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$C^2$ estimates for $k$-Hessian equations and a rigidity theorem (2408.10781v2)
Published 20 Aug 2024 in math.AP and math.DG
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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