Classification and Analysis of Mean Curvature Flow Self-Shrinkers

TL;DR

This paper analyzes the stability properties of specific mean curvature flow self-shrinkers, especially those of the form S^k×R^{n-k}, revealing their eigenvalues, eigenfunctions, and stability characteristics.

Contribution

It introduces a novel connection between the stability operator and the quantum harmonic oscillator to determine eigenvalues and eigenfunctions of these self-shrinkers.

Findings

Self-shrinkers of the form S^k×R^{n-k} have finite index.

These self-shrinkers have lower index than other complete self-shrinkers.

Ends of such self-shrinkers are stable.

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 $\mathbb S^k\times\R^{n-k}\subset \R^{n+1}$. 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.