Pogorelov interior estimates and a Liouville theorem for the complex Monge--Ampère equation

@misc{chen2026pogorelov,
  title = {{Pogorelov interior estimates and a Liouville theorem for the complex Monge--Ampère equation}},
  author = {Hongyu Chen and Jingchen Hu and Li Sheng},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18265},
  eprint = {2609.18265},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18265},
}

TY  - UNPB
TI  - Pogorelov interior estimates and a Liouville theorem for the complex Monge--Ampère equation
AU  - Hongyu Chen
AU  - Jingchen Hu
AU  - Li Sheng
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.18265
UR  - https://arxiv.org/abs/2609.18265
AB  - We prove a Pogorelov interior estimate for strictly plurisubharmonic solutions of the complex Monge-Ampère equation det(u_ij̅)=1 with homogeneous Dirichlet data under the condition that, for some constant κ≥ 1, (κu_ij̅−u_isu^st̅u_tj̅) is non-negative definite. For κ=1, this gives the estimate for real convex solutions. The estimate depends only on the dimension, κ, and the C^0 and C^1 norms of the solution. As an application, every smooth entire convex solution of det(u_ij̅)=1 on ℂ^n is a real quadratic polynomial. The same conclusion holds if real convexity is assumed only outside a compact subset of ℂ^n.
ER  -