Extrinsic flows are evolution equations whose speeds are determined by the extrinsic curvature of submanifolds in ambient spaces. Some of the well-known ones are mean curvature flow, Gauss curvature flow, and Lagrangian mean curvature flow.
We focus on the special case in which the speed of a flow is given by powers of mean curvature for smooth convex hypersurfaces of graph type, i.e., ones that can be represented as the graph of a function. Convergence and long-time existence of such flow will be discussed. Furthermore, C^2 estimates which are independent of height of the graph will be derived to see that the boundary of the domain of the graph is also a smooth solution for the same flow as a submanifold with codimension two in the classical sense. Some of the main ideas, notably a priori estimates via the maximum principle, come from the work of Huisken and Ecker on mean curvature evolution of entire graphs in 1989. This is a joint work with Ki-ahm Lee and Taehun Lee.