Asymptotic convergence for a class of anisotropic curvature flows (Q2123112)

The authors study a class of contracting flows of closed star-shaped hypersurfaces in \(\mathbb{R}^{n+1}\) with speed \(r^{\alpha/\beta}\sigma_k^{1/\beta}\) where \(\sigma_k\) is the \(k\)-th symmetric polynomial of the principal curvatures, \(\alpha>0, \beta>0\), are constants and \(r\) is the distance of the point on the hypersurface from the origin. When either (i) \(k\ge 2\), \(0<\beta\le 1\), and \(\alpha\ge\beta+k\) or (ii) \(k\ge 2\), \(\beta=k\) and \(\alpha\ge 2k\) holds, they prove that the \(k\)-convex solution of the flow exists for all time and converges smoothly to a sphere after rescaling. As a consequence they generalize Li-Sheng-Wang's result from uniformly convex to \(k\)-convex and Ling Xiao's result from \(k=2\) to \(k\ge 2\).