On the topological rigidity of self shrinkers in $\mathbb{R}^3$

TL;DR

This paper proves that all compact self-shrinkers in three-dimensional space are topologically equivalent to standard surfaces, showing that self-shrinking tori are unknotted and confirming their topological simplicity.

Contribution

It establishes the topological rigidity of compact self-shrinkers in bbr^3, demonstrating they are ambiently isotopic to standard embedded surfaces, a novel result in geometric analysis.

Findings

Self-shrinkers of genus g are topologically standard

Self-shrinking tori are unknotted

Compact self-shrinkers exhibit topological rigidity

Abstract

In this note we show that compact self shrinkers in $\mathbb{R}^3$ are "topologically standard" in that any genus $g$ compact self shrinker is ambiently isotopic to the standard genus $g$ embedded surface in $\mathbb{R}^3$. As a consequence self shrinking tori are unknotted.