Numerically computing the index of mean curvature flow self-shrinkers

TL;DR

This paper introduces a numerical method to compute the Morse index of rotationally symmetric self-shrinkers in mean curvature flow, applied to the Angenent torus, revealing an index of 5 and additional eigenvalue variations.

Contribution

The paper develops a novel numerical approach for calculating the index of self-shrinkers, specifically applied to the Angenent torus, providing new insights into its stability properties.

Findings

The index of the Angenent torus is 5, excluding symmetries.

Two additional eigenvalue variations with eigenvalue -1 were found.

The computed index aligns with existing theoretical bounds.

Abstract

Surfaces that evolve by mean curvature flow develop singularities. These singularities can be modeled by self-shrinkers, surfaces that shrink by dilations under the flow. Singularities modeled on classical self-shrinkers, namely spheres and cylinders, are stable under perturbations of the flow. In contrast, singularities modeled on other self-shrinkers, such as the Angenent torus, are unstable: perturbing the flow will generally change the kind of singularity. One can measure the degree of instability by computing the Morse index of the self-shrinker, viewed as a critical point of an appropriate functional. In this paper, we present a numerical method for computing the index of rotationally symmetric self-shrinkers. We apply this method to the Angenent torus, the first known nontrivial example of a self-shrinker. We find that, excluding dilations and translations, the index of the Angenent torus is $5$, which is consistent with the lower bound of $3$ from the work of Liu and the upper bound of $29$ from our earlier work. Also, we unexpectedly discover two additional variations of the Angenent torus with eigenvalue $-1$.