LAUSR

In this paper, we first investigate the flow of convex surfaces in the space form $$\mathbb {R}^3(\kappa )~(\kappa =0,1,-1)$$R3(κ)(κ=0,1,-1) expanding by $$F^{-\alpha }$$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 $$\alpha \in (0,1]$$α∈(0,1] for $$\kappa =0,-1$$κ=0,-1 and $$\alpha =1$$α=1 for $$\kappa =1$$κ=1. By deriving that the pinching ratio of the flow surface $$M_t$$Mt is no greater than that of the initial surface $$M_0$$M0, 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 $$\mathbb {H}^3$$H3 with $$\alpha \in (0,1)$$α∈(0,1), the limit shape may not be necessarily round after rescaling.