A Regularity Theorem for Curvature Flows

The regularity theory of minimal surfaces and minimizers of other elliptic functional is well known, due to DeGiorgi [DG], Federer and Fleming [FF], Reifenberg [R] and F. Almgren [A]. However the only regularity theory of evolutionary problem is due to K. Brakke [B] about mean curvature flow of unit density surfaces. Here we prove a regularity theory for general surface flows with a similar density condition (Section 2, Definition 9). The evolutions that we deal with do not necessarily come from a gradient flow of some functionals. This is the generalization of the paper [W2] where the corresponding elliptic problem was studied. The equations that we consider are the following