Polynomial growth of every entire minimal graph
An entire minimal graph is the graph of a solution $u : \mathbb{R}^n \to \mathbb{R}$ of the minimal surface equation $$\operatorname{div}\left(\frac{\nabla u}{\sqrt{1 + |\nabla u|^2}}\right) = 0.$$ For $n \le 7$ every such solution is affine, by the work of Bernstein, Fleming, De Giorgi, Almgren and Simons. For every $n \ge 8$ Bombieri, De Giorgi and Giusti constructed one that is not, and Simon constructed many more, whose graphs are asymptotic at infinity to cylinders over a variety of area-minimizing cones, including the area-minimizing Lawson cones. Mooney notes that these examples have polynomial growth.
Problem. For every integer $n \ge 8$ and every smooth solution $u : \mathbb{R}^n \to \mathbb{R}$ of the minimal surface equation, must there exist finite constants $C > 0$ and $p > 0$, allowed to depend on $u$, such that $$|u(x)| \le C(1 + |x|)^p \qquad \text{for all } x \in \mathbb{R}^n?$$
Source: A long-standing conjecture (Guo (2026), §1.1, citing Bombieri and Giusti (1972), Yau (1982) and Simon (1997)); see 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, Bernstein theorems, minimal graphs
References
@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{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{bombieri1972,
author = {Bombieri, E. and Giusti, E.},
title = {Harnack's inequality for elliptic differential equations on minimal surfaces},
journal = {Invent. Math.},
volume = {15},
number = {1},
year = {1972},
pages = {24--46},
doi = {10.1007/BF01418640},
}
@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{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-entire-minimal-graphs-polynomial-growth,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Polynomial growth of every entire minimal graph}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/407-entire-minimal-graphs-polynomial-growth.md},
note = {Entry 407 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-00859, "Polynomial growth of every entire minimal graph", clopen.solutions.