PreprintCC BY 4.0arXiv
The interpolation method for the complex Hessian equation on compact Kähler manifolds
Abstract
In current work, we delve into the study of complex Hessian equations on compact Kähler manifolds. By imposing a constraint on both the bisectional curvature and the oscillation of the solution, we demonstrate that the Laplacian estimate can be linearly controlled by its gradient estimate. Consequently, we are able to derive gradient estimate for the solution using a standard interpolation argument, without the need to employ the blow-up technique. As an application, we demonstrate that when the right-hand side of the complex Hessian equations approximate 1 in the L1 norm, the desired gradient estimate is obtained.
Related
- Smoothness of bounded solutions to complex Hessian equations on compact Hermitian manifolds
- Spline interpolation on compact riemannian manifolds
- Nakai-Moishezon type criteria for Hessian type equations on compact Kähler manifolds
- On Polynomial Complex Hessian Equations I: Non-Linear Non-Spectral Elliptic Hessian Equations on Riemannian Manifolds
- Regularization and Hölder continuity for complex Hessian equations on Hermitian manifolds
BibTeX
@misc{zhang2026interpolation,
title = {{The interpolation method for the complex Hessian equation on compact Kähler manifolds}},
author = {Jiaogen Zhang and Xi Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07852},
eprint = {2610.07852},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07852},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The interpolation method for the complex Hessian equation on compact Kähler manifolds
AU - Jiaogen Zhang
AU - Xi Zhang
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.07852
UR - https://arxiv.org/abs/2610.07852
AB - In current work, we delve into the study of complex Hessian equations on compact Kähler manifolds. By imposing a constraint on both the bisectional curvature and the oscillation of the solution, we demonstrate that the Laplacian estimate can be linearly controlled by its gradient estimate. Consequently, we are able to derive gradient estimate for the solution using a standard interpolation argument, without the need to employ the blow-up technique. As an application, we demonstrate that when the right-hand side of the complex Hessian equations approximate 1 in the L^1 norm, the desired gradient estimate is obtained.
ER -