Abstract
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 Sk−1×ℝn−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.

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