Inverse Curvature Flows of Rotation Hypersurfaces

We consider the inverse curvature flows of smooth, closed and strictly convex rotation hypersurfaces in space forms M k n + 1 with speed function given by F − α , where α ∈ (0, 1] for κ = 0, −1, α =1 for κ = 1 and F is a smooth, symmetric, strictly increasing and 1-homogeneous function of the principal curvatures of the evolving hypersurfaces. We show that the curvature pinching ratio of the evolving hypersurface is controlled by its initial value, and prove the long time existence and convergence of the flows. No second derivatives conditions are required on F .