The classification of the surfaces with parallel mean curvature vector in two-dimensional complex space forms

In this paper we find new surfaces with parallel mean curvature vectors in two-dimensional complex space forms that are not of constant Kähler angles and then classify all of them. The surface can be isometrically deformed preserving both the length of the mean curvature vector and the Kähler angle. From this classification, the deformations appear only when the curvature of the ambiant space is nonpositive.