Convexity estimates for surfaces moving by curvature functions

We consider the evolution of compact surfaces by fully nonlinear, parabolic curvature flows for which the normal speed is given by a smooth, degree one homogeneous function of the principal curvatures of the evolving surface. Under no further restrictions on the speed function, we prove that initial surfaces on which the speed is positive become weakly convex at a singularity of the flow. This generalises the corresponding result [26] of Huisken and Sinestrari for the mean curvature flow to the largest possible class of degree one homogeneous surface flows.