PreprintCC BY 4.0arXiv
Stability and gradient estimates for fully nonlinear elliptic equations on hermitian manifolds
Abstract
The known proofs of the gradient estimate for complex Hessian equations rely on the second order estimate of Hou-Ma-Wu and on a Liouville theorem of Dinew-Kolodziej. In this paper, we give a direct proof of the gradient estimate for concave fully nonlinear elliptic equations on compact Hermitian manifolds, which uses neither the Hou-Ma-Wu estimate nor the Liouville theorem. Instead, the proof combines a stability estimate with a comparison argument against nearby smooth admissible functions. We also give an independent proof of the complex Hessian estimate on compact Kähler manifolds, based on Dirichlet Green's functions, under additional assumptions on the operator.
Related
- Uniform estimates for complex Monge-Ampère equations II: Hermitian manifolds
- Concavity-Free L∞ Estimates and Stability for Fully Nonlinear Complex Equations
- Smoothness of bounded solutions to complex Hessian equations on compact Hermitian manifolds
- Regularization and Hölder continuity for complex Hessian equations on Hermitian manifolds
- Complex Monge-Ampère equations on Hermitian Manifolds: From bounded to smooth solutions
BibTeX
@misc{guo2026stability,
title = {{Stability and gradient estimates for fully nonlinear elliptic equations on hermitian manifolds}},
author = {Bin Guo and Jian Song},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39726},
eprint = {2609.39726},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39726},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Stability and gradient estimates for fully nonlinear elliptic equations on hermitian manifolds
AU - Bin Guo
AU - Jian Song
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39726
UR - https://arxiv.org/abs/2609.39726
AB - The known proofs of the gradient estimate for complex Hessian equations rely on the second order estimate of Hou-Ma-Wu and on a Liouville theorem of Dinew-Kolodziej. In this paper, we give a direct proof of the gradient estimate for concave fully nonlinear elliptic equations on compact Hermitian manifolds, which uses neither the Hou-Ma-Wu estimate nor the Liouville theorem. Instead, the proof combines a stability estimate with a comparison argument against nearby smooth admissible functions. We also give an independent proof of the complex Hessian estimate on compact Kähler manifolds, based on Dirichlet Green's functions, under additional assumptions on the operator.
ER -