In this talk, I will introduce our recent work on Gauss curvature flow with Xu-Jia Wang and Qi-Rui Li.

In this work we study a contracting flow of closed, convex hypersurfaces in the Euclidean space $\R^{n+1}$ with the 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. We prove that if $\alpha\ge n+1$, the flow exists for all time and converges smoothly after normalization to a hypersurface, which is a sphere if $f\equiv 1$.  Our argument provides a new proof for the classical Aleksandrov problem  ($\alpha = n+1$) and resolves the dual q-Minkowski problem introduced by Huang, Lutwak, Yang and Zhang recently, for the case q<0 ($\alpha>n+1$). 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.