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We study the conformal geometry of an oriented space-like surface in three-dimensional Lorentzian space forms. After introducing the conformal compactification of the Lorentzian space forms, we define the conformal Gauss map which is a conformally invariant two parameter family of oriented spheres. We use the area of the conformal Gauss map to define the Willmore functional and derive a Bemstein type theorem for parabolic Willmore surfaces. Finally, we study the stability of maximal surfaces for the Willmore functional.Mathematics Subject Classifications (1991): 53A30, 53C50.

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.

We study the stability of capillary surfaces without gravity for anisotropic free surface energies. For a large class of rotationally symmetric energy functionals, it is shown that the only stable equilibria supported on parallel planes are either cylinders or a part of the Wulff shape.

Abstract. We consider the functional of a hypersurface, given by a convex elliptic integrand with a volume constraint. We show that, up to homothety and translation, the only closed, oriented, stable critical point is the Wulff shape.