Classification and Analysis of Mean Curvature Flow Self-Shrinkers

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.