Quadratic-growth rigidity for the complex Monge–Ampère equation
Let $n \ge 2$, write $\partial_{z_j} = \tfrac12(\partial_{x_j} - i\partial_{y_j})$ and $u_{i\bar j} = \partial_{z_i}\partial_{\bar z_j} u$, and call $u \in C^\infty(\mathbb{C}^n; \mathbb{R})$ strictly plurisubharmonic if the Hermitian matrix $(u_{i\bar j})$ is positive definite. If moreover $\det(u_{i\bar j}) = 1$ on $\mathbb{C}^n$, the Kähler metric $(u_{i\bar j})$ is Ricci-flat. Wang proved that a plurisubharmonic solution that differs from a quadratic polynomial solution by $o(|z|^2)$ at infinity is itself a quadratic polynomial. Li and Sheng proved that a solution satisfying the two-sided bound below is quadratic when the metric $(u_{i\bar j})$ is complete, and Chen, Hu and Sheng proved that every smooth entire solution that is convex in the real coordinates is a real quadratic polynomial.
Problem. Let $u \in C^\infty(\mathbb{C}^n; \mathbb{R})$ be strictly plurisubharmonic with $\det(u_{i\bar j}) = 1$ on $\mathbb{C}^n$, and suppose that for some $C \ge 1$ $$C^{-1}(1 + |z|^2) \le u(z) \le C(1 + |z|^2) \qquad (z \in \mathbb{C}^n).$$ Must $u$ be a polynomial of degree two in the $2n$ real coordinates? Neither real convexity nor completeness of the metric $(u_{i\bar j})$ is assumed.
Source: Székelyhidi, AimPL: Nonlinear PDEs in real and complex geometry (2018), Monge–Ampère equations, Problem 1.4; see Mooney (2024), §7.4, and Chen, Hu and Sheng (2026).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- CV, DG
- Keywords
- AIM Open Applied Problems, complex Monge–Ampère equations, Liouville theorems
References
@misc{aimpl2018nonlinpdegeom,
editor = {Edwards, Gregory and Sz{\'e}kelyhidi, G{\'a}bor and Tosatti, Valentino and Weinkove, Ben},
title = {{AimPL}: Nonlinear {PDE}s in real and complex geometry},
howpublished = {American Institute of Mathematics problem list, from the workshop of 13 to 17 August 2018},
year = {2018},
url = {http://aimpl.org/nonlinpdegeom},
note = {Monge--Amp{\`e}re equations, Problem 1.4, posed by G{\'a}bor Sz{\'e}kelyhidi},
}
@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},
}
@misc{chen2026,
author = {Chen, Hongyu and Hu, Jingchen and Sheng, Li},
title = {Pogorelov interior estimates and a {L}iouville theorem for the complex {M}onge--{A}mp{\`e}re equation},
year = {2026},
eprint = {2609.18265},
archivePrefix = {arXiv},
}
@misc{wang2013,
author = {Wang, Yu},
title = {A {L}iouville theorem for the complex {M}onge--{A}mp{\`e}re equation},
year = {2013},
eprint = {1303.2403},
archivePrefix = {arXiv},
}
@article{li2021,
author = {Li, An-Min and Sheng, Li},
title = {A {L}iouville theorem on the {PDE} $\det(f_{i\bar j}) = 1$},
journal = {Math. Z.},
volume = {297},
number = {3-4},
year = {2021},
pages = {1623--1632},
doi = {10.1007/s00209-020-02571-z},
eprint = {1809.00824},
archivePrefix = {arXiv},
}
@misc{aim-complex-monge-ampere-quadratic-growth,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Quadratic-growth rigidity for the complex Monge–Ampère equation}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/410-complex-monge-ampere-quadratic-growth.md},
note = {Entry 410 at commit 24edaad, 2026-10-15},
}
Resolution claims
For the complex Monge--Ampère equation on $\mathbb{C}^{n}$, we prove the Liouville theorem for solutions with quadratic growth.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-20 | Needs verification → Solution claimed | — | Recorded here: arXiv:2610.26016v1. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00845, "Quadratic-growth rigidity for the complex Monge–Ampère equation", clopen.solutions.