Convex solutions to the mean curvature flow

In this paper we study the classification of ancient convex solutions to the mean curvature flow in ${\mathbb{R}}^{n+1}$. An open problem related to the classification of type II singularities is whether a convex translating solution is $k$-rotationally symmetric for some integer $2\le k\le n$, namely whether its level set is a sphere or cylinder $S^{k-1}\times {\mathbb{R}}^{n-k}$. In this paper we give an affirmative answer for entire solutions in dimension 2. In high dimensions we prove that there exist non-rotationally 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 ${\mathbb{R}}^{n+1}$ is a shrinking sphere or cylinder.