Complete biconservative surfaces in the hyperbolic space $\mathbb{H}^3$
Simona Nistor Cezar Oniciuc


Complete biconservative surfaces in the hyperbolic space ℍ^3]
Complete biconservative surfaces in the hyperbolic space ℍ^3



Faculty of Mathematics
 Al. I. Cuza University of Iasi

Bd. Carol I, 11 
 700506 Iasi, Romania nistor.simona@ymail.com

Faculty of Mathematics
 Al. I. Cuza University of Iasi

Bd. Carol I, 11 
 700506 Iasi, Romania oniciucc@uaic.ro

This work was supported by a grant of the “Alexandru Ioan Cuza” University of Iasi, within the Research Grants program, Grant UAIC, code GI-UAIC-2018-04

[2010]Primary 53A10; Secondary 53C40, 53C42




We construct simply connected, complete, non-CMC biconservative surfaces in the 3-dimensional hyperbolic space ℍ^3 in an intrinsic and extrinsic way. We obtain three families of such surfaces, and, for each surface, the set of points where the gradient of the mean curvature function does not vanish is dense and has two connected components. In the intrinsic approach, we first construct a simply connected, complete abstract surface and then prove that it admits a unique biconservative immersion in ℍ^3. Working extrinsically, we use the images of the explicit parametric equations and a gluing process to obtain our surfaces. They are made up of circles (or hyperbolas, or parabolas, respectively) which lie in 2-affine parallel planes and touch a certain curve in a totally geodesic hyperbolic surface ℍ^2 in ℍ^3.


[
    Simona Nistor, Cezar Oniciuc
    
================================



§ INTRODUCTION


In the last years the theory of biconservative submanifolds proved to be a very interesting research topic (see, for example, <cit.>). Since, in certain geometric contexts, finding biharmonic submanifolds is difficult, the interest in biconservative submanifolds, which generalize the biharmonic ones, has appeared naturally.

The biharmonic maps between two Riemannian manifolds (M^m,g) and (N^n,h) are characterized by the vanishing of the associated bitension field

    τ_2(φ)=-Δ^φτ(φ)-_g R^N(dφ,τ(φ))dφ,

and are critical points of the bienergy functional (see <cit.>).

When φ:(M^m,g)→(N^n,h) is an isometric immersion, i.e., M is a submanifold of (N,h), and φ is a biharmonic map, we say that M is a biharmonic submanifold. In this case, the biharmonic equation τ_2(φ)=0 splits into the tangent and normal part (see <cit.>). Submanifolds with (τ_2(φ))^⊤=0 are called biconservative submanifolds.

We note that submanifolds with divergence-free stress bienergy tensor are precisely the biconservative submanifolds (see <cit.>).

The biconservative submanifolds were studied for the first time in 1995 by Th. Hasanis and Th. Vlachos (see <cit.>). In that paper the biconservative hypersurfaces in the Euclidean space ℝ^n were called H-hypersurfaces and were fully classified in ℝ^3 and ℝ^4.

When the ambient space is a 3-dimensional space form N^3(c),  i.e., a 3-dimensional real space with constant sectional curvature c, it is easy to see that surfaces with constant mean curvature (CMC surfaces) are biconservative.  Indeed, a surface φ:M^2→ N^3(c) is biconservative if and only if

    A( f)=-f/2 f,

where A is the shape operator of M and f= A is its mean curvature function.

Therefore, we are interested in biconservative surfaces which are non-CMC, i.e.,f≠ 0 at any point of an open subset of M.

The explicit local parametric equations of biconservative surfaces in ℝ^3, 𝕊^3 and ℍ^3 were determined in <cit.> and <cit.>. We mention that, when the ambient space is ℝ^3, the result in <cit.> was reobtained in <cit.>. Also, some global and uniqueness results concerning biconservative surfaces in ℝ^3 and 𝕊^3 are given in <cit.>.

The aim of this paper is to obtain global results concerning non-CMC biconservative surfaces in the hyperbolic space ℍ^3. We start with a short section where we recall some known properties of biconservative surfaces in N^3(c) with a nowhere vanishing gradient of the mean curvature function. Then, in Section <ref>, working in an intrinsic way, we first construct a certain simply connected, complete abstract surface gluing by symmetry along their common boundary two abstract standard biconservative surfaces (see Theorem <ref>). These abstract standard biconservative surfaces were first determined in <cit.> but, in order to perform the gluing process, we change the coordinates to write the metric in the most appropriate form for our purpose. Then, we prove (in Theorem <ref>), that the above simply connected, complete abstract surface admits a unique biconservative immersion in ℍ^3. Moreover, for this immersion, f is different from zero on a dense set. We end the section by stating two conjectures. The first one claims the uniqueness of simply connected, complete, non-CMC biconservative surfaces in N^3(c), and the second one says that any compact biconservative surface in N^3(c) is CMC.

In Section <ref> we basically reobtain Theorem <ref> by constructing complete, non-CMC biconservative surfaces in ℍ^3. This construction is done using the images of the explicit parametric equations and a gluing process (see Theorem <ref>). More precisely, we begin with the known one-parameter family of standard biconservative surfaces with a nowhere vanishing f, indexed by a real constant, whose explicit parametric equations were given in <cit.>. Then, according to the sign of that constant, we obtain three families of complete, non-CMC biconservative surfaces in ℍ^3.

Conventions. We assume that all manifolds are connected and use the following sign conventions for the rough Laplacian acting on sections of φ^-1(TN) and for the curvature tensor field of N, respectively:

    Δ^φ=-_g(∇^φ∇^φ-∇^φ_∇)

and

    R^N(X,Y)Z=[∇^N_X,∇^N_Y]Z-∇^N_[X,Y]Z.


Acknowledgments. We would like to thank to  D. Fetcu and S. Moroianu for useful discussions and suggestions.



§ PRELIMINARIES
 



For the sake of completeness, we present some known results concerning biconservative surfaces in three-dimensional space forms N^3(c), that will be useful in the following sections.

First, we recall some properties of biconservative surfaces in N^3(c) with a nowhere vanishing f.


Let φ:M^2→ N^3(c) be a biconservative surface with f≠ 0 at any point of M. Then the Gaussian curvature K satisfies


  (i)
    
    K= A+c=-3f^2/4+c;


  (ii) f^2>0, i.e., c-K>0, K≠ 0 on M, and the level curves of K are circles in M with constant curvature
    
    κ=3| K|/8(c-K);


  (iii)

    (c-K)Δ K-| K|^2-8/3K(c-K)^2=0,

where Δ is the Laplace-Beltrami operator on M.



In particular, it follows that M is orientable and choosing H/|H| as the unit normal vector field, we have f>0.

From now on, we will assume that any abstract surface is oriented.


From (<ref>) we can see that the biconservative surfaces are closely related to the Ricci surfaces (see <cit.>) and the link between them was studied in <cit.>.


Next, we present the characterization theorem and an existence and uniqueness result concerning biconservative surfaces in N^3(c).


Let (M^2,g) be an abstract surface. Then M can be locally isometrically embedded in a space form N^3(c) as a biconservative surface with the gradient of the mean curvature different from zero everywhere if and only if the Gaussian curvature K satisfies c-K(p)>0, ( K)(p)≠ 0, for any p∈ M, and its level curves are circles in M with constant curvature

    κ=3| K|/8(c-K),

where c∈ℝ is a fixed constant.



Let (M^2,g) be an abstract surface and c∈ℝ an arbitrarily fixed constant. Assume that c-K>0 and K≠0 at any point of M, and the level curves of K are circles in M with constant curvature

    κ=3| K|/8(c-K).

Then, locally, there exists a unique biconservative embedding φ:(M^2,g)→ N^3(c). Moreover, the mean curvature function is positive and its gradient is different from zero at any point.


Next, we give some equivalent conditions with the hypothesis from the above theorem.


Let (M^2,g) be an abstract surface with Gaussian curvature K satisfying c-K(p)>0 and ( K)(p)≠ 0 at any point p∈ M, where c∈ℝ is arbitrarily fixed. Let X_1= K /| K| and X_2∈ C(TM) be two vector fields on M such that {X_1(p),X_2(p)} is a positively oriented basis at any point p∈ M. Then, the following conditions are equivalent:

  
  (i) the level curves of K are circles in M with constant curvature
  
    κ=3| K|/8(c-K)=3X_1K/8(c-K);

  
  (ii)
  
    X_2(X_1K)=0   and  ∇_X_2X_2=-3X_1K/8(c-K)X_1;

  
  (iii)
  
    ∇_X_1X_1=∇_X_1X_2=0,  ∇_X_2X_2=-3X_1K/8(c-K)X_1,  ∇_X_2X_1=3X_1K/8(c-K)X_2.

