# Bernstein theorem

2010 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.