Embedded self-similar shrinkers of genus 0

2014/11/17 by Simon Brendle, Brendle, S. · 9 citations
Mathematics · #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1411.4640

Abstract

We confirm a well-known conjecture that the round sphere is the only compact, embedded self-similar shrinking solution to the mean curvature flow with genus 0. More generally, we show that the only properly embedded self-similar shrinkers in ℝ³ with vanishing intersection form are the sphere, the cylinder, and the plane. This answers two questions posed by T. Ilmanen.

Cited by

• The curve shortening problem associated to a density
• On the rate of convergence of cylindrical singularity in mean curvature flow
• The Bounded Diameter Conjecture and Sharp Geometric Estimates for Mean Curvature Flow
• Mean curvature flow through singularities
• First eigenvalue and nodal domains of the drift Laplacian on symmetric self-shrinkers in ℝ³
• Self-expanders of positive genus
• Backwards uniqueness for Mean curvature flow with asymptotically conical singularities
• When can minimal hypersurfaces be connected by mean curvature flow?
• Smooth compactness of f-minimal hypersurfaces with bounded f-index

Related

• Self-shrinkers of the mean curvature flow in arbitrary codimension
• Self-shrinkers whose asymptotic cones fatten
• Spacelike Self-Similar Shrinking Solutions of the Mean Curvature Flow in Pseudo-Euclidean Spaces
• Smooth compactness of self-shrinkers
• Hopf type theorems for self-similar solutions of curvature flows in ℝ³