$C^2$ estimates for $k$-Hessian equations and a rigidity theorem

Ruijia Zhang

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arXiv · 2408.10781

$C^2$ estimates for $k$-Hessian equations and a rigidity theorem

Ruijia Zhang

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Abstract

We derive a concavity inequality for $k$-Hessian operators under the semi-convexity condition. As an application, we establish interior estimates for semi-convex solutions of the $k$-Hessian equations with vanishing Dirichlet boundary and obtain a Liouville-type result. Additionally, we provide new and simple proofs of Guan-Ren-Wang's results on global curvature estimates for $k$-curvature equations.

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Ruijia Zhang. 2024-08-20. $C^2$ estimates for $k$-Hessian equations and a rigidity theorem. https://arxiv.org/abs/2408.10781

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