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

TL;DR

This paper refines the classification of certain surfaces in complex space forms, identifying conditions under which new surfaces are characterized by harmonic functions and constants, extending previous results.

Contribution

It provides a new condition for classifying parallel mean curvature surfaces in complex space forms, including previously unconsidered surfaces, based on harmonic functions and constants.

Findings

Surfaces depend on one harmonic function and five constants

Classification applies when the ambient space is non-flat

Mean curvature vector is non-zero and Kaehler angle is non-constant

Abstract

We give a condition under which the findings of the paper cited above work well and determine the surfaces that were not considered before. In this paper, we show that a parallel mean curvature surface of a general type in a complex two-dimensional complex space form depends on one real-valued harmonic function on the surface and five real constants if the ambient space is not flat, the mean curvature vector does not vanish, and the Kaehler angle is not constant.