A Characterization of Weingarten Surfaces in Hyperbolic 3-space

We study 2-dimensional submanifolds of the space L(H) of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. Such a surface is Lagrangian iff there exists a surface in H orthogonal to the geodesics of Σ. We prove that the induced metric on a Lagrangian surface in L(H) has zero Gauss curvature iff the orthogonal surfaces in H are Weingarten: the eigenvalues of the second fundamental form are functionally related. We then classify the totally null surfaces in L(H) and recover the well-known holomorphic constructions of flat and CMC 1 surfaces in H. Recently the existence and uniqueness of a canonical neutral Kähler structure on the space L(H) of oriented geodesics of hyperbolic 3-space H has been established [3] [10]. The main purpose of this paper is to apply this work to the study of surfaces S in H. The oriented geodesics normal to S form a surface in L(H) which is Lagrangian with respect to this Kähler structure. In fact, the study of surfaces in H is equivalent, at least locally, to the study of Lagrangian surfaces in L(H). Special classes of surfaces in hyperbolic 3-space have been studied for many decades, with various constructions being developed and applied to the particular class of surfaces under consideration. For example, surfaces of constant mean curvature 1 have been investigated in [1], while flat surfaces have been treated in [5] [6] [9]. The common feature of these surfaces is that they are Weingarten [12]: they have some specified functional relationship between the eigenvalues of the second fundamental form. Other forms of this relationship have been considered [7] and uniqueness results obtained for Weingarten surfaces in R [2] [4]. In this paper we give a new characterization of the Weingarten condition for surfaces in H: Main Theorem: Let S ⊂ H be a C smooth oriented surface and Σ ⊂ L(H) be the Lagrangian surface formed by the oriented geodesics normal to S. Then S is Weingarten iff the Gauss curvature of the Lorentz metric induced on Σ by the neutral Kähler metric is zero. The proof of this result follows from a careful study of 2-dimensional submanifolds of L(H) using local coordinates, moving frames and the correspondence space. Date: 15th September, 2007. 1991 Mathematics Subject Classification. Primary: 51M09; Secondary: 51M30.