 1983, when Wu-yi Hsiang, in the framework of equivariant differential geometry, constructed infinitely many mutually non-congruent minimal embeddings of $\mathbb S^n$ into $\mathbb S^{n+1}$, for $n\in \{3,4,5,6,7,9,11,13\}$ (cf. {{Cite|H}}, {{Cite|H2}}). The basic idea of this approach, which was initiated by Hsiang and H.B. Lawson in {{Cite|HL}}, is to choose an orthogonal representation $(G, \Phi, \mathbb R^{n+1})$ with codimension-two principal orbits, and for minimal hypersurfaces which are invariant under the induced orthogonal transformation group $(G, \Phi+ 1, \mathbb S^{n+1})$. Thus, the original partial differential equation associated with the minimal hypersurface condition is reduced to an ordinary second-order differential equation on the $2$-dimensional orbit space $\mathbb S^{n+1}/G$, which is geometrically a spherical lune. Here, the "closed" (in an appropriate sense) solution curves represent minimal hypersurfaces, and they will have the topology of $\mathbb S^n$ thanks to the trivial summand added to $\Phi$.
 +
 
 +
During the 1980s, Hsiang and his collaborators obtained further results related to the spherical Bernstein problem, by investigating equivariant systems of the above type, as well as additional isoparametric foliations on $\mathbb S^{n+1}$. For instance, many new examples of embedded as well as immersed minimal hyperspheres $\Sigma$ in $\mathbb S^{n+1}$ have been constructed, and moreover, the stability of the singularity at the origin of the corresponding minimal cone $\Gamma$ in $\mathbb R^{n+1}$ has been investigated, see, e.g., {{Cite|HS}}. P. Tomter in {{Cite|To}} has shown the existence of a minimal and embedded hypersphere $\Sigma$ in $\mathbb S^{n+1}$ which is not an equator, for each odd $n$. On the other hand, although the methods give infinitely many non-congruent minimal immersions of $\mathbb S^n$ into $\mathbb S^{n+1}$ for each $n$, the problem of finding a non-equatorial embedded minimal hypersphere in $\mathbb S^{n+1}$ for $n\geq 8$ even remains open.
 +
 
