Second-order estimates for $\chi$-semiconvex solutions of Hessian equations on Hermitian manifolds

Xiaojuan Chen, Qiang Tu and Ni Xiang

Vol. 344 (2026), No. 2, 273–291

DOI: 10.2140/pjm.2026.344.273

Abstract

We establish the modified concavity inequality for complex Hessian equations under the semiconvexity assumption inspired by Lu (2023) and Zhang (2025) for the real case. Then second-order estimates for admissible solutions of complex Hessian equations on compact Hermitian manifolds with both sides of equations depending on gradient terms are obtained by taking advantage of this crucial inequality.

Keywords

complex Hessian equations, second-order estimates, semiconvex, Hermitian manifolds

Mathematical Subject Classification

Primary: 35B45, 35J15, 53C55, 58J05

Milestones

Received: 13 September 2025

Revised: 27 February 2026

Accepted: 29 May 2026

Published: 8 September 2026