Graphs with constant mean curvature in the 3-hyperbolic space

Abstracts

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 rho 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.

hyperbolic space; geodesic and horizontal graphs; constant mean curvature; elliptic partial differential equations

Consideramos superfícies com curvatura média constante no 3-espaço hiperbólico que sãodadas como gráfico de uma funçãosuave definida em um aberto limitado e simplesmente conexo contido em um hiperplano totalmente geodésico. Dos vários tipos de gráficos que podemos definir no espaço hiperbólico consideramos em particular o gráfico horizontal e o geodésico. Provamos que se a curvatura média é constante, entãotais gráficos são equivalentes no seguinte sentido: suponha que M é uma superfície de curvatura média constante no 3-espaço hiperbólico tal que M é o gráfico geodésico de uma função ro que se anula no bordo do seu domínio, entãoexiste uma outra funçãosuave f que também se anula no bordo e tal que M é o gráfico horizontal de f. Além disso, a recíproca é verdadeira.

espaço hiperbólico; gráfico geodésico e horizontal; curvatura média constante; equações diferenciais parciais elípticas

Graphs with constant mean curvature in the 3-hyperbolic space

PEDRO A. HINOJOSA

CCEN - Departamento de Matemática, Universidade Federal da Paraíba

58051-900 João Pessoa, PB, Brasil and

Pós Graduação em Matemática, Universidade Federal do Ceará, Centro de Ciências

Campus do Pici, Bloco 914, 60455-760 Fortaleza, CE, Brasil

Manuscript received on May 3, 2002; accepted for publication on June 10, 2002;

presented by J. LUCAS BARBOSA

ABSTRACT

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 r 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.

Key words: hyperbolic space, geodesic and horizontal graphs, constant mean curvature, elliptic partial differential equations.

To describe the hyperbolic space we consider the half-space model, that is, : = {(x₁, x₂, x₃) Î : x₃ > 0} with the metric given by .

Let be a totally geodesic hyperplane and W Ì an open, simply connected and bounded domain. We consider r : ® a real smooth function. Without loss of generality we fix : = {x₂ = 0}.

The horizontal graph of r in is the following set

(1)

Horizontal graphs with constant mean curvature in 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(r) has constant mean curvature H in , then rsatisfies the following partial differential equation

(2)

where r is the euclidean gradient of r and

The geodesic graph of r in is defined as the set

(3)

Little is known about geodesic graphs with constant mean curvature in . Some results can be found in (Nelli and Semmler 1999) and (Semmler 1997). When G_g(r) has mean curvature H in the function r satisfies the following equation

(4)

where W² = cosh²r+ v²½r½² and ½r½²=

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 W, then the horizontal and geodesic graphs of functions which are zero at the boundary of W 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 r.

THEOREM 1. Let M Í be a surface of constant mean curvature H ¹ 0 given as the geodesic graph of a smooth function r : ® such that r | _¶W = 0, that is, M = G_g(r). Then there exists f : ® with r | _¶W º 0 such that M = G_h(f).

SKETCH OF THE PROOF. If M is not a horizontal graph, then there exists p Î such that the horocicle L_p : = {p + t(0, 1, 0);t Î } intersects M at least in two points q₁ and q₂. But, as observed above, we can consider r ³ 0, so that q₁ and q₂ will be obtained with t > 0. Let a : = max{x₂ : (x₁, x₂, x₃) ÎM} the maximum value of x₂ when (x₁, x₂, x₃) ÎM. Note that 0 < a < ¥ since M ¹ W and M is compact. Now set

Note that (¶ \) È M is a topological closed submanifold in , then it bounds an open domain in .

For any real number t, lets P_t be a totally geodesic plane in given by P_t : = {x₂ = t}, M_t : = {(x₁, x₂, x₃) ÎM : x₂ > t} and the reflection of M_t with respect to the plane P_t.

It is clear that we have M_t ¹ f for 0 £ t < a. If t < a and a - t is sufficiently small, then Ì . Set t₀ : = inf{b Î : Ì ,"_t Î(b, a)}.

The existence of q₁ and q₂ in M Ç L_p implies t₀ > 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 ³ t₀, we have Ç P_-a = f. In particular, Ç P_-a = f. Moreover, Ç (¶ \ (P_-aÈ P₀)) = f, since r | _¶W = 0. Thus we have Ç M = f. Therefore there exists p₀Î such that Î Ç M, where denotes the reflection of p₀ with respect to the plane .

The following are all the possibilities for the relative positions of the points p₀ and .

1. p₀Î int (M_t0) and Î_¶M;

2. p₀Î int (M_t0) and Î int(M);

3. p₀ = Î_¶().

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 W Î be a bounded domain, such that the curve G := ¶W has geodesic curvature greater than or equal to 1. Let r, f : ® be functions such that r | _¶W = f | _¶W º 0. Denote by G : = G_g(r) and : = G_h(f) their graphs and by h e H their mean curvatures respectively. Represent by the domain of bounded by È where is the reflection of with respect to the plane . With this notation, we have the following proposition.

PROPOSITION 2. Suppose that H is a positive constant satisfying | h| < H. If h is constant or | h| < 1, we have G Ì .

Assuming that the geodesic curvature of ¶W is greater than or equal to one, the previous proposition allows to estimate |r| by |

f|. Moreover, it is easy to see that r is bounded in by a constant which does not depend on h. In fact, | sinh (r) | £ .

