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Self-shrinkers model singularities of the mean curvature flow; they are the special solutions that contract homothetically under the flow. Colding–Ilmanen–Minicozzi showed that any self-shrinker that is mean convex with uniformly bounded curvature on a large, but compact, set must be a round cylinder. This rigidity is fundamental in establishing the uniqueness of blowups at cylindrical singularities.

On the Rigidity of Mean Convex Self-Shrinkers