Convex solutions to the mean curvature flow
An open problem related to the classification of type II singularities is whether a convex translating solution is \(k\)--rotationally symmetric for some integer \(2 \leq k \leq n\), namely whether its level set is a sphere or a cylinder. The author gives an affirmative answer for entire solutions in dimension 2. In high dimensions it is also proved 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 and 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. The author surveys new progress made in this field, after this paper was finished in early 2003.
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