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arXiv · 2601.21133

Singularities of Mean Curvature Flow of Surfaces

Tom Ilmanen
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Abstract

This paper proves that, at the first singular time for a smoothly immersed surface moving by mean curvature flow in a n-manifold, each tangent flow is given by a smooth, branched shrinker, possibly with multiplicity. If n=3 and if the initial surface is embedded, then the shrinker is smoothly embedded without branch points, but possibly with multiplicity. A key ingredient of the proof is a new, local version of the Gauss-Bonnet formula.

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Tom Ilmanen. 2026-01-29. Singularities of Mean Curvature Flow of Surfaces. https://arxiv.org/abs/2601.21133

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