Marginally trapped surfaces in L4 and an extended Weierstrass-Bryant representation

TL;DR

This paper introduces a unified conformal representation for a class of spacelike surfaces in Lorentz-Minkowski 4-space, extending classical minimal, maximal, Bryant, and de Sitter surface representations.

Contribution

It provides a new extended Weierstrass-Bryant representation that encompasses several classical surface theories in different geometric contexts.

Findings

Unified conformal representation for spacelike surfaces in L^4.

Extension of classical surface representations to Lorentz-Minkowski 4-space.

Applicable to surfaces with lightlike or zero mean curvature vectors.

Abstract

We give a conformal representation in terms of meromorphic data for a certain class of spacelike surfaces in the Lorentz-Minkowski 4-space L^4 whose mean curvature vector is either lightlike or zero at each point. This representation extends simultaneously the Weierstrass representation for minimal surfaces in Euclidean 3-space and for maximal surfaces in the Lorentz-Minkowski 3-space, and the Bryant representation for mean curvature one surfaces in the hyperbolic 3-space and in the de Sitter 3-space.