2-dimensional complete self-shrinkers in $\mathbf{R}^3$

TL;DR

This paper classifies 2-dimensional complete self-shrinkers in three-dimensional Euclidean space without assuming polynomial volume growth, expanding understanding of their geometric properties and providing new examples.

Contribution

It removes the polynomial volume growth assumption from the classification of self-shrinkers, offering a complete classification under a broader condition.

Findings

Examples of complete self-shrinkers without polynomial volume growth

Complete classification of 2D self-shrinkers with constant second fundamental form norm

Extension of previous results by removing volume growth restrictions

Abstract

It is our purpose to study complete self-shrinkers in Euclidean space. First of all, we show some examples of complete self-shrinkers without polynomial volume growth. By making use of the generalized maximum principle for $\mathcal{L}$-operator, we give a complete classification for 2-dimensional complete self-shrinkers with constant squared norm of the second fundamental form in $\mathbb R^3$. In \cite{DX2}, Ding and Xin have proved this result under the assumption of polynomial volume growth, which is removed in our theorem.