Abstract

In this paper, the representation formula of maximal surfaces in a 3-dimensional lightlike cone Q3 is obtained by making use of the differential equation theory and complex function theory. Some particular maximal surfaces under a special induced metric are presented explicitly via the representation formula.

Highlights

  • There are very special submanifolds in some event horizons of the compact Cauchy horizons of Taub–NUT, which are called null submanifolds due to the degenerate induced metrics.Null submanifolds often seem to be some smooth parts of the achronal boundaries in general relativity.the degenerate submanifolds of Lorentzian manifolds may be of great use to explore the intrinsic structure of manifolds with degenerate metrics.It is well known that the pseudo Riemannian space forms are non-degenerate and complete pseudo Riemannian hypersurface with zero, positive, or negative constant sectional curvature, which are consisted by the pseudo Euclidean space Enq, the pseudo Riemannian sphere Snq (c, r ) and the pseudo Riemannian hyperbolic space Hnq (c, r )

  • Some geometers and experts have studied the geometry of submanifolds in degenerate pseudo Riemannian space form [3,4,5,6]

  • The surface x (u, v) can be written as x (u, v) =, which is a maximal surface with isothermal parameters;

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Summary

Introduction

There are very special submanifolds in some event horizons of the compact Cauchy horizons of Taub–NUT, which are called null submanifolds due to the degenerate induced metrics. Some geometers and experts have studied the geometry of submanifolds in degenerate pseudo Riemannian space form [3,4,5,6]. Taking the trapped surface as an example, which is a compact, spacelike 2-dimensional submanifold of spacetime on which the divergence of the outgoing null vector orthogonal to the surface converges. A spacelike submanifold is said to be a marginally trapped submanifold in the pseudo. Y. Chen tested the cone surface of Q3 is marginally trapped in E41 iff it is flat [7], one of the authors of this paper proved that the surface in Q3 is flat if the surface is maximal [3,6].

Preliminaries
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