  
  (iv) the metric g can be locally written, as g=e^2σ(du^2+dv^2), where (u,v) are positively oriented local coordinates, and σ=σ(u) satisfies the equation
      
    σ^''=e^-2σ/3-ce^2σ

      and the condition σ^'>0; moreover, the solutions of the above equation, u=u(σ), are
      
    u=∫_σ_0^σdτ/√(-3e^-2τ/3-ce^2τ+a)+u_0,

      where σ is in some open interval I, σ_0∈ I and a,u_0∈ℝ are constants;






§ THE INTRINSIC APPROACH


From Theorem <ref> and Theorem <ref>, we have the following local intrinsic characterization of biconservative surfaces in three-dimensional space forms, and in particular in the hyperbolic space ℍ^3 (c=-1). That is, if we consider an abstract surface (M^2,g) with -1-K(p)>0 and ( K)(p)≠ 0 at any point p∈ M, then locally it admits a (unique) biconservative immersion in ℍ^3 with a nowhere vanishing gradient of the mean curvature, if and only if, locally, the metric g can be written as g(u,v)=e^2σ(u)(du^2+dv^2), where σ'(u)≠ 0, for any u, and u=u(σ) is given by

    u(σ)=∫_σ_0^σdτ/√(-3e^-2τ/3+e^2τ+a)+u_0,      a,u_0∈ℝ.

With the new coordinates (σ,v) the metric g can be locally written as

    g(σ,v)=e^2σ(1/-3e^-2σ/3+e^2σ+adσ^2+dv^2),

and we have a one parameter family of such metrics. In order to find a more convenient expression for the metric g, we will change the coordinates twice. First we take (σ,v)=(log(3^3/4/ξ),v), ξ>0. Denoting C_-1=a/√(3)∈ℝ, one obtains

    g_C_-1(ξ,v)=1/ξ^2(3/-ξ^8/3+C_-1ξ^2+3dξ^2+3√(3)dv^2).

The second change of coordinates is given by (ξ,v)=(ξ,θ/3^3/4), and we have

    g_C_-1(ξ,θ)=1/ξ^2(3/-ξ^8/3+C_-1ξ^2+3dξ^2+dθ^2),

where C_-1, θ∈ℝ and ξ is positive and belongs to an open interval such that -ξ^8/3+C_-1ξ^2+3>0. In order to determine the largest interval for ξ, we define the function T:(0,∞)→ℝ such that

    T(ξ)=-ξ^8/3+C_-1ξ^2+3

and we try to find where T is positive. By some standard computations, we come to the following three cases:

  
  * if C_-1>0, we get that there exists a unique point ξ_01 depending on C_-1, ξ_01>(3C_-1/4)^3/2 such that the function T vanishes at this point, T(ξ)>0 for any ξ∈(0,ξ_01) and T(ξ)<0 for any ξ∈(ξ_01,∞).
  
  * if C_-1<0, one obtains that there exists a point ξ_01>0 (we keep the same notation for the vanishing point) such that the function T vanishes at this point and T is positive on the interval (0,ξ_01).
  
  * if C_-1=0, one gets that there exists a point ξ_01=3^3/8 such that the function T vanishes at this point and T is positive on (0,ξ_01).

Therefore, in all cases, we have,

    g_C_-1(ξ,θ)=1/ξ^2(3/-ξ^8/3+C_-1ξ^2+3dξ^2+dθ^2),      (ξ,θ)∈(0,ξ_01)×ℝ,

and we get the following result.


Let (M^2,g(u,v)=e^2σ(u)(du^2+dv^2)) be an abstract surface, where u=u(σ) is given by

    u(σ)=∫_σ_0^σdτ/√(-3e^-2τ/3+e^2τ+a)+u_0,     σ∈ I,

where a and u_0 are real constants and I is an open interval. Then (M^2,g) is isometric to

    (D_C_-1,g_C_-1)=((0,ξ_01)×ℝ, g_C_-1(ξ,θ)=1/ξ^2(3/-ξ^8/3+C_-1ξ^2+3dξ^2+dθ^2)
    ),

where C_-1 is a real constant and ξ_01 is the positive vanishing point of -ξ^8/3+C_-1ξ^2+3.



We call the surface (D_C_-1,g_C_-1) an abstract standard biconservative surface, and, in fact, we have a one-parameter family of abstract standard biconservative surfaces indexed by C_-1.



We note that

    lim_ξ↘ 03/ξ^2(-ξ^8/3+C_-1ξ^2+3)= lim_ξ↗ξ_013/ξ^2(-ξ^8/3+C_-1ξ^2+3)=∞,

and therefore, the metric g_C_-1 blows up at the boundary given by ξ=0 and ξ=ξ_01.


The surface (D_C_-1,g_C_-1) is not complete since the geodesic θ=θ_0 cannot be defined on the whole ℝ but only on a half line, and by standard computations it can be proved that its Gaussian curvature is given by

    K_C_-1(ξ,θ)=K(ξ)=-ξ^8/3/9-1

and

    K'(ξ)=-8/27ξ^5/3<0.

Therefore

    K=ξ^2(-ξ^8/3+C_-1ξ^2+3)/3 K'(ξ)∂/∂ξ

does not vanish at any point of D_C_-1 and

    lim_ξ↗ξ_01( K)(ξ,θ)=0,     θ∈ℝ.


As the metric g_C_-1 is not complete, in order to obtain a complete one, denoted by g̃_C_-1, or simply g̃, we will change the coordinates again and then glue, in a simple way, two (isometric) metrics g_C_-1. So, if we consider the change of coordinates given by (ξ,θ)=(ξ(ρ),θ), one obtains

    g_C_-1(ρ,θ)=1/ξ^2(ρ)dθ^2+dρ^2,

where ξ=ξ(ρ) is the inverse function of ρ,

    ρ(ξ)=-∫_ξ_00^ξ√(3/τ^2(-τ^8/3+C_-1τ^2+3)) dτ,

ξ_00 being an arbitrarily fixed constant in (0,ξ_01).

We are allowed to consider the above change as ρ is a strictly decreasing function. Moreover, we have the following lemma.


The function ρ:(0,ξ_01)→ℝ defined by

    ρ(ξ)=-∫_ξ_00^ξ√(3/τ^2(-τ^8/3+C_-1τ^2+3)) dτ

satisfies

    lim_ξ↘ 0ρ(ξ)= ∞,     lim_ξ↗ξ_01ρ(ξ)=ρ_1,

where ρ_1 is a negative real constant.



In order to compute the first limit, we change the variable τ=1/τ̃ in the integral

    I=-∫_ξ_00^ξ√(3/τ^2(-τ^8/3+C_-1τ^2+3)) dτ,

and obtain

    I=∫_1/ξ_00^1/ξF(τ̃) dτ̃,

where

    F(τ̃)=√(3)τ̃^1/3/√(3τ̃^8/3+C_-1τ̃^2/3-1),     τ̃∈[1/ξ_00,∞),

for ξ<ξ_00.

Since

    lim_τ̃→∞τ̃F(τ̃)=1∈(0,∞],

it follows that

    lim_ξ↘ 0ρ(ξ)= ∞.

In order to compute the second limit, we first note that ρ(ξ) is negative for any ξ∈(ξ_00,ξ_01) and

    ρ(ξ)>-√(3)/ξ_00∫_ξ_00^ξ1/√(T(
    τ)) dτ,     ξ∈(ξ_00,ξ_01),

where T(ξ)=-ξ^8/3+C_-1ξ^2+3.

We have that that lim_ξ↗ξ_01ρ(ξ) is finite if and only if

    lim_ξ↗ξ_01∫_ξ_00^ξ1/√(T(τ)) dτ <∞.

To prove this, we rewrite the function T as

    T(ξ)=     T(ξ_01)+(ξ-ξ_01)(T'(ξ_01)+α_0(ξ)) 
    
      =    (ξ-ξ_01)T_1(ξ),

where α_0 is a continuous function such that lim_ξ↗ξ_01α_0(ξ)=0 and

    T_1(ξ)= -8/3ξ^5/3_01+2C_-1ξ_01+α_0(ξ),     ξ∈[ξ_00,ξ_01].

Then, we have

    lim_ξ↗ξ_01√(ξ_01-ξ)1/√(T(ξ))=  lim_ξ↗ξ_011/√(-T_1(ξ))∈[0,∞),

and we come to the conclusion that

    lim_ξ↗ξ_01ρ(ξ)=ρ_1∈ℝ^∗_-.



Denoting h̃(ρ)=1/ξ(ρ), the metric g_C_-1 can be rewritten as

    g_C_-1(ρ,θ)=h̃^2(ρ) dθ^2+dρ^2,       (ρ,θ)∈(ρ_1,∞)×ℝ.

We can obtain a simpler form of the domain, i.e., (0,∞)×ℝ, considering a new change of coordinates given by (ρ,θ)=(ρ(ω)=ω+ρ_1,θ). Therefore, we have

    g_C_-1(ω,θ)=h^2(ω) dθ^2+dω^2,       (ω,θ)∈(0,∞)×ℝ,

where h(ω)=h̃(ρ(ω)).

We note that

    lim_ω↘ 0h^2(ω)= 1/ξ_01^2∈ℝ^∗_+,

and thus, the metric g_C_-1 can be smoothly extended to the boundary ω=0.


As the limit of the function h when ω approaches 0 is 1/ξ_01≠ 0, it is easy to see that ([0,∞)×ℝ, g_C_-1) can be viewed as a surface with boundary.

In order to obtain a complete surface, we extend the surface ((0,∞)×ℝ, g_C_-1) by “symmetry” with respect to its boundary and get the following result.


The surface (ℝ^2,g̃_C_-1(ω,θ)=Γ^2(ω)dθ^2+dω^2) is complete,
where the function Γ:ℝ→ℝ is given by

    Γ(ω)={[   h(ω),     ω>0; ; 1/ξ_01,     ω=0; ;  h(-ω),     ω<0 ]..




