Existence of a nonaffine polynomial minimal graph
The graph $\{(x, P(x)) : x \in \mathbb{R}^n\}$ of a polynomial $P$ has zero mean curvature exactly when $P$ satisfies the minimal surface equation, which for a polynomial is a polynomial identity. For $n \le 7$ every entire solution of the minimal surface equation is affine, by the work of Bernstein, Fleming, De Giorgi, Almgren and Simons; for $n \ge 8$ non-affine entire solutions were constructed by Bombieri, De Giorgi and Giusti and by Simon, none of them explicit. Guo proved that a polynomial solution of degree at least two has degree at least four and is not homogeneous, and that the tangent cone at infinity of its graph is not of the form $(C \times \mathbb{R}^{l-1}) \times \mathbb{R}$ with $C$ an isoparametric minimal cone and $l \ge 1$. Mooney constructed a quadratic polynomial on $\mathbb{R}^6$ that solves an anisotropic equation of minimal surface type.
Problem. Does there exist an integer $n \ge 8$ and a real polynomial $P : \mathbb{R}^n \to \mathbb{R}$ of degree at least two satisfying the polynomial identity $$\bigl(1 + |\nabla P|^2\bigr)\Delta P - \sum_{i,j=1}^n P_{x_i} P_{x_j} P_{x_i x_j} = 0?$$
Source: Simon (1997), as cited by Guo (2026), §1.1; also Mooney (2022), §1, and Mooney (2024), §7.1. Stated for n ≥ 8 by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DG
- Keywords
- AIM Open Applied Problems, minimal graphs, polynomial solutions
References
@incollection{simon1997,
author = {Simon, Leon},
title = {The minimal surface equation},
booktitle = {Geometry {V}},
editor = {Osserman, R.},
series = {Encyclopaedia Math. Sci.},
volume = {90},
publisher = {Springer},
address = {Berlin},
year = {1997},
pages = {239--266},
doi = {10.1007/978-3-662-03484-2_5},
}
@article{guo2026,
author = {Guo, Yifan},
title = {On polynomial solutions to the minimal surface equation},
journal = {Calc. Var. Partial Differential Equations},
volume = {65},
number = {5},
year = {2026},
pages = {141},
doi = {10.1007/s00526-026-03316-2},
eprint = {2404.00115},
archivePrefix = {arXiv},
}
@article{mooney2022,
author = {Mooney, Connor},
title = {Entire solutions to equations of minimal surface type in six dimensions},
journal = {J. Eur. Math. Soc. (JEMS)},
volume = {24},
number = {12},
year = {2022},
pages = {4353--4361},
doi = {10.4171/JEMS/1202},
eprint = {1910.11297},
archivePrefix = {arXiv},
}
@article{mooney2024,
author = {Mooney, Connor},
title = {Bernstein theorems for nonlinear geometric {PDE}s},
journal = {Commun. Pure Appl. Anal.},
volume = {23},
number = {12},
year = {2024},
pages = {1958--1989},
doi = {10.3934/cpaa.2024072},
eprint = {2407.11903},
archivePrefix = {arXiv},
}
@article{simons1968,
author = {Simons, James},
title = {Minimal varieties in {R}iemannian manifolds},
journal = {Ann. of Math. (2)},
volume = {88},
number = {1},
year = {1968},
pages = {62--105},
doi = {10.2307/1970556},
}
@article{bombieri1969,
author = {Bombieri, E. and De Giorgi, E. and Giusti, E.},
title = {Minimal cones and the {B}ernstein problem},
journal = {Invent. Math.},
volume = {7},
number = {3},
year = {1969},
pages = {243--268},
doi = {10.1007/BF01404309},
}
@article{simon1989,
author = {Simon, Leon},
title = {Entire solutions of the minimal surface equation},
journal = {J. Differential Geom.},
volume = {30},
number = {3},
year = {1989},
pages = {643--688},
doi = {10.4310/jdg/1214443827},
}
@misc{aim-isotropic-polynomial-minimal-graph,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Existence of a nonaffine polynomial minimal graph}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/414-isotropic-polynomial-minimal-graph.md},
note = {Entry 414 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00871, "Existence of a nonaffine polynomial minimal graph", clopen.solutions.