  Two necessary conditions for the induced metrics of parallel mean curvature
surfaces in a complex space form of complex two-dimension are observed. One is
similar to the Ricci condition of the classical surface theory in Euclidean
three-space and the other is related to the Liouville metric. Conversely, we
prove that a special type of the Liouville metric on a domain in the Euclidean
two-plane whose Gaussian curvature satisfies the differential equation similar
to the Ricci condition is explicitly determined by an elliptic function. We
have isometric immersions from a simply connected two-dimensional Riemannian
manifold with the special type of the Liouville metric satisfying the Ricci
condition to the complex hyperbolic plane with parallel mean curvature vector.