 +
Finally, as a generalization of the spherical Bernstein problem, it is also natural to replace the ambient space $\mathbb S^{n+1}$ by a simply-connected, compact symmetric space; see {{Cite|HHS}} for some results in this direction.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> F.J. Almgren, Jr., "Some interior regularity theorems for minimal surfaces and an extension of the Bernstein's theorem" ''Ann. of Math.'' , '''85''' (1966) pp. 277–292 {{MR|200816}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> S.N. Bernstein, "Sur une théorème de géometrie et ses applications aux dérivées partielles du type elliptique" ''Comm. Inst. Sci. Math. Mech. Univ. Kharkov'' , '''15''' (1915–17) pp. 38–45</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> E. Bombieri, E. DeGiorgi, E. Giusti, "Minimal cones and the Bernstein problem" ''Invent. Math.'' , '''7''' (1969) pp. 243–268 {{MR|0250205}} {{ZBL|0183.25901}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> E. Calabi, "Minimal immersions of surfaces in euclidean spaces" ''J. Diff. Geom.'' , '''1''' (1967) pp. 111–125</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> S.S. Chern, "Brief survey of minimal submanifolds" ''Tagungsbericht Oberwolfach'' (1969) {{MR|0358634}} {{MR|0358635}} {{ZBL|0218.53070}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> S.S. Chern, "Differential geometry, its past and its future" , ''Actes Congres Intern. Mathem.'' , '''1''' (1970) pp. 41–53 {{MR|0428217}} {{ZBL|0232.53001}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> W.H. Fleming, "On the oriented Plateau problem" ''Rend. Circ. Mat. Palermo'' , '''II''' (1962) pp. 1–22 {{MR|0157263}} {{ZBL|0107.31304}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> W.H. Fleming, "Flat chains over a finite coefficient group" ''Trans. Amer. Math. Soc.'' , '''121''' (1966) pp. 160–186 {{MR|0185084}} {{ZBL|0136.03602}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> E. DeGiorgi, "Una estensione del teoreme di Bernstein" ''Ann. Sc. Norm. Sup. Pisa'' , '''19''' (1965) pp. 79–85</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> W.Y. Hsiang, H.B. Lawson, Jr., "Minimal submanifolds of low cohomogeneity" ''J. Diff. Geom.'' , '''5''' (1971) pp. 1–38 {{MR|0298593}} {{ZBL|0219.53045}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> W.Y. Hsiang, "Minimal cones and the spherical Bernstein problem I" ''Ann. of Math.'' , '''118''' (1983) pp. 61–73 {{MR|0707161}} {{MR|0724010}} {{ZBL|0522.53051}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> W.Y. Hsiang, "Minimal cones and the spherical Bernstein problem II" ''Invent. Math.'' , '''74''' (1983) pp. 351–369 {{MR|0707161}} {{MR|0724010}} {{ZBL|0532.53045}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> W.Y. Hsiang, I. Sterling, "Minimal cones and the spherical Bernstein problem III" ''Invent. Math.'' , '''85''' (1986) pp. 223–247 {{MR|0846927}} {{ZBL|0615.53054}} </TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> W.Y. Hsiang, W.T. Hsiang, P. Tomter, "On the existence of minimal hyperspheres in compact symmetric spaces" ''Ann. Sci. Ecole Norm. Sup.'' , '''21''' (1988) pp. 287–305 {{MR|0956769}} {{ZBL|0652.53040}} </TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> P. Tomter, "The