Gaussian Curvature of Graph-Like Surfaces in 3-Dimensional Hyperbolic Space

Received: October 8, 2011 Accepted: October 31, 2011 Published: February 1, 2012 doi:10.5539/jmr.v4n1p30 URL: http://dx.doi.org/10.5539/jmr.v4n1p30 Abstract In this paper, we investigate the Gaussian curvature of graph-like surfaces in 3-dimensional hyperbolic space. We prove that the graph defined by a radially symmetric function with nonpositive Gaussian curvature is necessarily a surface with Gaussian curvature zero.


Introduction
The differential geometry of surfaces (or hypersurfaces) in hyperbolic space has been studied by many authors in recent years.Lin (Lin, 1989) and Sa Earp & Toubiana (Sa Earp & Toubiana, 2000) investigated the minimal graphs in hyperbolic space.De Silva & Spruck (De Silva & Spruck, 2009), Lopez (Lopez, 2001) and Nitsche (Nitsche, 2002) studied more generalized graphs in hyperbolic space with constant or arbitrary mean curvature.Rosenberg & Spruck (Rosenberg & Spruck, 1994) and Guan et al. (Guan, Spruck & Szapiel, 2009;Guan & Spruck, 2010) investigated the Gaussian curvature and other curvatures of hypersurfaces in hyperbolic space.But still little is known about the Gaussian curvature of surfaces in hyperbolic space.In this paper.we will investigate the Gaussian curvature of graph-like surfaces in 3-dimensional hyperbolic space.
In this paper we use a model of hyperbolic space which was first introduced by Nitsche (Nitsche, 2002).We call this model the Nitsche's model of hyperbolic space.One can also refer to Zhang (Zhang, 2005) for an explanation of the Nitsche's model.Now we give a quick review of this model.Let H n+1 be the upper halfspace model of hyperbolic space with curvature −1, that is, equipped with the hyperbolic metric where x = (x 1 , x 2 , . . ., x n ).Define mapping where | • | denotes the Euclidean norm, and Then Ψ is a diffeomorphism.Let ḡ be the pull-back metric of the hyperbolic metric ds 2 of H n+1 .Then D n × R equipped with metric ḡ is the Nitsche's model of hyperbolic space.From now on we denote by H n+1 the Nitsche's model of hyperbolic space.
In this paper we only consider 3-dimensional hyperbolic space Let Ω be an open subset of D 2 and φ be a smooth function defined on Ω.
is a surface of H 3 .We call Σ the graph of φ (Σ is also called a graph-like surface of H 3 ).If Ω = B R , the open disc of R 2 with center o (the origin of R 2 ) and (Euclidean) radius R < 1, and φ(x) = Φ(r(x)) for some function Φ(r) (x ∈ Ω), where r = r(x) denotes the hyperbolic distance from o to x, then we say that φ is radially symmetric (around o).If φ is radially symmetric, the graph Σ of φ is called a radial graph.
Let Ω ⊂ D 2 , φ ∈ C ∞ (Ω) and Σ be the graph of φ.We recall the definition of Gaussian curvature K of graph Σ.Let ∇ and ∇ be the Riemannian connections of H 3 and Σ, respectively.For any p ∈ Σ and u, v ∈ T p Σ (the tangent space of Σ at p), define where U, V are vector fields on Σ such that U| p = u, V| p = v.Define l by where ξ denotes a unit normal vector field of Σ.We call l the real valued second fundamental form of Σ.The Gaussian curvature K of Σ at a point p ∈ Σ is defined by where u, v ∈ T p Σ is linear independent (Gallot et al., 1987).
The main result of this paper is the following: Remark It is a well-known fact that D n × {t} is a totally geodesic submanifold of hyperbolic space Zhang, 2005).So, in the above Main Theorem, the claim that φ is constant means that the graph Σ defined by φ is totally geodesic in H 3 .

Preliminaries
Consider the Nitsche's model of 3-dimensional hyperbolic Space H 3 = D 2 × R. Let ḡ be the hyperbolic metric of H 3 .For every t ∈ R, it is easy to see that ḡ| D 2 ×{t} is just the hyperbolic metric of the unit Poincaré disk D 2 : where |dx| 2 denotes the Euclidean metric on D 2 .Note that where ρ ∈ [0, 1) and r ∈ [0, ∞).(r, θ) are the hyperbolic geodesic polar coordinates on D 2 .Then the hyperbolic metric of the unit Poincaré disk D 2 can be expressed as g = (dr) 2 + sinh 2 r dθ 2 .
(4) (Chavel, 1984).Hence the hyperbolic metric of H 3 can be expressed as (1 − |x| 2 ) 2 dt 2 (by Eq. (3) and Eq. ( 4)) = (dr) 2 + sinh 2 r dθ 2 + cosh 2 r dt 2 . (5) Now we recall some notations about product manifolds (O'Neill, 1983).Consider the product D 2 × R. Let be the projections of the product D 2 × R. For any (x, t) ∈ D 2 × R, π −1 (x) and σ −1 (t) are called the fibers and the leaves of the product D 2 × R, respectively.A vector at a point of H 3 tangent to the leaf at this point is called to be horizontal, and tangent to the fiber at this point is called to be vertical.Then functions or vector fields on the factor spaces (D 2 and R) can be viewed, by lifts (horizontal or vertical lifts) via the above projections , as functions or vector fields on For details see (O'Neill, 1983).That is to say, if X is a vector field on D 2 , then there exists a unique horizontal vector field X * on H 3 = D 2 × R such that dπ(X * ) = X (we call X * the horizontal lift of X); but, for convenience, we will use the same symbol X to denote both the vector fields X * and X (the exact meaning of X can be known by the context.).In a similar way, for a vector field U on R, we also have a unique vertical vector field U * on H 3 such that dσ(U * ) = U.We also use U to denote both U and U * .

Proof of the Main Theorem
Proof of the Main Theorem: At first we calculate the unit normal vector field ξ of Σ. Define mapping Let ∂ r and ∂ θ be the coordinate vector fields of geodesic polar coordinates (r, θ) on D 2 .Define A calculation shows that where ∂ t denotes the coordinate vector field of R (here viewed as a vector field on H 3 ).For simplicity, from now on we let f = cosh r.Then one can verifies directly that the unit normal vector field of Σ can be chosen as following where grad φ denotes the gradient of φ in the hyperbolic metric of D 2 .Now we calculate the Christoffel symbols of the Poincaré disk D 2 with respect to the geodesic polar coordinates.For convenience, we rewrite ∂ r = ∂ 1 and where ) .