Mean curvature flow via convex functions on Grassmannian manifolds

Using the convex functions on Grassmannian manifolds, the authors obtain the interior estimates for the mean curvature flow of higher codimension. Confinable properties of Gauss images under the mean curvature flow have been obtained, which reveal that if the Gauss image of the initial submanifold is contained in a certain sublevel set of the upsilon-function, then all the Gauss images of the submanifolds under the mean curvature flow are also contained in the same sublevel set of the upsilon-function. Under such restrictions, curvature estimates in terms of upsilon-function composed with the Gauss map can be carried out.