A Singular Example for the Averaged Mean Curvature Flow

Abstract

An embedded curve is presented which under numerical simulation of the averaged mean curvature flow develops first a loss of embeddedness and then a singularity where the curvature becomes infinite, all in finite time. This leads to the conjecture that not all smooth embedded curves persist for all times under the averaged mean curvature flow.

Keywords:

• Averaged mean curvature flow
• loss of embeddedness
• immersed curve
• blowup of curvature
• numerical experiments