Surfaces expanding by non-concave curvature functions

In this paper, we first investigate the flow of convex surfaces in the space form R3(κ)(κ=0,1,-1) expanding by F^-α, where F is a smooth, symmetric, increasing and homogeneous of degree one function of the principal curvatures of the surfaces and the power α∈ (0 , 1] for κ= 0 , - 1 and α= 1 for κ= 1. By deriving that the pinching ratio of the flow surface M_t is no greater than that of the initial surface M, we prove the long time existence and the convergence of the flow. No concavity assumption of F is required. We also show that for the flow in H³ with α∈ (0 , 1) , the limit shape may not be necessarily round after rescaling.