Having the estimate in the boundary given by the above proposition, it is necessary now to get a global estimate in W. That is given by the following lemma.

LEMMA 3. Let r, H : ® be real functions of class C³ and C¹, respectively, such that

where W² = cosh²r+ v²|r|² and |r|² = Let us assume that r is bounded in W and |r| is bounded in ¶W. Then |r| is bounded in W by a constant that depends only on and .

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.

THEOREM 4. Let W Ì be a bounded domain such that ¶W has curvature greater than or equal to 1. Let f : ® be a smooth function such that f | _¶W º 0 and = G_h(f) Ìbe the horizontal graph of f. Let us assume that has constant mean curvature H ¹ 0. Then there exists r : ® such that r| _¶W º 0 and the geodesic graph of r, G = G_g(r), has also mean curvature H.

COROLLARY 5. Let Ì be a horizontal graph of a smooth function f : Ì ® such that f | _¶W º 0, where W Ì is a bounded domain such that ¶W has curvature greater than or equal to 1. Let us assume that has constant mean curvature H ¹ 0, then there exists a smooth function r that is zero at the boundary ¶W and for which = G_g(r).

We are going to sketch the proof of the Theorem 4:

The existence of such function r: ® is equivalent to the existence of a solution for the following Dirichlet problem:

(5)

where W² = cosh²r + v²|r|² and |r|² = .

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 Î [0, 1] as follows.

The operator Q can be written in the following form

where

A¹¹ = cosh²r + v², A¹² = - 2v²r_ur_v, A²² = cosh²r + v²

and

B = - (vr_v + sinh r cosh r)|r|².

Now we consider the functions , : [0, 1] ® [0, 1] given by,

and

as well as the family of operators Q_t defined by

We have also the associated Dirichlet problems

(6)

In this way each solution of the Dirichlet problem given by the equations (6) represents a geodesic graph of mean curvature H_t, where

(7)

Therefore, we have

Nevertheless, we prove that

In this way

(8)

By comparison between the values of H_t and H while t varies we obtain the C¹ bound in the boundary, through the Proposition 2, for the solutions of the Dirichlet problems (6). The Lemma 3 supplies the C¹ uniform estimates in W, that is, there exists a constant C₁ > 0 such that

for any solution C² to the problem (6), where the constant C₁ is independent of t and H_t.

So, there exists a constant C > 0 independent of r and t such that any solution C^2,a ()of the Dirichlet problems (6) satisfies a priori

The Theorems 11.8 and 13.7 in (Gilbarg and Trudinger 1983) guarantee the existence of r Î C^2,a () 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.

RESUMO

Consideramos superfícies com curvatura média constante no 3-espaço hiperbólico que sãodadas como gráfico de uma funçãosuave definida em um aberto limitado e simplesmente conexo contido em um hiperplano totalmente geodésico. Dos vários tipos de gráficos que podemos definir no espaço hiperbólico consideramos em particular o gráfico horizontal e o geodésico. Provamos que se a curvatura média é constante, entãotais gráficos são equivalentes no seguinte sentido: suponha que M é uma superfície de curvatura média constante no 3-espaço hiperbólico tal que M é o gráfico geodésico de uma função r que se anula no bordo do seu domínio, entãoexiste uma outra funçãosuave f que também se anula no bordo e tal que M é o gráfico horizontal de f. Além disso, a recíproca é verdadeira.

Palavras-chave: espaço hiperbólico, gráfico geodésico e horizontal, curvatura média constante, equações diferenciais parciais elípticas.

Correspondence to: Universidade Federal do Ceará

E-mail: [e-mail]

• BARBOSA JLM AND EARP RS. 1997. Geometric Methods of Nonlinear Analysis Applied to Hypersurfaces Studies in II, preprint.
• BARBOSA JLM AND EARP RS. 1998a. Geometric Methods and Nonlinear Analysis in Hyperbolic Space. X Escola de Geom Dif UFMG, julho 1998.
• BARBOSA JLM AND EARP RS. 1998b. Prescribed Mean Curvature Hypersurfaces in with Convex Planar Boundary I. Geom Dedicata 71: 61-74.
• CAFFARELLI L, NIRENBERG L AND SPRUCK J. 1988. Nonlinear Second-Order Elliptic Equations V. The Dirichlet Problem for Weingarten Hypersurfaces. Comm on Pure and Applied Math, vol XLI: 47-70.
• GILBARG D AND TRUDINGER NS. 1983. Elliptic Partial Differential Equations of Second Order. 2nd edition, Springer-Verlag, Berlin.
• HINOJOSA PA. 2000. Unicidade de Superfícies com Curvatura Média Constante em e . Tese de Doutorado, Universidade Federal do Ceará.
• NELLI B AND SEMMLER B. 1999. Some Remarks on Compact Constant Mean Curvature Hypersurfaces in Halfspace of J Geom 64: 128-140.
• SEMMLER B. 1997. Surfaces de Courbure Moyenne Constante dans les Espaces Euclidien et Hiperbolique. Thèse de Doctorat, Université Paris 7.

Publication Dates

• Publication in this collection
  09 Oct 2002
• Date of issue
  Sept 2002

History

• Accepted
  10 June 2002
• Received
  03 May 2002