spherical Bernstein problem in even dimensions and related problems" ''Acta Math.'' , '''158''' (1987) pp. 189–212 {{MR|0892590}} {{ZBL|0631.53047}} </TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> Simons, J., "Minimal varieties in Riemannian manifolds" ''Ann. of Math.'' , '''88''' (1968) pp. 62–105 {{MR|233295}} {{ZBL|0181.49702}} </TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Al}}|| F.J. Almgren, Jr., "Some interior regularity theorems for minimal surfaces and an extension of the Bernstein's theorem" ''Ann. of Math.'' , '''85''' (1966) pp. 277–292 {{MR|200816}} {{ZBL|}}
 +
|-
 +
|valign="top"|{{Ref|Be}}||  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|}}
 +
|-
 +
|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}}
 +
|-
 +
|valign="top"|{{Ref|Ca}}|| E. Calabi, "Minimal immersions of surfaces in euclidean spaces" ''J. Diff. Geom.'' , '''1''' (1967) pp. 111–125
 +
|-
 +
|valign="top"|{{Ref|Ch}}|| S.S. Chern, "Brief survey of minimal submanifolds" ''Tagungsbericht Oberwolfach'' (1969) {{MR|0358634}} {{MR|0358635}} {{ZBL|0218.53070}}
 +
|-
 +
|valign="top"|{{Ref|Ch2}}|| S.S. Chern, "Differential geometry, its past and its future" , ''Actes Congres Intern. Mathem.'' , '''1''' (1970) pp. 41–53 {{MR|0428217}} {{ZBL|0232.53001}}
 +
|-
 +
|valign="top"|{{Ref|DG}}|| E. DeGiorgi, "Una estensione del teoreme di Bernstein" ''Ann. Sc. Norm. Sup. Pisa'' , '''19''' (1965) pp. 79–85
 +
|-
 +
|valign="top"|{{Ref|Fl}}|| W.H. Fleming, "On the oriented Plateau problem" ''Rend. Circ. Mat. Palermo'' , '''II''' (1962) pp. 1–22 {{MR|0157263}} {{ZBL|0107.31304}}
 +
|-
 +
|valign="top"|{{Ref|Fl2}}|| W.H. Fleming, "Flat chains over a finite coefficient group" ''Trans. Amer. Math. Soc.'' , '''121''' (1966) pp. 160–186 {{MR|0185084}} {{ZBL|0136.03602}}
 +
|-
 +
|valign="top"|{{Ref|H}}|| W.Y. Hsiang, "Minimal cones and the spherical Bernstein problem I" ''Ann. of Math.'' , '''118''' (1983) pp. 61–73 {{MR|0707161}} {{MR|0724010}} {{ZBL|0522.53051}}
 +
|-
 +
|valign="top"|{{Ref|H1}}||  W.Y. Hsiang, "Minimal cones and the spherical Bernstein problem II" ''Invent. Math.'' , '''74''' (1983) pp. 351–369 {{MR|0707161}} {{MR|0724010}} {{ZBL|0532.53045}}
 +
|-
 +
|valign="top"|{{Ref|HHS}}|| W.Y. Hsiang, W.T. Hsiang, P. Tomter, "On the existence of minimal  hyperspheres in compact symmetric spaces" ''Ann. Sci. Ecole Norm. Sup.'' , '''21''' (1988) pp. 287–305 {{MR|0956769}} {{ZBL|0652.53040}}
 +
|-
 +
|valign="top"|{{Ref|HL}}||  W.Y. Hsiang, H.B. Lawson, Jr., "Minimal submanifolds of low  cohomogeneity" ''J. Diff. Geom.'' , '''5''' (1971) pp. 1–38  {{MR|0298593}} {{ZBL|0219.53045}}
 +
|-
 +
|valign="top"|{{Ref|HS}}||  W.Y. Hsiang, I. Sterling, "Minimal cones and the spherical Bernstein problem III" ''Invent. Math.'' , '''85''' (1986) pp. 223–247  {{MR|0846927}} {{ZBL|0615.53054}}
 +
|-
 +
|valign="top"|{{Ref|Si}}||  J. Simons, "Minimal varieties in riemannian manifolds" ''Ann. of Math.'', '''88''' (1968) pp. 62-105 {{MR|233295}} {{ZBL|0181.49702}}
 +
|-
 +
|valign="top"|{{Ref|To}}|| P. Tomter, "The spherical Bernstein problem in even dimensions and  related problems" ''Acta Math.'' , '''158''' (1987) pp. 189–212  {{MR|0892590}} {{ZBL|0631.53047}}
 +
|-
 +
|}

