Hypersurfaces of prescribed curvature measure

2011/03/11 by Pengfei Guan, Junfang Li, YanYan Li · 79 citations
Mathematics · #A priori and a posteriori #Center of curvature #Curvature #Geometric Analysis and Curvature Flows #Mean curvature #Mean curvature flow #Measure (data warehouse) #Nonlinear Partial Differential Equations #Point processes and geometric inequalities #Total curvature #math.AP #math.DG #msc:35J60 #msc:53C23 #msc:53C42

paper · pdf · doi:10.1215/00127094-1645550

published in Duke Mathematical Journal 161(10) (Duke University Press)

arxiv created 2011/03/11 · openalex publication_date 2012/06/27 · openalex created_date 2016/06/24 · arxiv updated 2019/12/19 · openalex updated_date 2026/08/05

Abstract

We consider the corresponding Christoffel–Minkowski problem for curvature measures. The existence of star-shaped (n−k)-convex bodies with prescribed kth curvature measures (k>0) has been a longstanding problem. This is settled in this paper through the establishment of a crucial a priori C2-estimate for the corresponding curvature equation on Sn.

Citations

• The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian

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• Anisotropic fractional area measures
• Dirichlet problem for degenerate Hessian quotient type curvature equations
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• Curvature estimates for hypersurfaces of constant curvature in hyperbolic space II
• Parabolic approaches to curvature equations
• Pogorelov type C² estimates for sum Hessian equations
• Neumann Problems for Elliptic and Parabolic Sum Hessian Equations

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• Capillary Christoffel-Minkowski problem