We consider the evolution of compact hypersurfaces by fully nonlinear, parabolic
curvature flows for which the normal speed is given by a smooth, convex, degree-one
homogeneous function of the principal curvatures. We prove that solution
hypersurfaces on which the speed is initially positive become weakly convex at a
singularity of the flow. The result extends the convexity estimate of Huisken and
Sinestrari
[Acta Math.183:1 (1999), 45–70] for the mean curvature flow to a large
class of speeds, and leads to an analogous description of “type-II” singularities. We
remark that many of the speeds considered are positive on larger cones than the
positive mean half-space, so that the result in those cases also applies to
non-mean-convex initial data.