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

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.

Timelike surfaces with zero mean curvature in Minkowski 4-space

“…In [1] it is proved that the spacelike surfaces with H = 0 in Minkowski space R 4 1 are described by a system similar to (23): Thus the study of spacelike surfaces with zero mean curvature free of flat points is equivalent to the study of solutions of system (26).…”

“…with the property that the restriction of the metric , onto the tangent space T p M 2 is of signature (1,1), and the restriction of the metric , onto the normal space N p M 2 is of signature (2,0).…”

Minimal Lorentz surfaces in pseudo-Euclidean 4-space with neutral metric

“…Our aim is to characterize the minimal Lorentz surfaces in terms of a pair of smooth functions satisfying a system of two natural partial differential equations. This aim is motivated by similar results concerning minimal surfaces in the four-dimensional Euclidean space E 4 (see [14]) and spacelike or timelike surfaces with zero mean curvature in the Minkowski 4-space E 4 1 (see [1] and [8]). Our approach to the study of minimal Lorentz surfaces in E 4 2 is based on the introducing of special geometric parameters which we call canonical parameters.…”

Complete minimal hypersurfaces in with zero Gauss–Kronecker curvature

“…This class will play a crucial role in the classification of complete minimal hypersurfaces in the hyperbolic space with zero GaussKronecker curvature. We remind here that the totally geodesic submanifolds of S The following result is due to Alias and Palmer [1]. For the sake of completeness we give another short proof.…”

“…It is well known (cf. [1]) that there exists a holomorphic quadric differential on M 2 , the so called Hopf differential. A point x ∈ M 2 is a zero of the Hopf differential if and only if K (x) = 1 and K ⊥ (x) = 0.…”