An invariant theory of marginally trapped surfaces in the four-dimensional Minkowski space

TL;DR

This paper develops an invariant theory for marginally trapped surfaces in four-dimensional Minkowski space, characterizing them through seven invariants and classifying all such surfaces among meridian surfaces.

Contribution

It introduces a set of seven invariants that uniquely determine marginally trapped surfaces up to isometries in Minkowski space and classifies all such surfaces within meridian surfaces.

Findings

Seven invariants uniquely determine the surface.

Classification of all marginally trapped meridian surfaces.

Explicit description of these surfaces in Minkowski space.

Abstract

A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean curvature vector is lightlike at each point. We associate a geometrically determined moving frame field to such a surface and using the derivative formulas for this frame field we obtain seven invariant functions. Our main theorem states that these seven invariants determine the surface up to a motion in Minkowski space. We introduce meridian surfaces as one-parameter systems of meridians of a rotational hypersurface in the four-dimensional Minkowski space. We find all marginally trapped meridian surfaces.