By some standard computations it is easy to verify that the function Γ is at least of class C^3. In order to prove that the metric
g̃_C_-1 is complete, we first note that Γ(ω)≥ 1/ξ_01, for any ω∈ℝ, and then consider the metric

    g̃_0(ω,θ)=m_0(dθ^2+dω^2),      (ω,θ)∈ℝ^2,

where m_0 is the minimum between 1/ξ_01^2 and 1. As the metric g̃_0 is complete and g̃_C_-1-g̃_0 is non-negative at any point of the surface, it follows that g̃_C_-1 is complete (see <cit.>).



Since (K̃)(0,θ)=0, for any θ∈ℝ, where K̃ is the Gaussian curvature of (ℝ^2,g̃_C_-1), it follows that ∇_∂/∂θ∂/∂θ=0 along the boundary of ((0,∞)×ℝ, g_C_-1) and therefore this boundary becomes a geodesic in (ℝ^2,g̃_C_-1).



A similar construction is also possible when c=0 or c=1. In the first case, in order to obtain an abstract complete biconservative surface we glue two abstract standard biconservative surfaces, and in the second case, the gluing process must be performed infinitely many times. In fact, for c=0 we will reobtain Theorem 4.1.  from <cit.> (where the complete surface was obtained by working with isothermal coordinates), and for c=1 we will reobtain Proposition 4.17.  from <cit.> (where the main idea was that the abstract standard biconservative surface is isometric to a certain surface of revolution in the 3-dimensional Euclidean space ℝ^3).


We also note that, since the Gaussian curvature of the complete surface (ℝ^2,g̃_C_-1) satisfies (K̃_C_-1)(0,θ)=0, for any θ∈ℝ, the existence of a (non-CMC) biconservative immersion from (ℝ^2,g̃_C_-1) in ℍ^3 is not guaranteed. So, our aim is to construct such an immersion.

For the sake of simplicity, we will omit writing the index C_-1 in the following construction. Let us denote by

    ^1g(ω,θ)=h^2(ω)dθ^2+dω^2,      (ω,θ)∈ (0,∞)×ℝ

and

    ^2g(ω,θ)=h^2(-ω)dθ^2+dω^2,      (ω,θ)∈ (-∞,0)×ℝ.

It is easy to see that the Gaussian curvatures of the above two surfaces are given by

    ^1K(ω)=-h”(ω)/h(ω),  ω∈ (0,∞),      ^2K(ω)= ^1K(-ω)=-h”(-ω)/h(-ω), ω∈ (-∞,0)

and their derivatives are equal to

    ^1K'(ω)=-h”'(ω)h(ω)+h”(ω)h'(ω)/h^2(ω),     ω∈ (0,∞),


    ^2K'(ω)=- ^1K'(-ω)=h”'(-ω)h(-ω)-h”(-ω)h'(-ω)/h^2(-ω),     ω∈ (-∞,0),

respectively.

Let

    ^1X_1= ^1K/| ^1K|,       ^2X_1= ^2K/| ^2K|,

be two vector fields defined on (0,∞)×ℝ, respectively on (-∞,0)×ℝ. Clearly, as ^1K'>0, one obtains:

    ^1X_1=∂/∂ω and  ^2X_1=-∂/∂ω

on (0,∞)×ℝ, respectively on (-∞,0)×ℝ.

Now, let us define, on ℝ^2,

    X_1(ω,θ)={[     ^1X_1(ω,θ),  (ω,θ)∈ (0,∞)×ℝ; ;           ∂/∂ω,     (ω,θ)∈{0}×ℝ; ;    -^2X_1(ω,θ), (ω,θ)∈ (-∞,0)×ℝ ]..

Clearly, the vector field X_1 is given by X_1=∂/∂ω on ℝ^2.

Now, the vector field X_1 determines uniquely the global vector field X_2 by asking {X_1(ω,θ),X_2(ω,θ)} to be a positive orthonormal frame field in (ℝ^2,g̃), for any (ω,θ)∈ℝ^2. Obviously, X_2(ω,θ)=  ^1X_2(ω,θ), for any (ω,θ)∈ (0,∞)×ℝ, and X_2(ω,θ)=-  ^2X_2(ω,θ), for any (ω,θ)∈ (-∞,0)×ℝ, that is X_2(ω,θ)=1/Γ(ω)∂/∂θ on ℝ^2, where Γ is defined in (<ref>).

Further, we have the following properties of X_1 and X_2.


Let (ℝ^2,g̃) the above complete surface. Then, the Gaussian curvature K̃ of (ℝ^2,g̃) satisfies -1- K̃>0 at any point, and the vector fields X_1 and X_2 defined above, satisfy on ℝ^2

    ∇_X_1X_1=∇_X_1X_2=0,  ∇_X_2X_2=-3X_1K̃/8(-1-K̃)X_1,   ∇_X_2X_1=3X_1K̃/8(-1-K̃)X_2.




We recall that the surface ((0,∞)×ℝ, ^1g) has the following properties: -1-  ^1K>0, ^1K≠ 0 at any point, and on (0,∞)×ℝ one has

    ∇_^1X_1^1X_1=∇_^1X_1^1X_2=0, ∇_^1X_2^1X_2=-3 ^1X_1 ^1K/8(-1- ^1K) ^1X_1,
       ∇_^1X_2^1X_1=3 ^1X_1 ^1K/8(-1- ^1K) ^1X_2.

It is easy to see that ((-∞,0)×ℝ, ^2g) has the same properties as ((0,∞)×ℝ, ^1g).

Therefore, on (-∞,0)×ℝ, we have

    ∇_^2X_1^2X_1=∇_^2X_1^2X_2=0, ∇_^2X_2^2X_2=-3 ^2X_1 ^2K/8(-1- ^2K) ^2X_1,
       ∇_^2X_2^2X_1=3 ^2X_1 ^2K/8(-1- ^2K) ^2X_2.


From the definition of X_1 and X_2, we note that, on ℝ^∗×ℝ, we have

    ∇_X_1X_1=∇_X_1X_2=0,  ∇_X_2X_2=-3X_1K̃/8(-1-K̃)X_1,   ∇_X_2X_1=3X_1K̃/8(-1-K̃)X_2,

where

    K̃(ω,θ)=K̃(ω)={[          ^1K(ω),    (ω,θ)∈(0,∞)×ℝ; ;          ^2K(ω), (ω, θ)∈ (-∞,0)×ℝ ].,

K̃ being the Gaussian curvature of (ℝ^2,g̃). Then,

    K̃(0)=lim_ω↘ 0 ^1K(ω)=lim_ω↗ 0  ^2K(ω)=-1/9ξ_01^8/3-1.

Moreover, -1-K̃>0 on ℝ^2 and all the objects defined in (<ref>) are, in fact, defined on ℝ^2, and they are at least continuous. So, passing to the limit when ω approaches 0 we obtain that (<ref>) holds on whole ℝ^2.


Now, we can state the following existence and uniqueness result.


Let (ℝ^2,g̃) the above complete surface. Then, there exists a unique biconservative immersion Φ:(ℝ^2,g̃)→ℍ^3. Moreover, f≠ 0 at any point of ℝ^∗×ℝ, where f is the mean curvature function of the immersion Φ.



First, we note that, from Proposition <ref>, we have that the vector fields X_1 and X_2 on ℝ^2, previously defined, satisfy (<ref>) on ℝ^2.

In order to prove the existence of a biconservative immersion Φ:(ℝ^2,g̃)→ℍ^3, let us consider the operator A:C(Tℝ^2)→ C(Tℝ^2) defined by

    A(X_1)=-√(-1-K̃)/√(3)X_1,      A(X_2)=√(3(-1-K̃))X_2.

We will prove that A satisfies the Gauss and the Coddazi equations. Since the matrix of A with respect to {X_1,X_2} is

    A=
    (
    [ -√(-1-K̃)/√(3)              0;              0    √(3(-1-K̃)) ]),

it is easy to see that A=1+K̃, i.e., the Gauss equation is satisfied, and

    f= A=2/√(3)√(-1-K̃).

By some direct computations, also using (<ref>), one obtains that

    (∇_X_1A)(X_2)=(∇_X_2A)(X_1),

i.e., the Codazzi equation.

Therefore, from the fundamental theorem of surfaces in ℍ^3, it follows that there exists an unique isometric immersion Φ:(ℝ^2,g̃)→ℍ^3 such that A is its shape operator. Moreover, the operator A satisfies

    A( f)=-f/2 f,

which shows that Φ is biconservative.

Further, we will prove the uniqueness of biconservative immersions from (ℝ^2,g̃) in ℍ^3. Let Φ_1 and  Φ_2 two biconservative immersions from (ℝ^2,g̃) in ℍ^3. Obviously, Φ_1_|(0,∞)×ℝ. and Φ_1_|(-∞,0)×ℝ. are biconservative, and therefore using Theorem <ref>, it follows that f≠ 0 on ((0,∞)×ℝ,g̃) and on ((-∞,0)×ℝ,g̃) and these restrictions are unique (up to isometries of ℍ^3).

It follows that there exist two isometries ^1F and ^2F of ℍ^3, which preserve the orientation, such that

    Φ_2_|(0,∞)×ℝ.=  ^1F∘Φ_1_|(0,∞)×ℝ.

and

    Φ_2_|(-∞,0)×ℝ.=  ^2F∘Φ_1_|(-∞,0)×ℝ.

