Mean Curvature Flow of Higher Codimension in Hyperbolic Spaces

where H(x, t) is the mean curvature vector of Ft(M) and Ft(x) = F (x, t). We call F : M × [0, T ) → F(c) the mean curvature flow with initial value F . The mean curvature flow was proposed by Mullins [17] to describe the formation of grain boundaries in annealing metals. In [3], Brakke introduced the motion of a submanifold by its mean curvature in arbitrary codimension and constructed a generalized varifold solution for all time. For the classical solution of the mean curvature flow, most works have been done on hypersurfaces. Huisken [11, 12] showed that if the initial hypersurface in a Riemannian manifold is uniformly convex, then the mean curvature flow converges to a round point in finite time. Later, Huisken [13] extend this result to hypersurfaces satisfying a pinching condition in a sphere. Many other beautiful results have been obtained, and there are various approaches to study the mean curvature flow of hypersurfaces (see [6, 7], etc.). For the mean curvature flow of submanifolds in higher codimension, some special cases have been studied, see [19, 20, 21, 22, 23, 24] etc. for example. Recently, Andrews-Baker [1] proved a convergence theorem for the mean curvature flow of closed submanifolds satisfying a pinching condition in the Euclidean space. In [2], Baker proved a convergence result for the mean curvature flow of submanifolds in a sphere. In this paper, we study the mean curvature flow of closed submanifolds in hyperbolic