Graphs with constant mean curvature in the 3 – hyperbolic space

In this work we will deal with disc type surfaces of constant mean curvature in the three dimensional hyperbolic space which are given as graphs of smooth functions over planar domains. From the various types of graphs that could be defined in the hyperbolic space we consider in particular the horizontal and the geodesic graphs. We proved that if the mean curvature is constant, then such graphs are equivalent in the following sense: suppose that M is a constant mean curvature surface in the 3-hyperbolic space such that M is a geodesic graph of a function ρ that is zero at the boundary, then there exist a smooth function f, that also vanishes at the boundary, such that M is a horizontal graph of f. Moreover, the reciprocal is also true.

To describe the hyperbolic space we consider the half-space model, that is, H 3 := {(x 1 , x 2 , x 3 ) ∈ R 3 : x 3 > 0} with the metric given by ds 2 = 1 x 2 3 dx 2 1 + dx 2 2 + dx 2 3 .Let P ⊂ H 3 be a totally geodesic hyperplane and ⊂ P an open, simply connected and bounded domain.We consider ρ : → R a real smooth function.Without loss of generality we fix P := {x 2 = 0}.The horizontal graph of ρ in H 3 is the following set G h (ρ) := (x 1 , ρ(x 1 , x 3 ), x 3 ) , (x 1 , x 3 ) ∈ . (1) Horizontal graphs with constant mean curvature in H 3 had been studied by Barbosa and Earp (see, for example, (Barbosa and Earp 1997), (Barbosa and Earp 1998a,b)).We know that if G h (ρ) has constant mean curvature H in H 3 , then ρ satisfies the following partial differential equation where ∇ρ is the euclidean gradient of ρ and W 2 ρ = 1 + |∇ρ| 2 .The geodesic graph of ρ in H 3 is defined as the set Little is known about geodesic graphs with constant mean curvature in H 3 .Some results can be found in (Nelli and Semmler 1999) and (Semmler 1997).When G g (ρ) has mean curvature H in H 3 the function ρ satisfies the following equation where Note that the equations ( 2) and ( 4) are elliptic and quasilinear, so the maximum principle is true for both.Consequently, if the mean curvature H (u, v) is different to zero at any point of , then the horizontal and geodesic graphs of functions which are zero at the boundary of lie in only one of the halfspaces determined by the plane of their boundary.In this case, it is possible to choose a sign for ρ.
Theorem 1.Let M ⊆ H 3 be a surface of constant mean curvature H = 0 given as the geodesic graph of a smooth function ρ : → R such that ρ | ∂ = 0, that is, M = G g (ρ) .Then there exists Sketch of the proof.If M is not a horizontal graph, then there exists p ∈ P such that the horocicle L p := {p + t (0, 1, 0); t ∈ R} intersects M at least in two points q 1 and q 2 .But, as observed above, we can consider ρ ≥ 0, so that q 1 and q 2 will be obtained with t > 0. Let For any real number t, lets P t be a totally geodesic plane in H 3 given by P t the reflection of M t with respect to the plane P t .
It is clear that we have M t = φ for 0 ≤ t < a.If t < a and a − t is sufficiently small, then The existence of q 1 and q 2 in M ∩ L p implies t 0 > 0. Now we prove that this leads to the existence of a plane of symmetry of M. That is impossible since M is contained in only one of the sides of P t .
For any t ≥ t 0 , we have where p * 0 denotes the reflection of p 0 with respect to the plane P t 0 .
The following are all the possibilities for the relative positions of the points p 0 and p * 0 .
1. p 0 ∈ int M t 0 and p * 0 ∈ ∂M; We prove that the above possibilities do not occur.Details can be found in (Hinojosa 2000).So, the assumption that M is not a horizontal graph leads to a contradiction.
In order to show that a horizontal graph is also a geodesic one we study a Dirichlet problem for the equation 4. The necessary a priori estimates for this approach are given in the following results.
Let ⊂ P be a bounded domain, such that the curve := ∂ has geodesic curvature greater than or equal to 1. Let ρ, f : → R be functions such that and G := G h (f ) their graphs and by h e H their mean curvatures respectively.Represent by W the domain of H 3 bounded by G ∪ G * where G * is the reflection of G with respect to the plane P. With this notation, we have the following proposition.
Assuming that the geodesic curvature of ∂ is greater than or equal to one, the previous proposition allows to estimate |∇ρ| by |∇f | .Moreover, it is easy to see that ρ is bounded in by a constant which does not depend on h.In fact, | sinh (ρ) | ≤ |f | inf {x 3 } .Having the estimate in the boundary given by the above proposition, it is necessary now to get a global estimate in .That is given by the following lemma.
Lemma 3. Let ρ, H : → R be real functions of class C 3 and C 1 , respectively, such that where The proof of this lemma involves a long calculation based upon a suitable change of coordinates suggested by a corresponding result in (Caffarelli et al. 1988).The complete proof of Lemma 3 and Proposition 2 can be found in (Hinojosa 2000).
The theorem below gives a result concerning existence of geodesic graphs.The main corollary of this theorem affirms that, in the case of constant mean curvature, a horizontal graph is also geodesic.This allows to show the mentioned equivalence for these kind of graphs.(ρ).
We are going to sketch the proof of the Theorem 4: The existence of such function ρ : → R is equivalent to the existence of a solution for the following Dirichlet problem: where In order to apply the Continuity Method as stated in (Gilbarg and Trudinger 1983) we construct a family of quasilinear eliptic operators Q t , t ∈ [0, 1] as follows.
The operator Q can be written in the following form where

