Abstract
A closed surface evolving under mean curvature flow becomes singular in finite time. Near the singularity, the surface resembles a self-shrinker, a surface that shrinks by dilations under mean curvature flow. If the singularity is modeled on a self-shrinker other than a round sphere or cylinder, then the singularity is unstable under perturbations of the flow. One can quantify this instability using the index of the self-shrinker when viewed as a critical point of the entropy functional. In this work, we prove an upper bound on the index of rotationally symmetric self-shrinking tori in terms of their entropy and their maximum and minimum radii. While there have been a few lower bound results in the literature, we believe that this result is the first upper bound on the index of a self-shrinker. Our methods also give lower bounds on the index and the entropy, and our methods give simple formulas for two entropy-decreasing variations whose existence was proved by Liu. Surprisingly, the eigenvalue corresponding to these variations is exactly −1. Finally, we present some preliminary results in higher dimensions and six potential directions for future work.
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Berchenko-Kogan, Y. Bounds on the index of rotationally symmetric self-shrinking tori. Geom Dedicata 213, 83–106 (2021). https://doi.org/10.1007/s10711-020-00569-9
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DOI: https://doi.org/10.1007/s10711-020-00569-9