Increase in topological complexity along the mean curvature flow

Natasha Diederen

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arXiv · 2610.08357

Increase in topological complexity along the mean curvature flow

Natasha Diederen

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Abstract

We construct an example of a smooth embedding of $\mathbb{S}^{p+q-1}$ in $\mathbb{R}^{p+q}$ whose evolution under the mean curvature flow forms an isolated singularity at one point, after which it is a smoothly embedded copy of $\mathbb{S}^{p-1}\times \mathbb{S}^q$. In doing so, we prove some general results about $SO(p)\times SO(q)$-invariant mean curvature flow whose initial data is generated by the rotation of a graphical profile curve, including an analysis of bubblesheet singularities. The most involved part of our analysis is an instant smoothness result, which requires a delicate approximation and pseudolocality.

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Natasha Diederen. 2026-10-06. Increase in topological complexity along the mean curvature flow. https://arxiv.org/abs/2610.08357

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