Classification of spatial surfaces with parallel mean curvature vector in pseudo-Euclidean spaces of arbitrary dimension

Abstract

Spatial surfaces with parallel mean curvature vector play some important roles in general relativity, theory of harmonic maps, as well as in differential geometry. Recently, spatial surfaces in four-dimensional Minkowski space time with parallel mean curvature vector were classified by Chen and Van der Veken [“Complete classification of parallel surfaces in 4-dimensional Lorentzian space forms,” Tohoku Math. J. 61, 1 (2009)]. In this article, we completely classify spatial surfaces with parallel mean curvature vector in pseudo-Euclidean spaces of arbitrary dimension. The main result states that there exist 16 families of such surfaces. As by-product, we achieve the complete classification of spatial surfaces with parallel mean curvature vector in Minkowski space times of arbitrary dimension.