By continuity, one obtains that ^1F and ^2F coincide along the curve θ→Φ_1(0,θ), for any θ∈ℝ. By a straightforward computation, we have


    ^1F_∗,Φ_1(0,θ_0)(Φ_1∗,(0,θ_0)((∂/∂θ)_(0,θ_0)))=     ^2F_∗,Φ_1(0,θ_0)(Φ_1∗,(0,θ_0)((∂/∂θ)_(0,θ_0)))
    
    =    Φ_2∗,(0,θ_0)((∂/∂θ)_(0,θ_0)),

where θ_0∈ℝ is a given number.

We note that

    {[ Φ_2 (ω,θ_0)=  ^1F(Φ_1 (ω,θ_0)),                             ω>0; ; Φ_2 (ω,θ_0)=  ^2F(Φ_1 (ω,θ_0)),                             ω<0 ].,

and, again by continuity, we have

    {[ Φ_2 (ω,θ_0)=  ^1F(Φ_1 (ω,θ_0)),                            ω≥ 0; ; Φ_2 (ω,θ_0)=  ^2F(Φ_1 (ω,θ_0)),                            ω≤ 0 ]..

It follows that

    ^1F_∗,Φ_1(0,θ_0)(Φ_1∗,(0,θ_0)((∂/∂ω)_(0,θ_0)))=     ^2F_∗,Φ_1(0,θ_0)(Φ_1∗,(0,θ_0)((∂/∂ω)_(0,θ_0)))
    
    =    Φ_2∗,(0,θ_0)((∂/∂ω)_(0,θ_0)),

From equations (<ref>) and (<ref>), it is clear that ^1F_∗,Φ_1(0,θ_0) and ^2F_∗,Φ_1(0,θ_0) coincide on Φ_1∗,(0,θ_0)(ℝ^2). As ^1F and ^2F preserve the orientation of ℍ^3, it results that ^1F_∗,Φ_1(0,θ_0) and ^2F_∗,Φ_1(0,θ_0) coincide on [Φ_1∗,(0,θ_0)(ℝ^2)]^⊥. So,
^1F_∗,Φ_1(0,θ_0)= ^2F_∗,Φ_1(0,θ_0) and therefore, since ^1F and ^2F also agrees at least at one point, we come to the conclusion.



Similar results can be proved when c=0 or c=1. In fact, the existence part of these theorems were, essentially, already obtained in Theorem 4.1 and in Theorem 4.18 from <cit.>, respectively, by a direct construction.




We end this section with the following conjectures.



Conjecture 1. Let φ:M^2→ℍ^3 be a simply connected, complete, non-CMC biconservative surface. Then, up to isometries of the domain and codomain, M and φ are those given in Theorem <ref>.



In fact, Conjecture 1 applies for any 3-dimensional space form N^3(c).



Conjecture 2. Any compact biconservative surface in N^3(c) is CMC.




If the Conjecture 1 is true for N^3(c), the Conjecture 2 is also true for c=0 and for c=-1 (as we will see in the next section) by considering the universal cover and taking into account that the corresponding biconservative immersion is not double periodic. The only interesting case is c=1, where the domain quotients to a non-flat torus but it is not clear if the immersion is double periodic.

We note that the above conjectures were positively answered when the target manifold is the three-dimensional Euclidean space ℝ^3 and φ is an embedding (see <cit.>). Using the same technique, we expect the same kind of results to hold, i.e., if φ:(M^2,g)→ℍ^3 is an embedding and M is complete, then φ(M) is one of the complete biconservative surfaces constructed extrinsically in the next section.



§ THE EXTRINSIC APPROACH


The aim of this section is to construct complete, non-CMC biconservative surfaces in ℍ^3. The idea is to glue the images of two standard biconservative surfaces (the abstract domain of the two parametrizations remains the same), reobtaining basically, in an extrinsic way, the existence part of Theorem <ref>.

For the sake of completeness, we firs present some local extrinsic properties of biconservative surfaces.


Let M^2 be a biconservative surface in N^3(c) with a nowhere vanishing gradient of the mean curvature function f. Then, one has f>0 and

    fΔ f+| f|^2+4/3cf^2-f^4=0,

where Δ is the Laplace-Beltrami operator on M.


In the same paper <cit.>, it was proved that, from equation (<ref>), it follows that there exists a positively oriented local chart (U;u,v) such that f=f(u,v)=f(u) satisfies the following ODE:

    ff”=7/4f'^2+4/3cf^2-f^4.

Then, denoting κ(u)=f(u)/2, from equation (<ref>) one obtains that κ satisfies

    κκ”=7/4κ'^2+4/3cκ^2-4κ^4.

and

    κ'^2=-16/9cκ^2-16κ^4+C̃_-1κ^7/2,

where C̃_-1 is a real constant.

Next, we work in the 3-dimensional hyperbolic space ℍ^3, i.e., c=-1. As there exist several models for the hyperbolic space, in this paper we will consider, in each particular situation, the most appropriate model in order to obtain a complete biconservative surface.

We note that a local extrinsic characterization of biconservative surfaces in ℍ^3 was given in <cit.> where the authors considered the hyperboloid model of ℍ^3. Let us recall that the Minkowski space ℝ^4_1 is given by ℝ^4_1=(ℝ^4,⟨·,·⟩), where ⟨·,·⟩ is the bilinear form

    ⟨ x,y⟩=∑_i=1^3 x^iy^i-x^4y^4,      x=(x^1,x^2,x^3,x^4),   y=(y^1,y^2,y^3,y^4).

The hyperboloid model is

    ℍ^3={x∈ℝ^4_1   :   ⟨ x,x⟩=-1   and   x^4>0},

that is the the upper part of the hyperboloid of two sheets.

It is well known that the Levi-Civita connections ∇ of ℝ^4_1, and ∇^ℍ^3 of ℍ^3, are related by

    ∇_X Y=∇^ℍ^3_X Y+⟨ X,Y⟩ x,

for any X,Y∈ C(Tℍ^3) and x∈ℍ^3.

Further, let φ:M^2→ℍ^3 be a connected, oriented biconservative surface where on ℍ^3 we considered the Riemannian metric induced by the pseudo-Riemannian metric on ℝ^4_1. If we assume that f is nowhere vanishing and consider the global orthonormal frame field in ℍ^3 along M, {X_1= f/ | f|, X_2, η},  then the Levi-Civita connection ∇ on M, is given by

    [ ∇_X_1X_1=∇_X_1X_2=0,     ∇_X_2X_1=-3X_1f/4fX_2,     ∇_X_2X_2=3X_1f/4fX_1. ]


Using Gauss formula for M in ℍ^3, (<ref>) and (<ref>), by a straightforward computation we obtain

    [ ∇^ℍ^3_X_1X_1=-f/2η,     ∇^ℍ^3_X_2X_1=-3X_1f/4fX_2,     ∇^ℍ^3_X_1X_2=0,; ;                                         ∇^ℍ^3_X_2X_2=3X_1f/4fX_1+3f/2η ]

and then, using Gauss formula for ℍ^3 in ℝ^4_1 and (<ref>) we get

    [ ∇_X_1X_1=-f/2η +x,    ∇_X_2X_1=-3X_1f/4fX_2,     ∇_X_1X_2=0,; ;                                ∇_X_2X_2=3X_1f/4fX_1+3f/2η+x, ]

where x∈ℍ^3.

Now, it is easy to see that

    ⟨∇_X_2X_2, ∇_X_2X_2⟩ = 9(X_1f)^2/16f^2+9f^2/4-1.

The classification of biconservative surfaces in ℍ^3 will be done with respect to the sign of

    W=9(X_1f)^2/16f^2+9f^2/4-1=9| f|^2/16f^2+9f^2/4-1.

From Theorem <ref> we have that

    f^2=4/3(-1-K)>0,

and, therefore

    f=- K/√((3(-1-K))),

where K is the Gaussian curvature of M. It is known that the biconservative surface (M^2,φ^∗⟨,⟩) is isometric to a unique abstract standard biconservative surface  (D_C_-1,g_C_-1) defined in the previous section. So, to (M^2,φ^∗⟨,⟩) it corresponds a unique constant C_-1. Since the Gaussian curvature of (D_C_-1,g_C_-1) and its gradient are given in (<ref>) and (<ref>), by a straightforward computation one obtains an equivalent expression of W,

    W=9| K|^2/64(-1-K)^2-3K-4 =C_-1/3ξ^2,     ξ∈(0,ξ_01).

Therefore, the classification of biconservative surfaces in ℍ^3 will be done according the sign of the real constant C_-1.

As the gluing process will be done along the boundary given by ξ=ξ_01, we get from the above relation that we can glue only two standard biconservative surfaces corresponding to the same constant C_-1.

We denote by κ_2=√(|W|), i.e., κ_2 is the curvature of integral curves of X_2. If κ_2>0, let us consider

    N_2=1/κ_2(3X_1f/4fX_1+3f/2η+x),

and if κ_2=0, i.e., W=0, let us denote

    Ñ_2= 3X_1f/4fX_1+3f/2η+x.

We note that N_2, Ñ_2∈ C(Tℝ^4_1), |N_2|=1 and |Ñ_2|=0 . It is easy to see that X_2f=0 and

    X_2(X_1f)=     X_1(X_2f) - [X_1,X_2]f 
    
      =    (∇_X_1X_2-∇_X_2X_1)f 
    
      =     0.

