A class of anisotropic expanding curvature flows

In this paper, we consider an expanding flow of smooth, closed, uniformly convex hypersurfaces in Euclidean $ R^{n+1} $ with speed $ u^\alpha\sigma_k^\beta $ firstly, where $ u $ is support function of the hypersurface, $ \alpha, \beta \in R^1 $, and $ \beta>0 $, $ \sigma_k $ is the $ k $-th symmetric polynomial of the principal curvature radii of the hypersurface, $ k $ is an integer and $ 1\le k\le n $. For $ \alpha\le1-k\beta $, $ \beta>\frac{1}{k} $ we prove that the flow has a unique smooth and uniformly convex solution for all time, and converges smoothly after normalisation, to a sphere centered at the origin. Moreover, for $ \alpha\le1-k\beta $, $ \beta>\frac{1}{k} $, we prove that the flow with the speed $ fu^\alpha\sigma_k^\beta $ exists for all time and converges smoothly after normalisation to a soliton which is a solution of $ fu^{\alpha-1}\sigma_k^{\beta} = c $ provided that $ f $ is a smooth positive function on $ S^n $ and satisfies that $ (\nabla_i\nabla_jf^{\frac{1}{1+k\beta-\alpha}}+\delta_{ij}f^{\frac{1}{1+k\beta-\alpha}}) $ is positive definite. When $ \beta = 1 $, our argument provides a proof to the well-known $ L_p $ Christoffel-Minkowski problem for the case $ p\ge k+1 $ where $ p = 2-\alpha $, which is identify with Ivaki's recent result. Especially, we obtain the same result for $ k = n $ without any constraint on smooth positive function $ f $. Finally, we also give a counterexample for the two anisotropic expanding flows when $ \alpha>1-k\beta $.