Marginally trapped surfaces in spaces of oriented geodesics

TL;DR

This paper studies marginally trapped surfaces with null mean curvature in spaces of oriented geodesics of Euclidean and hyperbolic 3-space, revealing their properties and constructing explicit examples.

Contribution

It characterizes marginally trapped surfaces in these spaces, showing all rank one surfaces are marginally trapped and providing explicit constructions.

Findings

All rank one surfaces are marginally trapped.

Lagrangian rotationally symmetric sections are marginally trapped.

Constructed explicit families of marginally trapped surfaces.

Abstract

We investigate the geometric properties of marginally trapped surfaces (surfaces which have null mean curvature vector) in the spaces of oriented geodesics of Euclidean 3-space and hyperbolic 3-space, endowed with their canonical neutral Kaehler structures. We prove that every rank one surface in these four manifolds is marginally trapped. In the Euclidean case we show that Lagrangian rotationally symmetric sections are marginally trapped and construct an explicit family of marginally trapped Lagrangian tori. In the hyperbolic case we explore the relationship between marginally trapped and Weingarten surfaces, and construct examples of marginally trapped surfaces with various properties.