By a straightforward computation one obtains X_2 κ_2=0, i.e., the integral curves of X_2 are circles.

In order to compute ∇_X_1 N_2, ∇_X_2 N_2, ∇_X_1Ñ_2 and ∇_X_2Ñ_2, we first consider the orthonormal frame field {X_1,X_2, η, x} in ℝ^4_1 along M. Then, one has

    ∇_X_1η=f/2X_1,     ∇_X_2η=-3f/2X_2,     ∇_X_1N_2=0,     ∇_X_2Ñ_2=0,


    ∇_X_2N_2 = {[  κ_2N_2,   C_-1>0; ; -κ_2N_2,   C_-1<0 ].

and

    ∇_X_1Ñ_2 = (3/4f^2(X_1(X_1 f)· f-(X_1 f)^2)+3/4f^2+1)X_1+9/8(X_1f)η+3/4X_1f/fx.


As X_2f=0 and X_2(X_1f)=0, it follows that ∇_X_2Ñ_2=0.


Further, we consider the global problem and construct complete biconservative surfaces in ℍ^3 with f≠ 0 at any point of an open dense subset by using an extrinsic approach.

We will begin with a local extrinsic characterization of biconservative surfaces in ℍ^3, which has been found in <cit.>. These biconservative surfaces are called standard biconservative surfaces and, in order to reach our objective, we will glue two such surfaces.

As we have already announced, we will classify the biconservative surfaces in ℍ^3 with respect to the sign of the constant C_-1.



 §.§ Case C_-1>0



First, we recall the local extrinsic characterization of biconservative surfaces in ℍ^3.


Let M^2 be a biconservative surface in ℍ^3 with a nowhere vanishing gradient of the mean curvature function f. If C_-1>0, then, locally, M^2⊂ℝ^4_1 can be parametrized by

    X_C̃_-1(u,v)=σ(u)+4/3√(C̃_-1)κ^3/4(u)(c_1 cos v +c_2 sin v-c_1),

where C̃_-1>0 is a positive constant; c_1, c_2∈ℝ^4_1 are two constant vectors such that ⟨ c_i,c_j ⟩ = δ_ij; σ(u) is a curve parametrized by arc-length that satisfies

    ⟨σ (u),c_1 ⟩ = 4/3√(C̃_-1)κ^3/4(u),     ⟨σ (u),c_2 ⟩=0,

so σ(u) is a curve lying in the totally geodesic ℍ^2=ℍ^3∩Π, where Π is the linear hyperspace of ℝ^4_1 defined by ⟨r,c_2⟩ =0, while its curvature κ=κ(u) is a positive non-constant solution of the following ODE

    κκ”=7/4(κ')^2-4/3κ^2-4κ^4,

such that

    (κ')^2=16/9κ^2-16κ^4+C̃_-1κ^7/2.




The surface (M^2,X_C̃_-1^∗⟨,⟩) is isometric to the abstract standard biconservative surface (D_C_-1,g_C_-1), and the link between the constants C_-1 and C̃_-1 is

    C_-1=3^3/4/16C̃_-1>0.

Therefore, the above parametrization X_C̃_-1 gives a one-parameter family of biconservative surfaces with f nowhere vanishing indexed by C̃_-1.



We note that the biconservative surface defined by X_C̃_-1 is made up of circles which lie in 2-affine planes parallel with the 2-plane spanned  by c_1 and c_2, and which touch the curve σ. The surface is invariant under the actions of the 1-parameter group of isometries of ℝ^4_1 with positive determinant, which acts on the 2-plane spanned by the unit orthonormal spacelike constant vectors c_1 and c_2. In fact, the 1-parameter group of isometries that acts on M represents the flow of the Killing vector field X_C̃_-1,v which can be seen as a restriction to M of the following Killing vector field on ℝ^4_1

    Z(r)=-⟨r,c_2⟩ c_1+⟨r,c_1⟩ c_2,

r being the position vector of a point in ℝ^4_1.


The standard biconservative surface is “a surface of revolution” in ℝ^4_1 whose profile curve is σ=σ(u) which lies in ℝ^3_1. We also note that the immersion X_C̃_-1 is, in fact, an embedding and the profile curve σ does not have self-intersections; thus the image of X_C̃_-1 is a regular surface in ℍ^3. Therefore, in order to glue two standard biconservative surfaces in ℝ^4_1, it is enough to glue two profile curves defining them, in this way obtaining a complete biconservative regular surface in ℍ^3.

Our strategy is as follows: we reparametrize the profile curve σ in a more convenient way and get σ=σ(κ), then, since the gluing process of the curves σ implies all its components  (three components) it is more convenient to chose another model for ℍ^3 (the upper half space) such that, after that transformation, the curve σ would have two components. After the gluing process is performed, we will obtain a closed regular curve in the upper half plane and therefore, we will get a closed biconservative regular surface in ℍ^3 which has to be complete.

In the same paper <cit.>, it was proved that κ_2=(3√(C̃_-1)κ^3/4)/4 where κ is the geodesic curvature of σ in ℍ^2  given by κ(u)=f(u)/2 and κ_2 is the curvature of integral curves of X_2, using the same notations as in the previous section.

Choosing c_1=e_1 and c_2=e_2, where {e_1,e_2,e_3,e_4} is the canonical basis of ℝ^4_1, the curve σ can be rewritten as

    σ(u)=(4/3√(C̃_-1)κ^3/4(u),0,x(u),y(u)),

for some functions x=x(u) and y=y(u) which are solutions of the following system

    {[                      16/9C̃_-1κ^3/2(u)+x^2(u)-y^2(u)=-1; ;          16(1-9κ^2(u))/9C̃_-1κ^3/2(u)+x'^2(u)-y'^2(u)=0; ; 16(1+3κ^2(u))^2/9C̃_-1κ^3/2(u)+x”^2(u)-y”^2(u)=κ^2(u)-1 ]..

These equations are obtained from the relations σ(u)∈ℍ^3, | σ'(u)|^2=1 and | σ”(u)|^2=κ^2(u)-1, for any u. We can also assume that the function y is a positive function as σ lies in ℍ^3.

In order to prove that there exists a curve σ satisfying (<ref>), let us consider the change of coordinates

    {[ x(u)=R(u)sinh(μ(u)); ; y(u)=R(u)cosh(μ(u)) ].,

with R(u)>0 and μ(u)∈ (0,2π), for any u in an open interval I.

Then, from the first equation of (<ref>), one obtains

    R(u)=√(9C̃_-1κ^3/2(u)+16)/3√(C̃_-1)κ^3/4(u).

Since κ(u)>0, for any u, we can think u=u(κ), so R=R(κ), μ=μ(κ) and then, by a straightforward computation, we get from the second equation of (<ref>)

    μ(κ)=±∫_κ_00^κ36√(C̃_-1)τ^7/4/(9C̃_-1τ^3/2+16)√(16/9τ^2-16τ^4+C̃_-1τ^7/2) dτ +c_0,      c_0∈ℝ,

for any κ∈(0,κ_01), where κ_01 is the positive vanishing point of 16κ^2/9-16κ^4+C̃_-1κ^7/2, 16κ^2/9-16κ^4+C̃_-1κ^7/2>0, for any κ∈(0,κ_01), κ_01>(3C̃_-1)^2/2^12, and κ_00 is arbitrarily fixed in (0,κ_01).

Now, it is easy to see that the first two equations of (<ref>) imply the third one.


As the classification of biconservative surfaces is done up to isometries of ℍ^3, the sign of the integral and the constant c_0 in the expression of μ play an active role only in the gluing process.


The following lemma can be easily proved using similar arguments as in the proof of Lemma <ref>.

Consider

    μ_0(κ)=∫_κ_00^κ36√(C̃_-1)τ^7/4/(9C̃_-1τ^3/2+16)√(16/9τ^2-16τ^4+C̃_-1τ^7/2) dτ,

i.e., we fix the sign in (<ref>) and we choose c_0=0. Then

  
  (i) lim_κ↘ 0μ_0(κ)=μ_0,-1∈ (-∞,0) and lim_κ↗κ_01μ_0(κ)=μ_0,1∈ (0,∞).
  
  (ii) μ_0 is strictly increasing,
  
    lim_κ↘ 0μ'_0(κ)=0    and   lim_κ↗κ_01μ'_0(κ)=∞.

  
  (iii) lim_κ↘ 0μ”_0(κ)=0 and lim_κ↗κ_01μ”_0(κ)=-∞.



Now, we can give an explicit expression of the profile curve σ:

    σ(κ)=(4/3√(C̃_-1)κ^3/4, R(κ)sinhμ(κ), R(κ)coshμ(κ)),

for any κ∈(0,κ_01), where μ is given in equation (<ref>) and

    R(κ)=√(9C̃_-1κ^3/2+16)/3√(C̃_-1)κ^3/4.

Therefore, we can rewrite

    X_C̃_-1(κ,v)=(4cos v/3√(C̃_-1)κ^3/4,4sin v/3√(C̃_-1)κ^3/4, R(κ)sinhμ(κ),R(κ)coshμ(κ)),

for any (κ,v)∈(0,κ_01)×ℝ.

In order to obtain a complete biconservative surface in ℍ^3, first we reparametrize the profile curve σ as we announced earlier.
Through the standard diffeomorphism from hyperboloid model to upper half space model.

