Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems

2017/12/21 by Li, Qi-Rui, Sheng, Weimin, Wang, Xu-Jia · 8 citations
#35K96 #53C44 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1712.07774

Abstract

In this paper we study a contracting flow of closed, convex hypersurfaces in the Euclidean space \mathbb Rⁿ⁺¹ with speed f r^α K, where K is the Gauss curvature, r is the distance from the hypersurface to the origin, and f is a positive and smooth function. If α≥ n+1, we prove that the flow exists for all time and converges smoothly after normalisation to a soliton, which is a sphere centred at the origin if f ≡ 1. Our argument provides a parabolic proof in the smooth category for the classical Aleksandrov problem, and resolves the dual q-Minkowski problem introduced by Huang, Lutwak, Yang and Zhang (Acta Math. 216 (2016): 325-388), for the case q<0. If α< n+1, corresponding to the case q>0, we also establish the same results for even function f and origin-symmetric initial condition, but for non-symmetric f, counterexample is given for the above smooth convergence.

Cited by

• The p-th dual Minkowski problem for the k-torsional rigidity corresponding to a k-Hessian equation
• The L_p chord Minkowski problem for super-critical exponent
• Capillary L_p Minkowski Flows
• Long time behavior of a class of non-homogeneous anisotropic fully nonlinear curvature flows
• The L_p Dual Minkowski Problem for Group-Invariant Convex Bodies
• Compactness of the L_p dual Minkowski problem in ℝ³
• Capillary curvature images
• Uniqueness in the near isotropic Lp dual Minkowski problem

Related

• Flow by Gauss curvature to Dual Orlicz-Minkowski problems
• An Anisotropic shrinking flow and L_p Minkowski problem
• Flow by Gauss curvature to the L_p-Gaussian Minkowski problem
• Curvature bound for L_p Minkowski problem
• Flow by the power of the Gauss curvature