Interior curvature bounds for a class of curvature equations

2004/06/01 by Weimin Sheng, John Urbas, Xu-Jia Wang +1 · 75 citations
Mathematics · #Affine transformation #Boundary value problem #Class (philosophy) #Convexity #Curvature #Dirichlet problem #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Mathematical analysis #Mathematics #Mean curvature #Nonlinear Partial Differential Equations #Pure mathematics #Scalar curvature

paper · doi:10.1215/s0012-7094-04-12321-8

published in Duke Mathematical Journal 123(2) (Duke University Press)

openalex publication_date 2004/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/02

Abstract

We derive interior curvature bounds for admissible solutions of a class of curvature equations subject to affine Dirichlet data, generalizing a well-known estimate of Pogorelov for equations of Monge-Ampère type. For equations for which convexity of the solution is the natural ellipticity assumption, the curvature bound is proved for solutions with C1,1 Dirichlet data. We also use the curvature bounds to improve and extend various existence results for the Dirichlet and Plateau problems.

Citations

• Singularities of boundary value problems. I

Cited by

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• Prescribed k-symmetric curvature hypersurfaces in de Sitter space
• Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds
• Dirichlet problem for degenerate Hessian quotient type curvature equations
• Alexey Vasilyevich Pogorelov, the mathematician of an incredible power
• The Dirichlet problem for a class of Hessian quotient equations in Lorentz-Minkowski space ℝⁿ⁺¹₁
• Hypersurfaces of Constant Curvature in Hyperbolic Space I
• The global curvature estimate for the n-2 Hessian equation
• On the Dirichlet problem for a class of augmented Hessian equations
• Pogorelov type interior C² estimate for Hessian quotient equation and its application
• Curvature estimates for hypersurfaces of constant curvature in hyperbolic space II
• Interior curvature estimate for curvature quotient equations on convex hypersurfaces
• Pogorelov type C² estimates for sum Hessian equations

Related

• The Pogorelov estimates for degenerate curvature equations
• The Dirichlet Problem for Curvature Equations in Riemannian Manifolds
• The Dirichlet problem for a class of curvature equations in Minkowski space
• Interior gradient estimates for prescribed curvature equations in hyperbolic space
• Curvature estimates for the level sets of solutions of the Monge-Ampère equation det D² u=1