In this paper we study a contracting flow of closed, convex hypersurfaces in the Euclidean space with speed , where is the Gauss curvature, is the distance from the hypersurface to the origin, and is a positive and smooth function. If , we prove that the flow exists for all time and converges smoothly after normalisation to a soliton, which is a sphere if . Our argument provides a new proof in the smooth category for the classical Aleksandrov problem, and resolves the dual -Minkowski problem introduced by Huang, Lutwak, Yang and Zhang  for . If , corresponding to the case , we also establish the same results for even function and origin-symmetric initial condition, but for non-symmetric , counterexample is given for the above smooth convergence.
Cite this article
Qi-Rui Li, Weimin Sheng, Xu-Jia Wang, Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems. J. Eur. Math. Soc. 22 (2020), no. 3, pp. 893–923DOI 10.4171/JEMS/936