Classification and Analysis of Mean Curvature Flow Self-Shrinkers

2013/03/02 by Caleb Hussey, Hussey, Caleb
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals #math.DG

paper · pdf · doi:10.48550/arxiv.1303.0354

77 pages, dissertation

arxiv created 2013/03/02 · openalex publication_date 2013/03/02 · arxiv updated 2013/03/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate Mean Curvature Flow self-shrinking hypersurfaces with polynomial growth. It is known that such self shrinkers are unstable. We focus mostly on self-shrinkers of the form \mathbb S^k×\R^n-k⊂ \Rⁿ⁺¹. We use a connection between the stability operator and the quantum harmonic oscillator Hamiltonian to find all eigenvalues and eigenfunctions of the stability operator on these self-shrinkers. We also show self-shrinkers of this form have lower index than all other complete self-shrinking hypersurfaces. In particular, they have finite index. This implies that the ends of such self shrinkers must be stable. We look for the largest stable regions of these self shrinkers.

Related

• The Stability of Self-Shrinkers of Mean Curvature Flow in Higher Codimension
• Complete self-shrinkers of the mean curvature flow
• F-stability for self-shrinking solutions to mean curvature flow
• Eigenvalue estimates on shrinkers
• A maximum principle for self-shrinkers and some consequences