The log-concavity of eigenfunction to complex Monge-Amp\`ere operator in $\mathbb{C}^2$

Wei Zhang; Qi Zhou

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arXiv · 2505.12817

The log-concavity of eigenfunction to complex Monge-Amp\`ere operator in $\mathbb{C}^2$

Wei Zhang·Qi Zhou

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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\`{e}re operator. In this paper, we establish the $\log$-concavity of solutions to the Dirichlet eigenvalue problem for the complex Monge-Amp\`{e}re operator on bounded, smooth, strictly convex domain in $\mathbb{C}^2$. Our approach combines a constant rank theorem adapted to the study of real convexity in the complex setting with a deformation method.

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BibTeX RIS

Wei Zhang, Qi Zhou. 2025-05-19. The log-concavity of eigenfunction to complex Monge-Amp\`ere operator in $\mathbb{C}^2$. https://arxiv.org/abs/2505.12817

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