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Genus Increase in Closed Mean Curvature Flows

  • Jingwen Chen
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Abstract

We construct closed mean curvature flows with arbitrarily large genus increase in every smooth Riemannian three-manifold, giving counterexamples to Ilmanen's genus-reduction conjecture. We prove that a compact hypersurface with a single isolated conical singularity and a power rate of convergence to its cone admits a mean curvature flow whose forward rescaled limit is a prescribed expander in the tangent space at the singular point, smoothly asymptotic to that cone. Combining this theorem with closed mean curvature flows with prescribed tangent flows (arXiv:2405.15577, arXiv:2607.23879) and the noncompact counterexample (arXiv:2605.09657) gives the desired examples.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026genus,
  title = {{Genus Increase in Closed Mean Curvature Flows}},
  author = {Jingwen Chen},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35000},
  eprint = {2610.35000},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35000},
}
RIS
TY  - UNPB
TI  - Genus Increase in Closed Mean Curvature Flows
AU  - Jingwen Chen
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.35000
UR  - https://arxiv.org/abs/2610.35000
AB  - We construct closed mean curvature flows with arbitrarily large genus increase in every smooth Riemannian three-manifold, giving counterexamples to Ilmanen's genus-reduction conjecture. We prove that a compact hypersurface with a single isolated conical singularity and a power rate of convergence to its cone admits a mean curvature flow whose forward rescaled limit is a prescribed expander in the tangent space at the singular point, smoothly asymptotic to that cone. Combining this theorem with closed mean curvature flows with prescribed tangent flows (arXiv:2405.15577, arXiv:2607.23879) and the noncompact counterexample (arXiv:2605.09657) gives the desired examples.
ER  -