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 $f{r}^{\hat{I}}\pm K$, 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 $\hat{I}\pm \ge n+1$, 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 $f\equiv 1$. 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 $\hat{I}\pm <n+1$, 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.

34 pages