Non existence of Type II singularities for embedded and unknotted space curves

TL;DR

This paper proves that a specific class of embedded, unknotted space curves in three-dimensional space cannot develop Type II singularities under curve shortening flow before collapsing, using minimal surface theory and isoperimetric ratios.

Contribution

It establishes the non-existence of Type II singularities for certain unknotted space curves under curve shortening flow, a novel result in geometric analysis.

Findings

Type II singularities do not occur for the studied class of curves

Minimal surface theory tools are effective in analyzing curve evolution

The isoperimetric ratio plays a key role in the proof

Abstract

In this paper we prove that a certain class of embedded unknotted curves in $\mathbb{R}^3$ evolving under curve shortening flow do not form singularities Type II before collapsing to a point. Our proof uses tools of the minimal surface theory to study a suitable isoperimetric ratio.