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arXiv · 2606.25678

Brunn--Minkowski Inequality for the First Complex $\sigma_{2}$-Hessian Eigenvalue

Chuanqiang Chen·Jiahuan Li·Xi-Nan Ma
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Abstract

There are relatively few results on the convexity of solutions to complex equations. In this paper, We prove a strict real log-concavity theorem for the first eigenfunction of the complex $\sigma_{2}$-Hessian operator on smooth, bounded, real uniformly strictly convex domains in $\mathbb{C}^{n}$. As an application, we obtain a Brunn--Minkowski inequality for the first complex $\sigma_{2}$-Hessian eigenvalue. The proof combines a Bian--Guan constant-rank argument, a new inverse-convexity lemma for the compressed real Hessian, and Salani's viscosity admissible-test-function method.

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Chuanqiang Chen, Jiahuan Li, Xi-Nan Ma. 2026-06-24. Brunn--Minkowski Inequality for the First Complex $\sigma_{2}$-Hessian Eigenvalue. https://arxiv.org/abs/2606.25678

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