Rigidity of $4$-dimensional complete self-shrinkers in $\mathbb{R}^{5}$

TL;DR

This paper proves that 4-dimensional complete self-shrinkers in five-dimensional space with specific constant curvature conditions are isometric to Euclidean space, providing a classification of such geometric objects.

Contribution

It establishes a rigidity theorem for 4D self-shrinkers with constant squared second fundamental form norm and specific curvature conditions, leading to their classification.

Findings

Self-shrinkers with constant S, f3=0, and constant f4 are isometric to Euclidean space.

Provides a classification result for these self-shrinkers.

Demonstrates rigidity under given curvature constraints.

Abstract

We show that any $4$-dimensional complete self-shrinker in $\mathbb{R}^{5}$ with constant squared norm $S$ of the second fundamental form, $f_{3}=0$ and constant $f_{4}$ is isometric to $\mathbb{R}^{4}$, where $h_{ij}$ are components of the second fundamental form, $S=\sum h_{ij}^{2}$, $f_{3}=\sum h_{ij}h_{jk}h_{ki}$ and $f_{4}=\sum h_{ij}h_{jk}h_{kl}h_{li}$. As an application, we obtain a classification result.