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.

Keywords:

• mean curvature flow
• Angenent torus
• discrete Euler–Lagrange equations
• variational integrator

2010 Mathematics Subject Classification:

• 53C44
• 70H25
• 65-05
• 65Pxx

Acknowledgments

I would like to thank Jacob Bernstein for proposing this problem at the Geometric Analysis Conference at Rutgers. I would also like to thank Bill Minicozzi and Ari Stern for their comments on this work.