Polynomial solutions of the minimal surface equation

Bernstein's theorem says that a minimal surface which is the graph of a function over the whole plane must itself be a plane. The same holds for graphs over $\mathbb{R}^n$ when $n \le 7$ and fails when $n \ge 8$, but all known counterexamples are transcendental. Leon Simon asked whether a polynomial can do it: the task is to find one, or to show there is none.

Problem

A function $u\colon \mathbb{R}^n \to \mathbb{R}$ has a minimal graph in $\mathbb{R}^{n+1}$ exactly when it solves the minimal surface equation

$$\operatorname{div}\!\left(\frac{\nabla u}{\sqrt{1 + |\nabla u|^2}}\right) = 0.$$

Clearing the positive denominator, for a polynomial $u$ this is the polynomial identity

$$\bigl(1 + |\nabla u|^2\bigr)\,\Delta u \;-\; \sum_{i,j=1}^{n} u_i\, u_j\, u_{ij} \;=\; 0,$$

where $u_i = \partial u/\partial x_i$ and $u_{ij} = \partial^2 u/\partial x_i \partial x_j$.

Task. Find a non-affine polynomial $u$ (equivalently, one of degree at least 3) in $n \ge 8$ variables, with real algebraic coefficients, satisfying this identity; or prove that every polynomial solution is affine.

Background

Bernstein proved that entire minimal graphs over $\mathbb{R}^2$ are planes, and through the work of De Giorgi, Almgren and Simons the same was established for $n \le 7$ [2]. Bombieri, De Giorgi and Giusti then constructed non-affine entire minimal graphs over $\mathbb{R}^n$ for every $n \ge 8$ [1]. Their solutions are not polynomials. Simon raised the question of polynomial solutions in his survey of the minimal surface equation [3], which makes it an algebraic version of the Bernstein problem. A positive answer would give the first minimal graph of a non-affine polynomial and might help to build minimal hypersurfaces in high dimensions.

What is known

• For $n \le 7$ every entire solution is affine, so the question is open only for $n \ge 8$ [2].
• Non-affine entire solutions exist for $n \ge 8$, but none known is a polynomial [1].
• No non-affine polynomial solution is known in any dimension (Epoch).

What would count as progress

• An explicit polynomial: its terms with exponent vectors and coefficients in a number field, specified by an irreducible minimal polynomial and an isolating interval for the chosen real root. Substituting into the identity and expanding is exact symbolic computation.
• Epoch scores submissions with its own verifier, which receives $n \ge 8$, the coefficient field and the list of terms, and checks that the expression above is identically zero.
• A proof that polynomial solutions must be affine would answer Simon's question in the negative.

References

1. E. Bombieri, E. De Giorgi and E. Giusti, Minimal cones and the Bernstein problem, Invent. Math. 7 (1969), 243–268. doi:10.1007/BF01404309
2. J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. 88 (1968), 62–105. doi:10.2307/1970556
3. L. Simon, The minimal surface equation, in Encyclopaedia of Mathematical Sciences, Springer (1997), 239–266. doi:10.1007/978-3-662-03484-2_5

Source

Imported from FrontierMath: Open Problems, polynomial-minimal-surface, curated by Epoch AI. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.