Generic mean curvature flow I; generic singularities

Abstract

It has long been conjectured that starting at a generic smooth closed embedded surface in R^3, the mean curvature flow remains smooth until it arrives at a singularity in a neighborhood of which the flow looks like concentric spheres or cylinders. That is, the only singularities of a generic flow are spherical or cylindrical. We will address this conjecture here and in a sequel. The higher dimensional case will be addressed elsewhere. The key in showing this conjecture is to show that shrinking spheres, cylinders and planes are the only stable self-shrinkers under the mean curvature flow. We prove this here in all dimensions. An easy consequence of this is that every other singularity than spheres and cylinders can be perturbed away.

References in corpus (2)

• Evolution of curves and surfaces by mean curvature
• Smooth compactness of self-shrinkers

Cited by in corpus (14)

• Gap and rigidity theorems of $\lambda$-hypersurfaces
• On the self-shrinking systems in arbitrary codimension spaces
• The rigidity theorems of self shrinkers
• Classification and rigidity of self-shrinkers in the Mean curvature flow
• Volume estimate about shrinkers
• The Rigidity Theorems of Self Shrinkers via Gauss maps
• The Curvature of Gradient Ricci Solitons
• An immersed $S^2$ self-shrinker
• Entire spacelike translating solitons in Minkowski space
• Complete self-shrinkers of the mean curvature flow
• On second variation of Perelman's Ricci shrinker entropy
• Translating solitons for Lagrangian mean curvature flow in complex Euclidean plane
• A Bernstein Type Theorem For Self-similar Shrinkers
• The stability inequality for Ricci-flat cones