Singularities of Mean Curvature Flow of Surfaces
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.
Fields of Study & Subjects
Geometric Analysis and Curvature Flows
Nonlinear Partial Differential Equations
Geometry and complex manifolds
Publication details
- Journal / source
- arXiv (Cornell University)
- Publication date
- 29 January 2026
- Document type
- Preprint