Surfaces of prescribed mean curvature vector in semi-Riemannian manifolds

Solving a second order, quasilinear initial value problem, we prove existence and uniqueness of both space-like and time-like immersions of prescribed mean curvature vector in general semi-Riemannian manifolds. Such surfaces arise for example as critical points of generalized area functionals as well as self-similar solutions of the mean curvature flow.