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PreprintCC BY 4.0arXiv

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 š•Šp+qāˆ’1 in ā„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 š•Špāˆ’1Ɨ š•Šq. In doing so, we prove some general results about SO(p)Ɨ 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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Cite

BibTeXRIS
BibTeX
@misc{diederen2026increase,
  title = {{Increase in topological complexity along the mean curvature flow}},
  author = {Natasha Diederen},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08357},
  eprint = {2610.08357},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08357},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Increase in topological complexity along the mean curvature flow
AU  - Natasha Diederen
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.08357
UR  - https://arxiv.org/abs/2610.08357
AB  - We construct an example of a smooth embedding of š•Š^p+qāˆ’1 in ā„^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 š•Š^pāˆ’1Ɨ š•Š^q. In doing so, we prove some general results about SO(p)Ɨ 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.
ER  -