The log-concavity of eigenfunction to complex Monge-Amp\`ere operator in $\mathbb{C}^2$
Analysis of PDEs
2025-08-01 v2
Abstract
Following the authors' recent work \cite{Zhang-Zhou2025}, we further explore the convexity properties of solutions to the Dirichlet problem for the complex Monge-Amp\`ere operator. In this paper, we establish the -concavity of solutions to the Dirichlet eigenvalue problem for the complex Monge-Amp\`ere operator on bounded, smooth, strictly convex domain in . The key ingredients consist of the constant rank theorem and the deformation method.
Cite
@article{arxiv.2505.12817,
title = {The log-concavity of eigenfunction to complex Monge-Amp\`ere operator in $\mathbb{C}^2$},
author = {Wei Zhang and Qi Zhou},
journal= {arXiv preprint arXiv:2505.12817},
year = {2025}
}
Comments
The proof of the constant rank theorem in this paper (Section 3) is incomplete. Therefore, we are withdrawing the submission. A revised version will be uploaded once the issue has been corrected