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arXiv · 2608.17405

A concavity inequality and interior $C^2$ estimate for Hessian quotient equations

Zhisu Li·Ke Wu
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Abstract

We establish a concavity inequality for the Hessian quotient operators $\frac{\sigma_k}{\sigma_l}$ in the cases $k-l\in\{1,2\}$, and then derive the corresponding Jacobi inequality. Combining this with the framework developed by Lu and Tsai, we obtain an interior Hessian estimate for convex solutions of $\frac{\sigma_k(D^2u)}{\sigma_l(D^2u)}=f$. As an application, we prove that any entire convex solution in $\mathbb R^n$ with quadratic growth must be a quadratic polynomial.

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Zhisu Li, Ke Wu. 2026-08-18. A concavity inequality and interior $C^2$ estimate for Hessian quotient equations. https://arxiv.org/abs/2608.17405

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