The entropy of the Angenent torus is approximately 1.85122

TL;DR

This paper numerically estimates the entropy of the Angenent torus, a key self-shrinker in mean curvature flow, providing insights into singularity formation where no explicit entropy formula exists.

Contribution

The paper introduces a numerical method to estimate the entropy of the Angenent torus, filling a gap where no explicit formula is known.

Findings

Estimated the entropy of the Angenent torus as approximately 1.85122.

Demonstrated the effectiveness of discrete Euler-Lagrange equations in geometric analysis.

Provided a numerical benchmark for future theoretical studies.

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.