On maximal surfaces in the space of oriented geodesics of hyperbolic 3-space

2010/01/13 by Nikos Georgiou, Georgiou, Nikos
Mathematics · #51M09 #51M30 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.1001.2179

openalex publication_date 2010/01/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study area-stationary, or maximal, surfaces in the space \mathbb L(\mathbb H³) of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. We prove that every holomorphic curve in \mathbb L(\mathbb H³) is a maximal surface. We then classify Lagrangian maximal surfaces Σ in \mathbb L(\mathbb H³) and prove that the family of parallel surfaces in \mathbb H³ orthogonal to the geodesics γ∈Σ form a family of equidistant tubes around a geodesic.

Citations

• On the partial differential equations of mathematical physics

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