On an extension of the H k mean curvature flow

In this note, we generalize an extension theorem in Le-Sesum] and Xu-Ye-Zhao] of the mean curvature flow to the H ^k mean curvature flow under some extra conditions. The main difficulty in proving the extension theorem is to find a suitable version of Michael-Simon inequality for the H ^k mean curvature flow, and to do a suitable Moser iteration process. These two problems are overcome by imposing some extra conditions which may be weakened or removed in our forthcoming paper. On the other hand, we derive some estimates for the generalized mean curvature flow, which have their own interesting.