Curvature estimates for semi-convex solutions of asymptotic Plateau problem in $\mathbb{H}^{n+1}$

Han Hong; Ruijia Zhang

Curvature estimates for semi-convex solutions of asymptotic Plateau problem in $\mathbb{H}^{n+1}$

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arxiv 2408.09428 v1 pith:GLOPGAAO submitted 2024-08-18 math.DG

Han Hong , Ruijia Zhang This is my paper

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keywords asymptoticcurvatureestimatesplateauproblemsemi-convexsigmaboundary

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abstract

In this paper, we consider the asymptotic $\sigma_k$ Plateau problem in hyperbolic space. We establish $C^2$ estimates for semi-convex complete hypersurfaces satisfying constant $\sigma_k$ curvature with a prescribed asymptotic boundary at the infinity for $2\leq k\leq n-2$ . The result is based on a new crucial concavity inequality derived for hessian equations.

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Hypersurfaces of constant sum Hessian curvature in Hyperbolic space
math.DG 2025-07 conditional novelty 5.0 of 10

For the constant sum Hessian curvature equation in hyperbolic space, a curvature estimate and conditional existence are proved under an extra lower-bound assumption on sigma_n; the unconditional problem remains open.

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