On area stationary surfaces in the space of oriented geodesics of hyperbolic 3-space

Abstract

We study area-stationary surfaces in the space L(H 3 ) of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. We prove that every holomorphic curve in L(H 3 ) is an area-stationary surface. We then classify Lagrangian area-stationary surfaces in L(H 3 ) and prove that the family of parallel surfaces in H 3 orthogonal to the geodesics γ ∈ form a family of equidistant tubes around a geodesic. Finally we find an example of a two parameter family of rotationally symmetric area-stationary surfaces that are neither Lagrangian nor holomorphic.The last two decades has seen increasing interest in spaces L(M) of oriented geodesics of a manifold M, with particular attention to their rich geometric structure. In the case of the space L(E 3 ) of oriented affine lines in Euclidean 3-space this interest can be traced back over a hundred years to Weierstrass's construction of minimal surfaces [16] and Whittaker's solutions to the wave equation [17].A natural complex structure on L(E 3 ) was considered by Hitchin to construct monopoles in E 3 [11], and then Guilfoyle and Klingenberg understood that the canonical symplectic structure on L(E 3 ) is compatible with this complex structure [7], [8] and that the associated Kähler metric is of neutral signature. Salvai subsequently proved that this neutral Kähler metric is (up to addition of the round metric) the unique metric on L(E 3 ) that is invariant under Euclidean motions [12]. This Kähler structure has recently been used by Guilfoyle and Klingenberg to solve an 80 year old conjecture of Carathéodory [10].More recently, Anciaux, Guilfoyle and Romon [2] have studied Lagrangian area-stationary surfaces in TN, with N being an oriented Riemannian surface and the neutral Kähler structure generalising that of the space of oriented geodesics in Euclidean and Lorentzian 3-space.In addition, Salvai constructed a neutral Kähler metric on the space L(H 3 ) of oriented geodesics in hyperbolic 3-space [13], while the geometry of L(H 3 )

On the Gauss map of equivariant immersions in hyperbolic space

“…In this paper, we are interested in the case of hyperbolic 𝑛-space ℍ 𝑛 , whose space of oriented geodesics is denoted here by (ℍ 𝑛 ). The geometry of (ℍ 𝑛 ) has been addressed in [39] and, for 𝑛 = 3, in [21][22][23]25]. For the purpose of this paper, the most relevant geometric structure on (ℍ 𝑛 ) is a natural para-Kähler structure (𝔾, 𝕁, Ω) (introduced in [1,4]), a notion which we will describe in Subsection 1.4 of this introduction and more in detail in Subsection 2.3.…”

“…Let us start by defining our condition of small principal curvatures. Recall that the principal curvatures of an immersion of a hypersurface in a Riemannian manifold (in our case the ambient manifold is ℍ 𝑛+1 ) are the eigenvalues of the shape operator, which was defined in (21).…”

“…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 $$\mathcal {G}(\mathbb {H}^{n})$$. The geometry of $$\mathcal {G}(\mathbb {H}^{n})$$ has been addressed in [39] and, for $$n=3$$, in [21–23, 25]. For the purpose of this paper, the most relevant geometric structure on $$\mathcal {G}(\mathbb {H}^{n})$$ is a natural para‐Kähler structure $$(\mathbb {G}, \mathbb {J}, \Omega )$$ (introduced in [1, 4]), a notion which we will describe in Subsection 1.4 of this introduction and more in detail in Subsection 2.3.…”

On minimal Lagrangian surfaces in the product of Riemannian two manifolds

“…In particular, if S is a smoothly immersed surface in M, the set of oriented geodesics normal to S form a Lagrangian surface in L(M 3 ). A Lagrangian surface Σ in L(M 3 ) is G-minimal if and only if Σ is locally the set of normal oriented geodesics of an equidistant tube along a geodesic in M [3] [6] [12]. Oh in [16] has introduced a natural variational problem, apart from the classical variational problem of minimizing the volume functional in a homology class, consisting of minimizing the volume with respect to Hamiltonian compactly supported variations.…”