Marginally trapped surfaces in spaces of oriented geodesics

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.Comment: 15 pages, 1 figur

On the Gauss map of equivariant immersions in hyperbolic space

“…In Theorem C, the conclusion that 𝐺 = 𝐺 𝜎 for 𝜎 a proper embedding follows from the fact that 𝜎 is complete, which is an easy consequence of Equation (24) relating the first fundamental forms of 𝜎 and 𝐺 𝜎 , and the non-trivial fact that complete immersions in ℍ 𝑛+1 with small principal curvatures are proper embeddings (Proposition 4.15).…”

“…The condition that the Gauss map is a Riemannian immersion is equivalent to I(𝑊, 𝑊) > 0 for every 𝑊 ≠ 0. By Equation (24), this is equivalent to ‖𝐵(𝑊)‖ Recall that we introduced in Definition 3.4 the normal evolution 𝜎 𝑡 of an immersion 𝜎 ∶ 𝑀 → ℍ 𝑛+1 , for 𝑀 an oriented 𝑛-manifold. An immediate consequence of Proposition 4.2 is the following.…”

“…In the groundbreaking paper [28], Hitchin observed the existence of a natural complex structure on the space of oriented geodesics in Euclidean three‐space. A large interest has then grown on the geometry of the space of oriented (maximal unparameterized) geodesics of Euclidean space of any dimension (see [24, 26, 38, 40]) and of several other Riemannian and pseudo‐Riemannian manifolds (see [1, 4, 7, 14, 44]). In this paper, we are interested in the case of hyperbolic $$n$$‐space $$\mathbb {H}^n$$, whose space of oriented geodesics is denoted here by $$\mathcal {G}(\mathbb {H}^{n})$$.…”

Marginally trapped submanifolds in space forms with arbitrary signature

“…have been studied, while Lagrangian marginally trapped surfaces of complex space forms of complex dimension two were characterized in [Chen and Dillen 2007]. Recently marginally trapped surfaces of certain spaces of oriented geodesics have been investigated [Georgiou and Guilfoyle 2014].…”