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

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.