Asymptotic convergence for a class of anisotropic curvature flows

In this paper, by using new auxiliary functions, we study a class of contracting flows of closed, star-shaped hypersurfaces in $\mathbb{R}^{n+1}$ with speed $r^{\frac{\alpha}{\beta}}\sigma_k^{\frac{1}{\beta}}$, where $\sigma_k$ is the $k$-th elementary symmetric polynomial of the principal curvatures, $\alpha$, $\beta$ are positive constants and $r$ is the distance from points on the hypersurface to the origin. We obtain convergence results under some assumptions of $k$, $\alpha$, $\beta$. When $k\geq2$, $0<\beta\leq 1$, $\alpha\geq \beta+k$, we prove that the $k$-convex solution to the flow exists for all time and converges smoothly to a sphere after normalization, in particular, we generalize Li-Sheng-Wang's result from uniformly convex to $k$-convex. When $k \geq 2$, $\beta=k$, $\alpha \geq 2k$, we prove that the $k$-convex solution to the flow exists for all time and converges smoothly to a sphere after normalization, in particular, we generalize Ling Xiao's result from $k=2$ to $k \geq 2$.