PreprintCC BY 4.0arXiv
Interior C2 Estimates and a Liouville Theorem for the Complex Monge--Ampère Equation
Abstract
We establish an interior C2 estimate for locally bounded plurisubharmonic Bedford-Taylor solutions of the complex Monge--Ampère equation det(uij̅)=f under weak quadratic growth bounds on sufficiently large balls, where f is positive and Dini-continuous. We assume a distributional lower bound for the Laplacian of f1/n. The main ingredient is a pseudo-C2 estimate using the Green-function averaging error, valid also when f is nonnegative. As an application, we prove that every entire Bedford-Taylor solution of det(uij̅)=1 with weak quadratic growth is a real quadratic polynomial, resolving a conjecture of Gábor Székelyhidi.
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BibTeX
@misc{hu2026interior,
title = {{Interior $C^2$ Estimates and a Liouville Theorem for the Complex Monge--Ampère Equation}},
author = {Jingchen Hu and Li Sheng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28375},
eprint = {2610.28375},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28375},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Interior C^2 Estimates and a Liouville Theorem for the Complex Monge--Ampère Equation
AU - Jingchen Hu
AU - Li Sheng
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28375
UR - https://arxiv.org/abs/2610.28375
AB - We establish an interior C^2 estimate for locally bounded plurisubharmonic Bedford-Taylor solutions of the complex Monge--Ampère equation det(u_ij̅)=f under weak quadratic growth bounds on sufficiently large balls, where f is positive and Dini-continuous. We assume a distributional lower bound for the Laplacian of f^1/n. The main ingredient is a pseudo-C^2 estimate using the Green-function averaging error, valid also when f is nonnegative. As an application, we prove that every entire Bedford-Taylor solution of det(u_ij̅)=1 with weak quadratic growth is a real quadratic polynomial, resolving a conjecture of Gábor Székelyhidi.
ER -