The Dirichlet Problem in Hyperbolic Space
We consider the Dirichlet problem for the constant mean curvature equation in hyperbolic space \(\mathbb{H}^{3}\) . Due to the type of umbilical surfaces in \(\mathbb{H}^{3}\) as well as the different notions of graphs, there is a variety of problems of D
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The Dirichlet Problem in Hyperbolic Space
We consider the Dirichlet problem for the cmc equation in hyperbolic space H3 . Due to the type of umbilical surfaces in H3 as well as the different notions of graphs, there is a variety of problems of Dirichlet type. We may consider Killing graphs on a geodesic plane, such as those introduced in Sect. 10.4. Results of existence appear in [ALR12, DR05, GE05, Lir02b, Nit02, Zha05]. If the domain is unbounded, we may use the Perron method as in Chap. 9 (see [ET99, ET00]). For parabolic graphs, see [AM09, BE98a]. For other methods, such as geometric measure theory or a variational approach, see [AR97, Cus09, SS09]. In this chapter we study geodesic graphs defined in a domain Ω of a horosphere, a geodesic plane and an equidistant surface. In order to describe the techniques, we consider the Dirichlet problem when Ω is a bounded domain and the boundary curve is ∂Ω. As in Euclidean space, we shall prove the existence of such graphs provided there is a certain relation between H and the value of the mean curvature H∂Ω of ∂Ω as submanifold of Ω. The main results are taken from [Lop01a, LM99].
11.1 Preliminaries and Definitions Denote by Q a geodesic plane, an equidistant surface or a horosphere in H3 . Fix an orientation ξ on Q. We define a geodesic graph on a domain of Q that extends the given one in Definition 10.4.1. Definition 11.1.1 Let Ω ⊂ Q be a bounded domain and f ∈ C 2 (Ω) ∩ C 0 (Ω). The geodesic graph of f is the set {γq (f (q)) : q ∈ Ω}, where γq = γq (s) is the unique geodesic orthogonal to Ω at q in the direction of ξ(q) and s is the arc parameter. Note that each point p of the graph is determined by a unique point q ∈ Ω and the number f (q). This is because for an equidistant surface and for a horosphere, two geodesics through two distinct points of Ω and perpendicular to Ω do not intersect. If f = 0 along ∂Ω, the boundary of the graph is ∂Ω. R. López, Constant Mean Curvature Surfaces with Boundary, Springer Monographs in Mathematics, DOI 10.1007/978-3-642-39626-7_11, © Springer-Verlag Berlin Heidelberg 2013
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The Dirichlet Problem in Hyperbolic Space
When Q is a horosphere, this problem has received special attention because in the upper half-space model of H3 , a horosphere is given by a horizontal plane L(c) and a geodesic graph on a domain Ω ⊂ L(c) is a Euclidean graph of a function z = u(x, y). The function u satisfies an elliptic equation of divergence type given in Eq. (11.8) below. Moreover, we may assume that Ω is included in the plane z = 0 and u = 0 on ∂Ω. In such case, Ω ⊂ ∂∞ H3 and the graph represents a surface in H3 with non-empty asymptotic boundary. We refer to [DuHi94, DuHi95, DS94, EN96, ET01, GS00, Lie00, NeSp96, Sal89, Ton96]. In this chapter we work with the Minkowski model of H3 (see Sect. 10.6). Umbilical surfaces in this model are determined by the intersections between H3 as a hyperquadric of L4 with hyperplanes of R4 . Depending on the causal character of the hyperplane, we have the following description of umbilical surfac
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