Singularities of Mean Curvature Flow of Surfaces
Differential Geometry
2026-01-30 v1
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.
Cite
@article{arxiv.2601.21133,
title = {Singularities of Mean Curvature Flow of Surfaces},
author = {Tom Ilmanen},
journal= {arXiv preprint arXiv:2601.21133},
year = {2026}
}
Comments
37 pages. This unpublished paper, written by Tom Ilmanen (1961-2025) in 1993 or 1994, has been posted on arXiv with the permission of his family.