    δ(x^1,x^2,x^3,x^4)=(1, 2x^2/x^1+x^4,2x^3/x^1+x^4,2/x^1+x^4)

the profile curve σ becomes

    σ(κ)=    (1,2√(9C̃_-1κ^3/2+16)sinhμ(κ)/4+√(9C̃_-1κ^3/2+16)coshμ(κ), 6√(C̃_-1)κ^3/4/4+√(9C̃_-1κ^3/2+16)coshμ(κ)) 
    ≡   (2√(9C̃_-1κ^3/2+16)sinhμ(κ)/4+√(9C̃_-1κ^3/2+16)coshμ(κ), 6√(C̃_-1)κ^3/4/4+√(9C̃_-1κ^3/2+16)coshμ(κ)).

Choosing appropriate values of the constant c_0 and of the sign in the expression of μ(k), we can find two profile curves σ_1 and σ_2 such that we can glue them smoothly.

So, let us consider the following two curves

    σ_1(κ)= (2√(9C̃_-1κ^3/2+16)sinhμ_0(κ),6√(C̃_-1)κ^3/4)/4+√(9C̃_-1κ^3/2+16)coshμ_0(κ),

i.e., we take, for the sake of simplicity, c_0=0 and the sign “+” in (<ref>),
and then

    σ_2(κ)=    (2√(9C̃_-1κ^3/2+16)sinh(-μ_0(κ)+2μ_0,1),
       6√(C̃_-1)κ^3/4)/4+√(9C̃_-1κ^3/2+16)cosh(-μ_0(κ)+2μ_0,1),

i.e., we take c_0=2μ_0,1 and the sign “-” in (<ref>). If we choose a different value for c_0 and a different sign in (<ref>), then the new curve σ_2 cannot be glued at the C^1 smoothness level with σ_1.

It is easy to see that

    lim_κ↗κ_01σ_1(κ)=lim_κ↗κ_01σ_2(κ)=(6κ_01sinhμ_0,1/1+3κ_01coshμ_0,1,3√(C̃_-1)κ_01^3/4/2+6κ_01coshμ_0,1)∈ℝ^2.

Let us denote by

    x_1(κ)=2√(9C̃_-1κ^3/2+16)sinhμ(κ)/4+√(9C̃_-1κ^3/2+16)coshμ(κ),     y_1(κ)=6√(C̃_-1)κ^3/4/4+√(9C̃_-1κ^3/2+16)coshμ(κ).

It is easy to see that x_1'(κ)>0 in a neighborhood of κ_01, (κ_01-ε,κ_01). If we denote by x_0,1=lim_κ↗κ_01x_1(κ) and x_0,-1=lim_κ↘κ_01-εx_1(κ), it follows that there exists the inverse function x_1^-1=κ_1:(x_0,-1,x_0,1)→(κ_01-ε,κ_01).

So, we can consider y_1=y_1(x) and by a straightforward computation one obtains

    lim_x↗ x_0,1dy_1/dx=-3√(C̃_-1)κ_01^3/4sinhμ_0,1/4coshμ_0,1+12 κ_01.

Similarly, we denote by

    x_2(κ)=2√(9C̃_-1κ^3/2+16)sinh(-μ_0(κ)+2μ_0,1)/4+√(9C̃_-1κ^3/2+16)cosh(-μ_0(κ)+2μ_0,1),

and

    y_2(κ)=6√(C̃_-1)κ^3/4/4+√(9C̃_-1κ^3/2+16)cosh(-μ_0(κ)+2μ_0,1).

It is easy to see that x_2'(κ)<0 in a neighborhood of κ_01, (κ_01-ε,κ_01). Since lim_κ↗κ_01x_2(κ)=x_0,1 and denoting by x_1,-1=lim_κ↘κ_01-εx_2(κ), it follows that there exists the inverse function x_2^-1=κ_2:(x_0,1,x_1,-1)→(κ_01-ε,κ_01).

So, we can also consider y_2=y_2(x) and to glue σ_1 and σ_2 at the C^1 smoothness level means that y_1=y_1(x) and y_2=y_2(x) yield a C^1 smooth function around x_0,1. We note that this is equivalent to the fact that σ_1 and σ_2 have the same tangent space at the gluing point.

By a straightforward computation one obtains

    lim_x↘ x_0,1dy_2/dx=lim_x↗ x_0,1dy_1/dx=-3√(C̃_-1)κ_01^3/4sinhμ_0,1/4coshμ_0,1+12 κ_01∈ℝ.

One can also show that the second and the third derivative of y_1 exist at x_0,1.

Thus, gluing the two curves σ_1 and σ_2, we obtain at least a C^3 smooth curve. Moreover, the curve obtained by the gluing process is a closed regular curve in the upper half plane and therefore, we get a closed biconservative regular surface in ℍ^3 which has to be complete. In Figure 1 we represent the curves σ_1  and σ_2, with the colors red and blue, respectively, for the constant C̃_-1=1, and in Figure 2 we represent the corresponding surfaces to σ_1  and σ_2 in the upper half space (with the Euclidean metric).



    < g r a p h i c s >

   

    < g r a p h i c s >



Figure 1. The profile curves
   
Figure 2. The corresponding surfaces


σ_1  and σ_2
   
to σ_1  and σ_2



If we denote by S^±_C̃_-1,c_0 the image of X_C̃_-1 (given in (<ref>)) corresponding to the sign “+” or “-” in (<ref>), we obtain S^-_C̃_-1,2μ_0,1=T(S^+_C̃_-1,0), where T is the symmetry of ℝ^4_1 with respect to the (Ox^1x^2x^4) hyperplane followed by a linear orthogonal transformation with positive determinant. Moreover, the boundary of S^+_C̃_-1,0 is given by the circle

    v→ ( 4cos v/3√(C̃_-1)κ^3/4_01,4sin v/3√(C̃_-1)κ^3/4_01, R(κ_01)sinhμ_0,1,R(κ_01)coshμ_0,1),

and this circle is invariant by T.



Now it is not difficult to give the explicit expression of the biconservative immersion F in Theorem <ref>. Moreover, if we want to use Theorem <ref> in the gluing process, then it is enough to assure that the gluing process in at least of C^1 smoothness as, once the first standard biconservative surface is fixed, the C^1 gluing process determines uniquely the second standard biconservative surface (the sign “+” or “-” and the constant c_0 in (<ref>)). As the two standard biconservative surfaces give F, the gluing process is in fact at least of the class C^3.




 §.§ Case C_-1<0



First, we recall a local extrinsic result which provides a characterization of biconservative surfaces in ℍ^3 when C_-1<0.


Let M^2 be a biconservative surface in ℍ^3 with f nowhere vanishing. If C_-1<0, then, locally, M^2⊂ℝ^4_1 can be parametrized by

    X_C̃_-1(u,v)=σ(u)+4/3√(-C̃_-1)κ^3/4(u)(c_1sinh v+c_2cosh v-c_2),

where C̃_-1<0 is a negative constant; c_1, c_2∈ℝ^4_1 are two constant vectors such that ⟨ c_1,c_1 ⟩=1, ⟨ c_2,c_2 ⟩=-1 and ⟨ c_1,c_2 ⟩ =0; σ(u) is a curve parameterized by arc-length that satisfies

    ⟨σ (u),c_1 ⟩ =0,    ⟨σ (u),c_2 ⟩= -4/3√(-C̃_-1)κ^3/4(u),

so σ(u) is a curve lying in the totally geodesic ℍ^2=ℍ^3∩Π, where Π is the linear hyperspace of ℝ^4_1 defined by ⟨r,c_1⟩ =0, while its curvature κ=κ(u) is a positive non-constant solution of the following ODE

    κκ”=7/4(κ')^2-4/3κ^2-4κ^4,

such that

    (κ')^2=16/9κ^2-16κ^4+C̃_-1κ^7/2.




The surface (M^2,X_C̃_-1^∗⟨,⟩) is isometric to the abstract standard biconservative surface (D_C_-1,g_C_-1), and the link between the constants C_-1 and C̃_-1 is

    C_-1=3^3/4/16C̃_-1<0.




We note that the biconservative surface defined by X_C̃_-1 is made up of (branches of) “hyperbolas” which lie in 2-affine planes parallel with the 2-plane spanned  by c_1 and c_2, and whose vertices belong to the curve σ. The surface is invariant under the actions of the 1-parameter group of isometries of ℝ^4_1 with positive determinant, which acts on the Minkowski 2-plane spanned by the constant vectors c_1 and c_2. In fact, the 1-parameter group of isometries that acts on M represents the flow of the Killing vector field X_C̃_-1,v which can be seen as a restriction to M of the following Killing vector field on ℝ^4_1

    Z(r)=-⟨r,c_2⟩ c_1+⟨r,c_1⟩ c_2,

r being the position vector of a point in ℝ^4_1.



We note that we slightly corrected the expression of the above local parametrization: the multiplicative coefficient in formula (44), in the original paper <cit.>, should be 2√(2)/(3√(-C)k(u)^3/4) and not 4/(3√(-C)k(u)^3/4).


