Bernstein theorem

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