Bounds on the index of rotationally symmetric self-shrinking tori
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Bounds on the index of rotationally symmetric self-shrinking tori Yakov Berchenko-Kogan1 Received: 23 April 2020 / Accepted: 7 September 2020 © Springer Nature B.V. 2020
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. Keywords Mean curvature flow · Self-shrinkers · Angenent torus Mathematics Subject Classification (2020) 53E10
1 Introduction Mean curvature flow is a well-studied geometric flow under which a hypersurface ⊂ Rn+1 evolves in such a way as to decrease its area as fast as possible. Under mean curvature flow, each point on the surface moves in the inward normal direction with velocity equal to the mean curvature of the surface at that point. Mean curvature flow has applications to image denoising, and the rich features of mean curvature flow provide a good testing ground for studying geometric flows and nonlinear parabolic partial differential equations more generally. Some surfaces, such as spheres and cylinders, evolve under mean curvature by dilations. These surfaces, known as self-shrinkers, will shrink until they disappear in finite time. Self-
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Yakov Berchenko-Kogan [email protected] University of Hawaii at Manoa, Honolulu, HI 96822, USA
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Geometriae Dedicata
shrinkers are particularly important in the study of mean curvature flow because they model the singularities that develop as a surface evolves under mean curvature flow: as the flow approaches a singularity, the surface will locally resemble a self-shrinker. For example, if the initial surface is convex, then the surface will become rounder and rounder as it shrinks to a point, resembling a sphere right before it disappears. Meanwhile, if the initial surface is shaped like a dumbbell, with two large lobes connected by a thin tube, then the thin tube will collapse, splitting the surface into two. Right before the singular time, the surface will locally resemble a cylinder near the singular point. The world of self
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