Revision as of 08:47, 3 November 2013

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

It is a well-known and elementary fact in complex analysis that a bounded and holomorphic function on the whole plane (cf. also entire function) must be a constant (cf. Liouville theorems). In fact, more generally an holomorphic function with polynomial growth is necessarily a polynomial. The same property can be then inferred for (real-valued) harmonic functions.

S.N. Bernstein proved an analogous result in differential geometry (cf. [Be]), saying that, if the graph of a $C^2$ function $f: \mathbb R^2\to \mathbb R$ is a minimal surface (i.e. its mean curvature vanishes everywhere), then $f$ must be an affine function (see Bernstein theorem). The question whether the corresponding statement holds or not in higher dimensions became famous in the last century as the Bernstein problem and was completely solved at the end of the sixties. More precisely, since the mean curvature of a graph can be explicitely computed in terms of first and second derivatives of $f$, the Bernstein problem can be stated in the following terms.

Problem 1 Is it true that $C^2$ solutions $f: \mathbb R^n \to \mathbb R$ of the minimal surface equation \[ {\rm div}\, \frac{\nabla f}{\sqrt{1+|\nabla f|^2}} \] are necessarily affine functions?

The answer to the above question is affirmative in the range $n \leq 7$. The proof of this is the result of the successive efforts of W.H. Fleming [Fl], [Fl2], E. DeGiorgi [DG], F. Almgren [Al], and J. Simons [Si]. The answer is instead negative when $n\geq 8$, as was shown by E. Bombieri, De Giorgi and E. Giusti in [BDG]. In particular:

  • Fleming remarked that an affirmative answer in $\mathbb R^n$ can be concluded from the statement that area-minimizing hypercones in $\mathbb R^{n+1}$ are necessarily planar (thereby giving a new proof of Bernstein's original theorem).
  • De Giorgi relaxed Fleming's condition to the nonexistence of area-minimizing hypercones in $\mathbb R^n$, extending Bernstein's statement to $n=3$.
  • Almgren proved that all area-minimizing hypercones in $\mathbb R^4$ are planar, thereby solving Bernstein's problem for $n=4$.
  • Simons proved that area-minimizing hypercones in $\mathbb R^n$ are planar for any $n\leq 7$, hence generalizing Bernstein's statement to $n\leq 7$. Indeed Simons proved the stronger statement that any stable minimal hypercone in $\mathbb R^n$ must be planar for $n\leq 7$. His proof is based on an inequality for the norm of the second fundamental form of minimal surfaces, see Simons inequality, which has several deep consequences in the theory of minimal hypersurfaces. Simons also pointed out the existence of a minimal nonplanar $7$-dimensional cone in $\mathbb R^8$ which is stable. Such a cone is nowadays known as the Simons cone.
  • Finally, Bombieri, De Giorgi and Giusti showed that Simons' cone is indeed area minimizing and obtained as a consequence a non-affine entire solution $f: \mathbb R^8 \to \mathbb R$ of the minimal surface equation.

The complete solution of the classical Bernstein problem constitutes, indeed, an exciting chapter of global differential geometry, involving geometric measure theory and nonlinear analysis (cf. also Plateau problem).

Spehrical Bernstein problem

Among the various generalizations of Problem 1, the so-called spherical Bernstein problem is a natural and challenging one in the realm of global differential geometry. It is due to S.S. Chern (cf. [Ch]; he also proposed it at the International Congress of Mathematicians at Nice (1970), cf. [Ch2]).

Problem 2 Are the equators in $\mathbb S^{n+1}$ the only smooth embedded minimal hypersurfaces which are topological $n$-dimensional spheres?

As it happens for the classical Bernstein problem, the question has a direct bearing on the possible local structures of isolated singularities of minimal hypersurfaces $\Sigma$ in a general Riemannian manifold $N$ of dimension $n+1$. Indeed, the tangent cone of $\Sigma$ at an isolated singularity is a minimal cone in the Euclidean space $\mathbb R^{n+1}$, whose intersection with $\mathbb S^n$ is a minimal hypersurface.

By the Almgren–Calabi theorem ([Al] and [Ca]), any immersion (cf. Immersion of a manifold) of $\mathbb S^2$ into $\mathbb S^3$ must, in fact, be an equator. Thus, at least the beginning case $n=2$ of the spherical Bernstein problem was known to have a positive answer. However, no further progress was made until 1983, when Wu-yi Hsiang, in the framework of equivariant differential geometry, constructed infinitely many mutually non-congruent minimal embeddings of $\mathbb S^n$ into $\mathbb S^{n+1}$, for $n\in \{3,4,5,6,7,9,11,13\}$ (cf. [H], [H2]). The basic idea of this approach, which was initiated by Hsiang and H.B. Lawson in [HL], is to choose an orthogonal representation $(G, \Phi, \mathbb R^{n+1})$ with codimension-two principal orbits, and for minimal hypersurfaces which are invariant under the induced orthogonal transformation group $(G, \Phi+ 1, \mathbb S^{n+1})$. Thus, the original partial differential equation associated with the minimal hypersurface condition is reduced to an ordinary second-order differential equation on the $2$-dimensional orbit space $\mathbb S^{n+1}/G$, which is geometrically a spherical lune. Here, the "closed" (in an appropriate sense) solution curves represent minimal hypersurfaces, and they will have the topology of $\mathbb S^n$ thanks to the trivial summand added to $\Phi$.

