View source for Bernstein problem in differential geometry

{{MSC|53A10}}
[[Category:Differential geometry]]
[[Category:Partial differential equations]]
{{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.

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 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'''
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
\begin{equation}\label{e:min_surf}
{\rm div}\, \frac{\nabla f}{\sqrt{1+|\nabla f|^2}
\end{equation}
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]]).

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?

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.

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" />.

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.

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.

====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>