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.
A strong Frankel Theorem for shrinkers
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We prove a strong Frankel theorem for mean curvature flow shrinkers in all dimensions: Any two shrinkers in a sufficiently large ball must intersect. In particular, the shrinker itself must be connected in all large balls. The key to the proof is a strong Bernstein theorem for incomplete stable Gaussian surfaces.
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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.