Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems

Abstract

In this paper we study a contracting flow of closed, convex hypersurfaces in the Euclidean space \mathbb R^{n+1} with speed f r^{\alpha} 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 \alpha \ge n+1 , we prove that the flow exists for all time and converges smoothly after normalisation to a soliton, which is a sphere if f \equiv 1 . Our argument provides a new proof in the smooth category for the classical Aleksandrov problem, and resolves the dual q -Minkowski problem introduced by Huang, Lutwak, Yang and Zhang [30] for q < 0 . If \alpha < 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.