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

Published online by Cambridge University Press:  01 September 1998

LUIS J. ALÍAS
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, 30100 Espinardo, Murcia, Spain; e-mail: ljalias@fcu.um.es
BENNETT PALMER
Affiliation:
Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, England; e-mail: Bennett.Palmer@durham.ac.uk

Abstract

We study the global behaviour of Gaussian curvature K and normal curvature K 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[ges ]0 are those with K≡0≡K 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.

Type
Research Article
Copyright
Cambridge Philosophical Society 1998

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