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arXiv · 2610.35000

Genus Increase in Closed Mean Curvature Flows

Jingwen Chen
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Abstract

We construct counterexamples to Ilmanen's genus-reduction conjecture for closed mean curvature flows. 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 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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Jingwen Chen. 2026-10-26. Genus Increase in Closed Mean Curvature Flows. https://arxiv.org/abs/2610.35000

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