Complete parallel mean curvature surfaces in two-dimensional complex space-forms

TL;DR

This paper classifies complete parallel mean curvature surfaces in two-dimensional complex space-forms, identifying unique solutions in positive curvature spaces and a family generated by elliptic functions in negative curvature spaces.

Contribution

It explicitly characterizes all complete parallel mean curvature surfaces in complex projective and hyperbolic planes, revealing uniqueness in positive curvature and a family in negative curvature.

Findings

Unique complete surfaces in positive curvature space

Family of surfaces generated by Jacobi elliptic functions in negative curvature

Complete classification of such surfaces in both space-forms

Abstract

The purpose of this article is to determine explicitly the complete surfaces with parallel mean curvature vector, both in the complex projective plane and the complex hyperbolic plane. The main results are as follows: When the curvature of the ambient space is positive, there exists a unique such surface up to rigid motions of the target space. On the other hand, when the curvature of the ambient space is negative, there are `non-trivial' complete parallel mean curvature surfaces generated by Jacobi elliptic functions and they exhaust such surfaces.