Mathematical Proceedings of the Cambridge Philosophical Society

Curvature properties of zero mean curvature surfaces in four-dimensional Lorentzian space forms

a1 Departamento de Matemáticas, Universidad de Murcia, 30100 Espinardo, Murcia, Spain; e-mail:
a2 Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, England; e-mail:


We study the global behaviour of Gaussian curvature K and normal curvature K[bot bottom] of zero mean curvature spacelike surfaces (stationary surfaces) in a four-dimensional Lorentzian space form L4(c). In particular, we show that the only complete stationary surfaces in Minkowski space E41 with K[gt-or-equal, slanted]0 are those with K[identical with]0[identical with]K[bot bottom] and we give an explicit description of them. More general results are obtained for stationary surfaces in L4(c). We also discuss applications to Willmore surfaces in both Lorentzian and Riemannian three spaces. We give new examples of complete stationary surfaces in E41 with finite total curvature.

(Received November 29 1996)