Title: Parallel mean curvature surfaces in four-dimensional homogeneous spaces,
Abstract: We survey different classification results for surfaces with parallel mean
curvature immersed into some Riemannian homogeneous four-manifolds, including
real and complex space forms, and product spaces. We provide a common framework
for this problem, with special attention to the existence of holomorphic
quadratic differentials on such surfaces. The case of spheres with parallel
mean curvature is also explained in detail, as well as the state-of-the-art
advances in the general problem.