Hypersurfaces of Prescribed Curvature Measure

TL;DR

This paper solves a longstanding geometric problem by establishing key estimates that guarantee the existence of star-shaped convex bodies with prescribed curvature measures, advancing the understanding of curvature measure equations.

Contribution

It provides the first proof of existence for star-shaped convex bodies with prescribed curvature measures for all positive indices, using new a priori estimates.

Findings

Existence of star-shaped $(n-k)$-convex bodies with prescribed $k$-th curvature measures.

Established crucial $C^2$ a priori estimates for the curvature equation.

Resolved a longstanding problem in geometric analysis.

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 $\mathbb S^n$.