For closed smooth mean curvature flows in R^{n+1}, 3≤n≤6, at the first singular time either the mean curvature or the Morse index must blow up.
Asymptotic structure of self-shrinkers
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We show that each end of a noncompact self-shrinker in $\mathbb{R}^3$ of finite topology is smoothly asymptotic to either a regular cone or a self-shrinking round cylinder.
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Singularities of mean curvature flow with bounded mean curvature and Morse index
For closed smooth mean curvature flows in R^{n+1}, 3≤n≤6, at the first singular time either the mean curvature or the Morse index must blow up.