Parabolic approaches to curvature equations

Abstract

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 $L_p$-Aleksandrov and dual Minkowski problems as well as their counterparts, namely the $L_p$-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.

arXiv comment32 pages

References in corpus (17)

• The Logarithmic Minkowski Problem
• Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems
• Hypersurfaces of Prescribed Curvature Measure
• The $L_p$-Minkowski problem for $-n<p<1$
• Deforming a hypersurface by principal radii of curvature and support function
• Curvature flows in the sphere
• Locally constrained inverse curvature flows
• A unified flow approach to smooth, even $L_p$-Minkowski problems
• Alexandrov-Fenchel inequalities for convex hypersurfaces with free boundary in a ball
• Smoothness in the $l_p$ Minkowski problem for $p<1$
• New pinching estimates for Inverse curvature flows in space forms
• Global $C^2$ estimates for convex solutions of curvature equations
• Isotropic functions revisited
• Minkowski type problems for convex hypersurfaces in hyperbolic space
• Harnack inequalities for curvature flows in Riemannian and Lorentzian manifolds
• Iterations of curvature images
• Prescribed $k$-symmetric curvature hypersurfaces in de Sitter space

Cited by in corpus (4)

• Orlicz-Minkowski flows
• Constant rank theorems for curvature problems via a viscosity approach
• Christoffel-Minkowski flows
• Prescribed $L_p$ curvature problem