A classification of complete $3$-dimensional self-shrinkers in the Euclidean space $\mathbb R^{4}$

TL;DR

This paper provides a complete classification of 3-dimensional complete self-shrinkers in Euclidean 4-space with constant second fundamental form norm and constant third fundamental form component, advancing understanding of their geometric structure.

Contribution

It offers a full classification of certain self-shrinkers in four-dimensional space under specific curvature conditions, which was previously unknown.

Findings

Classification of self-shrinkers with constant S and f_3

Explicit descriptions of geometric structures

Advancement in understanding self-shrinker geometry

Abstract

In this paper, we completely classify $3$-dimensional complete self-shrinkers with constant norm $S$ of the second fundamental form and constant $f_{3}$ in Euclidean space $\mathbb R^{4}$, where $h_{ij}$ are components of the second fundamental form, $S=\sum_{i,j}h^{2}_{ij}$ and $f_{3}=\sum_{i,j,k}h_{ij}h_{jk}h_{ki}$.