Hypersurfaces of Prescribed Curvature Measure

Abstract

We consider the corresponding Christoffel-Minkowski problem for curvature measures. The existence of star-shaped $(n-k)$-convex bodies with prescribed $k$-th curvature measures ($k>0$) has been a longstanding problem. This is settled in this paper through the establishment of a crucial $C^2$ a priori estimate for the corresponding curvature equation on ${\mathbb{S}}^n$.

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• Curvature estimates for immersed hypersurfaces in Riemannian manifolds
• On the planar Lp-Minkowski problem
• Parabolic approaches to curvature equations
• The prescribed curvature problem for entire hypersurfaces in Minkowski space
• Two problems related to prescribed curvature measures
• On the Minkowski problem for p-harmonic measures
• Prescribed $L_p$ curvature problem