Parallel mean curvature surfaces in four-dimensional homogeneous spaces

摘要

We survey different classification results for surfaces with parallel meancurvature immersed into some Riemannian homogeneous four-manifolds, includingreal and complex space forms, and product spaces. We provide a common frameworkfor this problem, with special attention to the existence of holomorphicquadratic differentials on such surfaces. The case of spheres with parallelmean curvature is also explained in detail, as well as the state-of-the-artadvances in the general problem.