Abstract:We consider mean curvature flow with free boundary through cylindrical or half-cylindrical singularities, namely singularities of the types $\mathbb{R}^k\times S^{n-k}$, $\mathbb{R}^k_+\times S^{n-k}$ or $\mathbb{R}^k\times S^{n-k}_+$. Using the foundational results for free boundary Brakke flows by Edelen and the first author, and the recent classification of ancient asymptotically cylindrical flows by Bamler-Lai, we prove that all these singularities have a mean-convex neighborhood. Moreover, generalizing work of Hershkovits-White to the free boundary setting we show that the free boundary level set flow is nonfattening provided all singularities have a mean-convex neighborhood. We conclude that free boundary flow through singularities is well-posed as long as all singularities are of cylindrical or half-cylindrical type.
From: Robert Haslhofer [view email]
[v1]
Mon, 15 Jun 2026 22:53:07 UTC (17 KB)
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