# Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems

### Qi-Rui Li

The Australian National University, Canberra, Australia and Zhejiang University, Hangzhou, China### Weimin Sheng

Zhejiang University, Hangzhou, China and The Australian National University, Canberra, Australia### Xu-Jia Wang

The Australian National University, Canberra, Australia

## 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.

## 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. 893–923

DOI 10.4171/JEMS/936