The entropy of the Angenent torus is approximately 1.85122

Abstract

To study the singularities that appear in mean curvature flow, one must understand self-shrinkers, surfaces that shrink by dilations under mean curvature flow. The simplest examples of self-shrinkers are spheres and cylinders. In 1989, Angenent constructed the first nontrivial example of a self-shrinker, a torus. A key quantity in the study of the formation of singularities is the entropy, defined by Colding and Minicozzi based on work of Huisken. The values of the entropy of spheres and cylinders have explicit formulas, but there is no known formula for the entropy of the Angenent torus. In this work, we numerically estimate the entropy of the Angenent torus using the discrete Euler-Lagrange equations.

arXiv comment14 pages, 5 figures, Jupyter code included as ancillary file. Journal of Experimental Mathematics, 2019

References in corpus (2)

• Self-shrinkers with a rotational symmetry
• The index of shrinkers of the mean curvature flow

Cited by in corpus (6)

• A finite element error analysis for axisymmetric mean curvature flow
• Numerical approximation of boundary value problems for curvature flow and elastic flow in Riemannian manifolds
• Noncompact self-shrinkers for mean curvature flow with arbitrary genus
• Bounds on the Index of Rotationally Symmetric Self-Shrinking Tori
• Genus one singularities in mean curvature flow
• Entropy Bounds, Compactness and Finiteness Theorems for Embedded Self-shrinkers with Rotational Symmetry