Convex solutions to the mean curvature flow

In this paper we study the classification of ancient convex solutions to the mean curvature flow in Rⁿ⁺¹. An open problem related to the classification of type II singularities is whether a convex translating solution is k-rotationally symmetric for some integer 2 ≤ k ≤ n, namely whether its level set is a sphere or cylinder S^k-1 × R^n-k. In this paper we give an affirmative answer for entire solutions in dimension 2. In high dimensions we prove that there exist nonrotationally symmetric, entire convex translating solutions, but the blow-down in space of any entire convex translating solution is k-rotationally symmetric. We also prove that the blow-down in space-time of an ancient convex solution which sweeps the whole space Rⁿ⁺¹ is a shrinking sphere or cylinder.