Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems

Original · EN

In this paper we study a contracting flow of closed, convex hypersurfaces in the Euclidean space Rⁿ⁺¹ with speed f rα 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.