During the 1980s, Hsiang and his collaborators obtained further results related to the spherical Bernstein problem, by investigating equivariant systems of the above type, as well as additional isoparametric foliations on $\mathbb S^{n+1}$. For instance, many new examples of embedded as well as immersed minimal hyperspheres $\Sigma$ in $\mathbb S^{n+1}$ have been constructed, and moreover, the stability of the singularity at the origin of the corresponding minimal cone $\Gamma$ in $\mathbb R^{n+1}$ has been investigated, see, e.g., [HS]. P. Tomter in [To] has shown the existence of a minimal and embedded hypersphere $\Sigma$ in $\mathbb S^{n+1}$ which is not an equator, for each odd $n$. On the other hand, although the methods give infinitely many non-congruent minimal immersions of $\mathbb S^n$ into $\mathbb S^{n+1}$ for each $n$, the problem of finding a non-equatorial embedded minimal hypersphere in $\mathbb S^{n+1}$ for $n\geq 8$ even remains open.

Finally, as a generalization of the spherical Bernstein problem, it is also natural to replace the ambient space $\mathbb S^{n+1}$ by a simply-connected, compact symmetric space; see [HHS] for some results in this direction.

References

[Al] F.J. Almgren, Jr., "Some interior regularity theorems for minimal surfaces and an extension of the Bernstein's theorem" Ann. of Math. , 85 (1966) pp. 277–292 MR200816
[Be] 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
[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
[Ca] E. Calabi, "Minimal immersions of surfaces in euclidean spaces" J. Diff. Geom. , 1 (1967) pp. 111–125
[Ch] S.S. Chern, "Brief survey of minimal submanifolds" Tagungsbericht Oberwolfach (1969) MR0358634 MR0358635 Zbl 0218.53070
[Ch2] S.S. Chern, "Differential geometry, its past and its future" , Actes Congres Intern. Mathem. , 1 (1970) pp. 41–53 MR0428217 Zbl 0232.53001
[DG] E. DeGiorgi, "Una estensione del teoreme di Bernstein" Ann. Sc. Norm. Sup. Pisa , 19 (1965) pp. 79–85
[Fl] W.H. Fleming, "On the oriented Plateau problem" Rend. Circ. Mat. Palermo , II (1962) pp. 1–22 MR0157263 Zbl 0107.31304
[Fl2] W.H. Fleming, "Flat chains over a finite coefficient group" Trans. Amer. Math. Soc. , 121 (1966) pp. 160–186 MR0185084 Zbl 0136.03602
[H] W.Y. Hsiang, "Minimal cones and the spherical Bernstein problem I" Ann. of Math. , 118 (1983) pp. 61–73 MR0707161 MR0724010 Zbl 0522.53051
[H1] W.Y. Hsiang, "Minimal cones and the spherical Bernstein problem II" Invent. Math. , 74 (1983) pp. 351–369 MR0707161 MR0724010 Zbl 0532.53045
[HHS] W.Y. Hsiang, W.T. Hsiang, P. Tomter, "On the existence of minimal hyperspheres in compact symmetric spaces" Ann. Sci. Ecole Norm. Sup. , 21 (1988) pp. 287–305 MR0956769 Zbl 0652.53040
[HL] W.Y. Hsiang, H.B. Lawson, Jr., "Minimal submanifolds of low cohomogeneity" J. Diff. Geom. , 5 (1971) pp. 1–38 MR0298593 Zbl 0219.53045
[HS] W.Y. Hsiang, I. Sterling, "Minimal cones and the spherical Bernstein problem III" Invent. Math. , 85 (1986) pp. 223–247 MR0846927 Zbl 0615.53054
[Si] J. Simons, "Minimal varieties in riemannian manifolds" Ann. of Math., 88 (1968) pp. 62-105 MR233295 Zbl 0181.49702
[To] P. Tomter, "The spherical Bernstein problem in even dimensions and related problems" Acta Math. , 158 (1987) pp. 189–212 MR0892590 Zbl 0631.53047
How to Cite This Entry:
Bernstein problem in differential geometry. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bernstein_problem_in_differential_geometry&oldid=30652
This article was adapted from an original article by E. Straume (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article