2020/03/31 by Paul Bryan, Mohammad N. Ivaki, Julian Scheuer
Mathematics · #Convexity #Curvature #Euclidean space #Gaussian curvature #Geometric Analysis and Curvature Flows #Geometry #Mathematical analysis #Mathematical physics #Mathematics #Minkowski space #Nonlinear Partial Differential Equations #Point processes and geometric inequalities #Pure mathematics #Scalar curvature #math.AP #math.DG
paper · pdf · doi:10.1016/j.na.2020.112174
published as Nonlinear Anal. 203 (2021), art. 112174 · 32 pages
arxiv created 2020/10/24 · openalex publication_date 2020/11/02 · arxiv updated 2021/01/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We employ curvature flows without global terms to seek strictly convex, spacelike solutions of a broad class of elliptic prescribed curvature equations in the simply connected Riemannian spaceforms and the Lorentzian de Sitter space, where the prescribed function may depend on the position and the normal vector. In particular, in the Euclidean space we solve a class of prescribed curvature measure problems, intermediate Lp-Aleksandrov and dual Minkowski problems as well as their counterparts, namely the Lp-Christoffel-Minkowski type problems. In some cases we do not impose any condition on the anisotropy except positivity, and in the remaining cases our condition resembles the constant rank theorem/convexity principle due to Caffarelli-Guan-Ma (Commun. Pure Appl. Math. 60 (2007), 1769--1791). Our approach does not rely on monotone entropy functionals and it is suitable to treat curvature problems that do not possess variational structures.