AFFINE TRANSLATION SURFACES IN 3-DIMENSIONAL EUCLIDEAN SPACE SATISFYING $ \Delta r_{i}=\lambda _{i}r_{i}$

Abstract

In this paper we study the affine translation surfaces in 3-dimensional Euclidean space $\mathbb{E}^{3}$ under the condition $\Delta r_{i}=\lambda _{i}r_{i}$, where $\lambda _{i}\in \mathbb{R}$ and $\Delta $ denotes the Laplace operator. We obtain the complete classification for those ones.