Hypersurfaces of prescribed curvature measure

By Pengfei Guan, Junfang Li, Yanyan Li. 2011

124 citations recorded by the source corpus.

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 $\mathbb S^n$.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/4e4e1d36d47b32a205cf7ce74338aa8c5ac43b28

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

Citations in the corpus

The global curvature estimate for the n-2 Hessian equation (2020, methodology): “The curvature estimate was also established for the equation of the prescribing curvature measure problem by Guan-Li-Li [24] and Guan-Lin-Ma [23].”
Existence of solutions to the even dual Minkowski problem (2018, unknown): “Progress on this problem was made by Guan, Li & Li [24] and Guan, Lin & Ma [26].”
A class of curvature type equations. (2019, unknown): “This problem has been extensively studied and important progress has been made in Guan-Guan [21] and Guan-Li-Li [24].”
On the curvature estimates for Hessian equations (2016, background): “The proof of the first Lemma can be found in [22, 24].”
Global C2‐Estimates for Convex Solutions of Curvature Equations (2013, background): “C2 estimate was also proved for equation of prescribing curvature measures problem in [15, 14], where f(X, ν) = 〈X, ν〉f̃(X).”
Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems (2016, unknown): “See Guan–Lin–Ma [35] and the recent work of Guan–Li–Li [33] on this problem.”