Affine Bernstein rigidity in dimensions three through nine
For an integer $n \ge 2$ and $u \in C^4(\mathbb{R}^n)$ with positive-definite Hessian everywhere, let $U^{ij}$ be the cofactor matrix of $D^2 u$ and $$w = (\det D^2 u)^{-(n+1)/(n+2)}.$$ The affine maximal surface equation $$\sum_{i,j=1}^n U^{ij} \partial_{ij} w = 0$$ is the Euler–Lagrange equation of the affine area $\int (\det D^2 u)^{1/(n+2)} \, dx$. Chern conjectured that an entire convex solution in $\mathbb{R}^2$ is a quadratic polynomial. Trudinger and Wang proved that every locally uniformly convex affine maximal surface in $\mathbb{R}^3$ that is complete in the induced Euclidean metric is an elliptic paraboloid, and conjectured the same for Euclidean complete hypersurfaces in $\mathbb{R}^{n+1}$ when $n \le 9$. They also showed that $\sqrt{|x'|^9 + x_{10}^2}$, with $x' = (x_1, \dots, x_9)$, is affine maximal on $\mathbb{R}^{10}$; it has a singular point.
Problem. Let $3 \le n \le 9$. If $u \in C^4(\mathbb{R}^n)$ has positive-definite Hessian everywhere and satisfies the affine maximal surface equation on $\mathbb{R}^n$, must $u$ be a quadratic polynomial with positive-definite quadratic part? No growth, bounded-Hessian or affine completeness assumption is made.
Source: Chern (1979), for entire surfaces; Trudinger and Wang (2008), the conjecture in §6.3, for Euclidean complete hypersurfaces in dimensions n ≤ 9. Stated by AIM — Open Applied Problems for entire graphs of C^4 functions, 3 ≤ n ≤ 9.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DG
- Keywords
- affine maximal hypersurfaces, AIM Open Applied Problems, Bernstein problem, fourth-order elliptic equations
References
@incollection{chern1979,
author = {Chern, Shiing-Shen},
title = {Affine minimal hypersurfaces},
booktitle = {Minimal Submanifolds and Geodesics ({P}roc. {J}apan--{U}nited {S}tates {S}em., {T}okyo, 1977)},
editor = {Obata, Morio},
publisher = {North-Holland},
address = {Amsterdam},
year = {1979},
pages = {17--30},
}
@article{trudinger2000,
author = {Trudinger, Neil S. and Wang, Xu-Jia},
title = {The {B}ernstein problem for affine maximal hypersurfaces},
journal = {Invent. Math.},
volume = {140},
number = {2},
year = {2000},
pages = {399--422},
doi = {10.1007/s002220000059},
}
@incollection{trudinger2008,
author = {Trudinger, Neil S. and Wang, Xu-Jia},
title = {The {M}onge--{A}mp{\`e}re equation and its geometric applications},
booktitle = {Handbook of Geometric Analysis, {N}o. 1},
series = {Adv. Lect. Math.},
volume = {7},
publisher = {International Press},
address = {Somerville, MA},
year = {2008},
pages = {467--524},
url = {https://maths-people.anu.edu.au/~wang/publications/MA.pdf},
note = {Section 6.3, the conjecture following Theorem 6.4},
}
@misc{aim-affine-bernstein-entire-low-dimensions,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Affine Bernstein rigidity in dimensions three through nine}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/391-affine-bernstein-entire-low-dimensions.md},
note = {Entry 391 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We prove the Euclidean-complete affine Bernstein conjecture in dimensions three through nine: a smooth locally uniformly convex affine maximal graph is an elliptic paraboloid whenever its induced Euclidean metric is complete. The same conclusion holds, without an initial graph assumption, for connected smooth open (noncompact and without boundary) affine-complete locally uniformly convex immersed hypersurfaces that are classically affine maximal in …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00660, "Affine Bernstein rigidity in dimensions three through nine", clopen.solutions.