As in the first case, it can be shown that κ_2=(3√(-C̃_-1)κ^3/4)/4 where κ is the geodesic curvature of σ in ℍ^2  given by κ(u)=f(u)/2 and κ_2 is the curvature of integral curves of X_2. Choosing

    c_1=e_2,     c_2=e_1+√(2)e_4,

where {e_1,e_2,e_3,e_4} is the canonical basis of ℝ^4_1, the curve σ can be rewritten as

    σ(u)=(√(2)y(u)-4/3√(-C̃_-1)κ^3/4(u),0,x(u),y(u)),

for some functions x=x(u) and y=y(u) that have to be solutions of the system

    {[    (y(u)-4√(2)/3√(-C̃_-1)κ^3/4(u))^2+x^2(u)=-1+16/-9C̃_-1κ^3/2(u),      y(u)>0; ;         (y(u)-4√(2)/3√(-C̃_-1)κ^3/4(u))'^2+x'^2(u)=1+(4/3√(-C̃_-1)κ^3/4(u))'^2; ; (y(u)-4√(2)/3√(-C̃_-1)κ^3/4(u))”^2+x”^2(u)=κ^2(u)-1+(4/3√(-C̃_-1)κ^3/4(u))”^2. ].


In order to prove that there exists a curve σ which satisfies (<ref>), let us consider the change of coordinates

    {[                          x(u)=R(u)cos(μ(u)); ; y(u)=R(u)sin(μ(u))+4√(2)/3√(-C̃_-1)κ^3/4(u) ].,

with R(u)>0 and μ(u)∈ (0,2π), for any u in an open interval I.

Then, from the first equation of (<ref>), one obtains

    R(u)=√(9C̃_-1κ^3/2(u)+16)/3√(-C̃_-1)κ^3/4(u).

Further, it is easy to see that 16κ^2/9-16κ^4+C̃_-1κ^7/2>0, for any κ∈(0,κ_01) where κ_01>0 is the positive vanishing point of the function 16κ^2/9-16κ^4+C̃_-1κ^7/2, for C̃_-1 a negative fixed scalar. Then, it is clear that 16+9C̃_-1κ^3/2>0, for any κ∈(0,κ_01). Using this inequality and (<ref>) it is possible to verify that the change of coordinates is correct, i.e., R(u)sin(μ(u))+(4√(2))/(3√(-C̃_-1)κ^3/4(u))>0, for any u∈ I.

In this case, our strategy is similar to that used in the previous case: in order to obtain a complete biconservative surface in ℍ^3, we will glue two standard biconservative surfaces and for this it is enough to glue the two “profile curves” defining them.  We will obtain a closed regular curve in the upper half plane, so a closed biconservative surface in ℍ^3.

First, we reparametrize the profile curve σ in a more convenient way and choose the appropriate model for ℍ^3 (the upper half space). Since κ(u)>0, for any u, we can consider u=u(κ), so R=R(κ), μ=μ(κ) and then, by a similar computation as in the C_-1>0 case, we get


    μ(κ)=±∫_κ_00^κ36√(-C̃_-1)τ^7/4/(9C̃_-1τ^3/2+16)√(16/9τ^2-16τ^4+C̃_-1τ^7/2) dτ +c_0,      c_0∈ℝ,

for any κ∈(0,κ_01) and κ_00 arbitrarily fixed in (0,κ_01), where κ_01 is the vanishing point of 16κ^2/9-16κ^4+C̃_-1κ^7/2.

Further, we denote by

    μ_0(κ)=∫_κ_00^κ36√(-C̃_-1)τ^7/4/(9C̃_-1τ^3/2+16)√(16/9τ^2-16τ^4+C̃_-1τ^7/2) dτ,

and then μ(κ)=±μ_0(κ)+c_0. We will also preserve the same notations for the limits of μ_0 in 0 and in κ_01, i.e.,

    μ_0,-1=lim_κ↘ 0μ_0(κ),    μ_0,1=lim_κ↗κ_01μ_0(κ),

where μ_0,-1∈ (-∞,0) and μ_0,1∈ (0,∞).

Thus, the explicit expression of the profile curve σ is


    σ(κ)=(√(2)R(κ)sinμ(κ)+4/3√(-C̃_-1)κ^3/4, R(κ)cosμ(κ), R(κ)sinμ(κ)+4√(2)/3√(-C̃_-1)κ^3/4),


for any κ∈(0,κ_01), where μ is given in equation (<ref>) and

    R(κ)=√(9C̃_-1κ^3/2+16)/3√(-C̃_-1)κ^3/4.

and therefore,

    X_C̃_-1(κ,v)    =   (√(2)R(κ)sinμ(κ)+4cosh v/3√(-C̃_-1)κ^3/4, 4sinh v/3√(-C̃_-1)κ^3/4,. 
          .    R(κ)cosμ(κ), R(κ)sinμ(κ)+4√(2)cosh v/3√(-C̃_-1)κ^3/4),

for any (κ,v)∈(0,κ_01)×ℝ.

Through the standard diffeomorphism from hyperboloid model to upper half space model (<ref>), the profile curve σ becomes

    σ(κ)=(2√(9C̃_-1κ^3/2+16)cosμ(κ), 6√(-C̃_-1)κ^3/4)/(1+√(2))(4+√(9C̃_-1κ^3/2+16)sinμ(κ)).

In order to find two curves that we will glue such that the gluing process to be smooth (in fact, at least of class C^3), we make the same choices of the constant c_0 and of the sign in (<ref>), as in C_-1>0 case, so let us consider

    σ_1(κ)= (2√(9C̃_-1κ^3/2+16)cosμ_0(κ), 6√(-C̃_-1)κ^3/4)/(1+√(2))(4+√(9C̃_-1κ^3/2+16)sinμ_0(κ))

and

    σ_2(κ)= (2√(9C̃_-1κ^3/2+16)cos(-μ_0(κ)+2μ_0,1),6√(-C̃_-1)κ^3/4)/(1+√(2))(4+√(9C̃_-1κ^3/2+16)sin(-μ_0(κ)+2μ_0,1)).

By computations similar to the first case, we can see that gluing the two curves σ_1 and σ_2, one obtains at least a C^3 smooth curve. Moreover, this is a closed regular curve in the upper half plane, so the corresponding biconservative surface is a closed one in ℍ^3. In Figure 3 we represent the curves σ_1  and σ_2, with the colors red and blue, respectively, for the constant C̃_-1=-1, and in Figure 4 we represent the corresponding surfaces to σ_1  and σ_2 in the upper half space (with the Euclidean metric).



    < g r a p h i c s >

   

    < g r a p h i c s >



Figure 3. The profile curves
   
Figure 4. The corresponding surfaces


σ_1  and σ_2
   
to σ_1  and σ_2





 §.§ Case C_-1=0


We mention that a local extrinisic characterization of biconservative surfaces when C_-1=0 is given in <cit.>. However, using a similar technique as in <cit.>, we prove the following local extrinsic result concerning biconservative surfaces in ℍ^3 when C_-1=0.


Let M^2 be a biconservative surface in ℍ^3 with a nowhere vanishing gradient of the mean curvature function f. If C_-1=0, then, locally, M^2⊂ℝ^4_1 can be parametrized by

    X(u,v)=σ(u)+2^3/4κ^3/4(u)v^2c_1+vc_2,

where c_1, c_2∈ℝ^4_1 are two constant vectors such that ⟨ c_1,c_1⟩=⟨ c_1,c_2⟩=0, ⟨ c_2,c_2 ⟩=1; σ(u) is a curve parameterized by arc-length that satisfies

    ⟨σ(u),c_1⟩=-1/2^7/4κ^3/4(u),     ⟨σ (u),c_2⟩=0,

so σ(u) is a curve lying in the totally geodesic ℍ^2=ℍ^3∩Π, where Π is the linear hyperspace of ℝ^4_1 defined by ⟨r,c_2⟩ =0, while its curvature κ=κ(u) is a positive non-constant solution of the following ODE

    κκ”=7/4(κ')^2-4/3κ^2-4κ^4,

such that

    (κ')^2=16/9κ^2-16κ^4.




Let γ be an integral curve of X_2 parametrized by arc-length and

    Ñ_2=3X_1f/4fX_1+3f/2η+x,

where X_1, X_2 and Ñ_2 are defined in the previous section. Then, as

    γ”(s)=∇_γ'γ'=Ñ_2(s),

where Ñ_2(s)=Ñ_2(γ(s)), and ∇_X_2Ñ_2=0, it follows that

    γ”'(s)= ∇_γ'γ”=0.

Therefore, γ can be parametrized by

    γ(s)=1/2a_1s^2+a_2 s+a_3,      a_1,a_2,a_3∈ℝ^4_1

with

    ⟨ a_1, a_1⟩=0,     ⟨ a_2, a_2⟩=1,     ⟨ a_1,a_2⟩=0.

Let p_0∈ M be an arbitrary point, and σ=σ(u) be an integral curve of X_1, with σ(0)=p_0. Considering the flow ϕ of the vector field X_2 around the point p_0, i.e., ϕ'_σ(u)(v)=X_2(ϕ_σ(u)(v)), we obtain

    ϕ_σ(u)(s)= 1/2a_1(u)s^2+a_2(u) s+a_3(u),

for any u∈(-δ,δ) and for any s∈(-ε,ε), where the functions a_1, a_2 and a_3 satisfy

    ⟨ a_1(u), a_1(u)⟩=0,     ⟨ a_2(u), a_2(u)⟩=1,     ⟨ a_1(u),a_2(u)⟩=0,      u∈(-δ,δ).

