A rigidity theorem of self-expander

TL;DR

This paper classifies 3-dimensional complete self-expanders in Euclidean space with constant second fundamental form norm and third fundamental form component, providing a comprehensive rigidity theorem for these geometric objects.

Contribution

It offers a complete classification of 3D self-expanders with constant second fundamental form norm and third fundamental form component, advancing understanding of their geometric structure.

Findings

Complete classification of such self-expanders.

Identification of specific geometric conditions for rigidity.

Extension of previous results in self-expander theory.

Abstract

In this paper, we completely classify $3$-dimensional complete self-expanders 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 $f_{3}=\sum_{i,j,k}h_{ij}h_{jk}h_{ki}$.