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Now we consider the functions T 1 , T 2 : [0, 1] → [0, 1] given by, as well as the family of operators Q t defined by We have also the associated Dirichlet problems In this way each solution of the Dirichlet problem given by the equations (6) represents a geodesic graph of mean curvature H t , where Therefore, we have Nevertheless, we prove that In this way The Theorems 11.8 and 13.7 in (Gilbarg and Trudinger 1983) guarantee the existence of ρ ∈ C 2,α solution of the problem

ACKNOWLEDGMENTS
This work is part of my Doctoral Thesis at the Universidade Federal do Ceará -UFC.I want to thank my advisor, Professor J. Lucas Barbosa, and also Professor A.G. Colares, by the encouragement and many helpful conversations.Special gratefulness to my colleage J. Herbert for the innumerable conversations and help.This work was supported by a scholarship of CAPES.
us assume that ρ is bounded in and |∇ρ| is bounded in ∂ .Then |∇ρ| is bounded in by a constant that depends only on sup |ρ| ∂|∇ρ| .
Theorem 4. Let ⊂ P be a bounded domain such that ∂ has curvature greater than or equal to 1. Let f : → R be a smooth function such that f | ∂ ≡ 0 and G = G h (f ) ⊂ H 3 be the horizontal graph of f.Let us assume that G has constant mean curvature H = 0. Then there exists ρ : → R such that ρ| ∂ ≡ 0 and the geodesic graph of ρ, G = G g (ρ), has also mean curvature H.Corollary 5. Let G ⊂ H 3 be a horizontal graph of a smooth function f : ⊂ P → R such that f | ∂ ≡ 0, where ⊂ P is a bounded domain such that ∂ has curvature greater than or equal to 1. Let us assume that G has constant mean curvature H = 0, then there exists a smooth function ρ that is zero at the boundary ∂ and for which G = G g By comparison between the values of H t and H while t varies we obtain the C 1 bound in the boundary, through the Proposition 2, for the solutions of the Dirichlet problems (6).The Lemma 3 supplies the C 1 uniform estimates in , that is, there exists a constant C 1 > 0 such that |ρ| 1, := sup |ρ| + sup |∇ρ| ≤ C 1 for any solution C 2 to the problem (6), where the constant C 1 is independent of t and H t .So, there exists a constant C > 0 independent of ρ and t such that any solution C 2,α of the Dirichlet problems (6) satisfies a priori|ρ| C 1 < C .