Then, the surface can be locally parametrized by

    X(u,s)=ϕ_σ(u)(s).

Since X(u,0)= ϕ_σ(u)(0), it is clear that a_3(u)=σ(u). Moreover, from ϕ'_σ(u)(0)=X_2(ϕ_σ(u)(0)), we have a_2(u)=X_2(u), where X_2(u)=X_2(σ(u)).

We know that ϕ”_σ(u)(0)=a_1(u) and γ”(0)=Ñ_2(ϕ_σ(u)(0))=Ñ_2(σ(u)). Then, we get a_1(u)=Ñ_2(u), where Ñ_2(u)=Ñ_2(σ(u)).

Now, we can see that a_2(u) is, in fact, a constant vector.  Indeed, as ∇_X_1X_2=0 (from (<ref>)) and as σ is an integral curve of X_1, it follows that a_2 is a constant vector, i.e., it does not depend on u.

In order to see that a_1 is constant, we use the fact that W=0, i.e.,

    (X_1f)^2=16/9f^2(1-9/4f^2).

and therefore f∈ (0,2/3) and

    X_1f=±4/3f√(1-9/4f^2).


As X_2f=0, the last equation can be rewritten as


    f'/f√(1-9/4f^2)=±4/3.

Replacing (<ref>) in (<ref>), by a straightforward computation, we obtain

    ∇_X_1Ñ_2=±√(1-9/4f^2)Ñ_2.

Thus, as a_1(u)=Ñ_2(u), using (<ref>) and the above relation, it follows that

    d a_1/du=∇_σ'Ñ_2=3/4f'/fa_1.

Integrating the above equation, one gets a_1(u)=c_3 f^3/4(u), where c_3∈ℝ^4_1. As ⟨ a_1,a_1⟩=0, it follows that the constant vector c_3 satisfies ⟨ c_3,c_3⟩=0. Denoting c_1=c_3/2∈ℝ^4_1 and c_2=a_2∈ℝ^4_1, the local parametrization of M can be rewritten as

    X(u,v)=σ(u)+f^3/4(u)v^2c_1+vc_2,      u∈ (-δ,δ), v∈(-ε,ε),

where |c_1|=0, |c_2|=1, ⟨ c_1,c_2⟩=0 and

    ⟨σ(u),c_1⟩=-1/2f^3/4(u),     ⟨σ (u),c_2⟩=0.

We note that the curve σ is parametrized by arc-length and, as a curve in ℍ^2=ℍ^3∩Π, where Π is the linear hyperspace of ℝ^4_1 defined by ⟨r,c_2⟩ =0, has the geodesic curvature κ(u)=f(u)/2>0, for any u∈ (-δ,δ).

Since f(u)=2κ(u), from (<ref>) one gets

    κ'(u)=±4/3κ(u)√(1-9κ^2(u)),

with κ(u)∈ (0,1/3), for any u∈ (-δ,δ). Now, it is easy to see that κ'≠ 0, and the solution of the above equation is given by

    u(κ)=±3/4log(κ/1+√(1-9κ^2))+C,     κ∈(0,1/3), C∈ℝ.












We note that the surface (M^2,X^∗⟨,⟩) is isometric to the abstract standard biconservative surface (D_0,g_0), that means C_-1=0.



We note that the biconservative surface defined by X_C̃_-1 is a “parabola” which lies in a 2-affine plane parallel with the 2-plane spanned  by c_1 and c_2, and its vertex belongs to the curve σ.


Further, considering c_1=e_1+e_4 and c_2=e_2, where {e_1,e_2,e_3,e_4} is the canonical basis of ℝ^4_1, the curve σ can be rewritten as

    σ(u)=(y(u)-1/2^7/4κ^3/4(u),0,x(u),y(u)),

where the functions x=x(u) and y=y(u) must satisfy

    {[ (y(u)-1/2^7/4κ^3/4(u))^2+x^2(u)-y^2(u)=-1,      y(u)>0; ;            (y(u)-1/2^7/4κ^3/4(u))'^2+x'^2(u)-y'^2(u)=0; ;    (y(u)-1/2^7/4κ^3/4(u))”^2+x”^2(u)-y”^2(u)=κ^2(u)-1. ].


From the first equation of (<ref>), we obtain

    y(u)=2^3/4κ^3/4(u)x^2(u)+2^3/4κ^3/4(u)+1/2^11/4κ^3/4(u).

Replacing the first derivative of y respect to u from above equation in the second equation of (<ref>) and using  (<ref>), we get

    (x'(u)+√(1-9κ^2(u)) x(u))^2=9κ^2(u).

As κ=κ(u)>0, for any u, we can consider u=u(κ), so x=x(u(κ))=x(κ), and if we denote the derivative with respect to κ by “·”, we get

    (4/3κẋ(κ)+x(κ))^2=9κ^2/1-9κ^2.

From this equation, it follows that

    ẋ(κ)=-3/4κx(κ)±9/4√(1-9κ^2)

and, one obtains

    x(κ)=1/κ^3/4μ(κ),     κ∈(0,1/3),

where

    μ(κ)=±9/4∫_κ_00^κτ^3/4/√(1-9τ^2) dτ+c_0,

with c_0∈ℝ. If we denote by

    μ_0(κ)=9/4∫_κ_00^κτ^3/4/√(1-9τ^2) dτ,     κ∈(0,1/3),

where κ_00 is arbitrarily fixed in (0,1/3), then we can write μ(κ)=±μ_0(κ)+c_0.

Now, it is easy to verify that the last equation of (<ref>) is automatically satisfied.

Next, we can write the explicit parametrization of the curve σ

    σ(κ)=    (y(κ)-1/2^7/4κ^3/4,0,x(κ),y(κ)) 
    ≡   (y(κ)-1/2^7/4κ^3/4,x(κ),y(κ)),     κ∈(0,1/3)

where the function x=x(κ) is given in (<ref>) and y=y(κ) is defined as

    y(κ)=2^3/4κ^3/4(x^2(κ)+1)+1/2^11/4κ^3/4.

Thus,

    X(κ,v)    =   (2^3/4κ^3/4(1+x^2(κ)+v^2)-1/2^11/4κ^3/4,v,x(κ),. 
          .    2^3/4κ^3/4(1+x^2(κ)+v^2)+1/2^11/4κ^3/4),

for any (κ,v)∈(0,1/3)×ℝ.

Through the standard diffeomorphism from hyperbolid model to upper half space model, the “profile curve” σ becomes

    σ(κ)=(μ(κ),κ^3/4)/2^3/4(κ^3/2+μ^2(κ)),     κ∈(0,1/3).


Further, we prove that the limits of μ_0 as κ approaches 0 and 1/3 are finite.

In order to show that the limit of μ_0 as κ approaches 0 is finite, first, let us consider the change of variables τ̃=1/τ. The integral μ_0 becomes

    μ_0(κ)=∫_1/κ^1/κ_009/4τ̃^7/4√(τ̃^2-9) dτ̃.

Since

    lim_τ̃→∞τ̃^7/4·9/4τ̃^7/4√(τ̃^2-9)=0∈ [0,∞),

we get that the integral

    ∫_1/κ_00^∞9/4τ̃^7/4√(τ̃^2-9) dτ̃<∞,

so we obtain

    lim_κ↘ 0μ_0(κ)=μ_0,-1∈ (-∞,0).

In order to show that the limit of μ_0 as κ approaches 1/3 is finite, we note that

    lim_κ↗ 1/3√(1/3-τ)9τ^3/4/4√(1-9τ^2)=3^3/4/2^5/2∈[0,∞),

thus,

    ∫_κ_00^1/39τ^3/4/4√(1-9τ^2) dτ<∞,

and then

    lim_κ↗ 1/3μ_0(κ)=μ_0,1∈ (0,∞).

As in the previous cases, our aim is to find two profile curves of the standard surfaces and glue them at least of class C^3 in order to obtain a closed regular curve in the upper half plane, which would define the complete biconservative surface in ℍ^3.

Considering the curves σ_1 and σ_2 given by

    σ_1(κ)=(μ_0(κ),κ^3/4)/2^3/4(κ^3/2+μ_0^2(κ)),  σ_2(κ)=(2μ_0,1-μ_0(κ),κ^3/4)/2^3/4(κ^3/2+(2μ_0,1-μ_0(κ))^2),

that means we choose c_0=0 and the sign “+”, and c_0=2μ_0,1 and the sign “-”, respectively in (<ref>), by a direct computation we can see that gluing the curves σ_1 and σ_2, we obtain at least a C^3 smooth closed curve in the upper half plane.

We note that the immersion X is in fact an embedding and the profile curve has no self-intersections. Therefore, we get a closed biconservative surface in ℍ^3, which has to be complete. In Figure 5 we represent the curves σ_1  and σ_2, with the colors red and blue, respectively, and in Figure 6 we represent the corresponding surfaces to σ_1  and σ_2 in the upper half space (with the Euclidean metric).



    < g r a p h i c s >

   

    < g r a p h i c s >



Figure 5. The profile curves
   
Figure 6. The corresponding surfaces


σ_1  and σ_2
   
to σ_1  and σ_2


We can conclude with our last theorem.


By gluing two standard biconservative surfaces along their common boundary we get a complete biconservative regular surface in ℍ^3. Moreover, the gradient of its mean curvature vanishes along the initial boundary which now is a geodesic of the surface.


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