Singularity Behavior of the Mean Curvature Flow

It was proved that a blow-up solution to the mean curvature flow with positive mean curvature is an ancient convex solution, that is a convex solution which exists for time t from −∞. In this paper we study the geometry of ancient convex solutions. Our main results are contained in Theorems 1-3 below. Theorem 1 asserts that after normalization, the solution converges to a sphere or cylinder as t → −∞. Theorem 2 shows that in any dimension n ≥ 3, there exists ancient convex solutions which are not rotationally symmetric. But Theorem 3 shows that a translating convex solution in R3 must be rotationally symmetric if it is a blow-up solution. These results are contained in paper [W].