Global properties of biconservative surfaces in $\mathbb{R}^3$ and $\mathbb{S}^3$
Simona Nistor Cezar Oniciuc


Global properties of biconservative surfaces in ℝ^3 and 𝕊^3]
Global properties of biconservative surfaces 
 in ℝ^3 and 𝕊^3




Faculty of Mathematics - Research Department
 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



The authors' work was supported by a grant of the Romanian National Authority for Scientific Research and Innovation, CNCS - UEFISCDI, project number PN-II-RU-TE-2014-4-0004.

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





We survey some recent results on biconservative surfaces in 3-di­mension­al space forms N^3(c) with a special emphasis on the c=0 and c=1 cases. We study the local and global properties of such surfaces, from extrinsic and intrinsic point of view. We obtain all non-CMC complete biconservative surfaces in ℝ^3 and 𝕊^3.


[
    Cezar Oniciuc
    
=================



§ INTRODUCTION


The study of submanifolds with constant mean curvature, i.e., CMC submanifolds, and, in particular, that of CMC surfaces in 3-dimensional spaces, represents a very active research topic in Differential Geometry for more than 50 years.

There are several ways to generalize these submanifolds. For example, keeping the CMC hypothesis and adding other geometric hypotheses to the submanifold or, by contrast, in the particular case of hypersurfaces in space forms, studying the hypersurfaces which are “highly non-CMC”.

The biconservative submanifolds seem to be an interesting generalization of CMC submanifolds. Biconservative submanifolds in arbitrary manifolds (and in particular, biconservative surfaces) which are also CMC have some remarkable properties (see, for example <cit.>). CMC hypersurfaces in space forms are trivially biconservative, so more interesting is the study of biconservative hypersurfaces which are non-CMC; recent results in non-CMC biconservative hypersurfaces were obtained in <cit.>.

The biconservative submanifolds are closely related to the biharmonic submanifolds. More precisely, let us consider the bienergy functional defined for all smooth maps between two Riemannian manifolds (M^m,g) and (N^n,h) and given by

    E_2(φ)=1/2∫_M|τ(φ)|^2 v_g,     φ∈ C^∞(M,N),

where τ(φ) is the tension field of φ. A critical point of E_2 is called a biharmonic map and is characterized by the vanishing of the bitension field τ_2(φ) (see <cit.>).

A Riemannian immersion φ:M^m→(N^n,h) or, simply, a submanifold M of N, is called biharmonic if φ is a biharmonic map.

Now, if φ:M→(N,h) is a fixed map, then E_2 can be thought as a functional defined on the set of all Riemannian metrics on M. This new functional's critical points are Riemannian metrics determined by the vanishing of the stress-bienergy tensor S_2. This tensor field satisfies

    S_2=-⟨τ_2(φ),dφ⟩.

If S_2=0 for a submanifold M in N, then M is called a biconservative submanifold and it is characterized by the fact that the tangent part of its bitension field vanishes. Thus we can expect that the class of biconservative submanifolds to be much larger than the class of biharmonic submanifolds.

The paper is organized as follows. After a section where we recall some notions and general results about biconservative submanifolds, we present in Section 3 the local, intrinsic characterization of biconservative surfaces. The local, intrinsic characterization theorem provides the necessary and sufficient conditions for an abstract surface (M^2,g) to admit, locally, a biconservative embedding with positive mean curvature function f and f≠ 0 at any point.

Our main goal is to extend the local classification results for biconservative surfaces in N^3(c), with c=0 and c=1, to global results, i.e., we ask that biconservative surfaces to be complete, with f>0 everywhere and | f|>0 on an open dense subset.

In Section 4 we consider the global problem and construct complete biconservative surfaces in ℝ^3 with f>0 on M and f≠ 0 at any point of an open dense subset of M. We determine such surfaces in two ways. One way is to use the local, extrinsic characterization of biconservative surfaces in ℝ^3 and “glue” two pieces together in order to obtain a complete biconservative surface. The other way is more analytic and consists in using the local, intrinsic characterization theorem in order to obtain a biconservative immersion from (ℝ^2,g_C_0) in ℝ^3 with f>0 on ℝ^2 and | f|> 0 on an open dense subset of ℝ^2 (the immersion has to be unique); here, C_0 is a positive constant and therefore we obtain a one-parameter family of solutions. It is worth mentioning that, by a simple transformation of the metric g_C_0, (ℝ^2,√(-K_C_0)g_C_0) is (intrinsically) isometric to a helicoid.

In the last section we consider the global problem of biconservative surfaces in 𝕊^3 with f>0 on M and f≠ 0 at any point of an open dense subset of M. As in the ℝ^3 case, we use the local, extrinsic classification of biconservative surfaces in 𝕊^3, but now the “gluing” process is not as clear as in ℝ^3. Further, we change the point of view and use the local, intrinsic characterization of biconservative surfaces in 𝕊^3. We determine the complete Riemannian surfaces  (ℝ^2,g_C_1,C^∗_1) which admit a biconservative immersion in 𝕊^3 with f>0 everywhere and | f|>0 on an open dense subset of ℝ^2 and we show that, up to isometries, there exists only a one-parameter family of such Riemannian surfaces indexed by C_1.

We end the paper with some figures, obtained for particular choices of the constants, which represent the non-CMC complete biconservative surfaces in ℝ^3 and the way how these surfaces can be obtained in 𝕊^3.



§ BICONSERVATIVE SUBMANIFOLDS; GENERAL PROPERTIES


Throughout this work, all manifolds, metrics, maps are assumed to be smooth, i.e. in the C^∞ category, and we will often indicate the various Riemannian metrics by the same symbol ⟨,⟩. All surfaces are assumed to be connected and oriented.

A harmonic map φ:(M^m,g)→(N^n,h) between two Riemannian manifolds is a critical point of the energy functional

    E:C^∞(M,N)→ℝ,   E(φ)=1/2∫_M|dφ|^2 v_g,

and it is characterized by the vanishing of its tension field

    τ(φ)=_g ∇ dφ.


The idea of the stress-energy tensor associated to a functional comes from D. Hilbert (<cit.>). Given a functional E, one can associate to it a symmetric 2-covariant tensor field S such that S=0 at the critical points of E. When E is the energy functional, P. Baird and J. Eells (<cit.>), and A. Sanini (<cit.>), defined the tensor field

    S=e(φ)g-φ^∗ h=1/2|dφ|^2g-φ^∗ h,

and proved that

    S=-⟨τ(φ),dφ⟩.

Thus, S can be chosen as the stress-energy tensor of the energy functional. It is worth mentioning that S has a variational meaning. Indeed, we can fix a map φ:M^m→(N^n,h) and think E as being defined on the set of all Riemannian metrics on M. The critical points of this new functional are Riemannian metrics determined by the vanishing of their stress-energy tensor S.

More precisely, we assume that M is compact and denote

    𝒢={g :  g  is a Riemannian metric on  M}.

For a deformation {g_t} of g we consider ω=.d/dt|_t=0g_t∈ T_g𝒢=C(⊙^2T^∗ M). We define the new functional

    ℱ:𝒢→ℝ,   ℱ(g)=E(φ)

and we have the following result.


Let φ:M^m→(N^n,h) and assume that M is compact. Then

    .d/dt|_t=0ℱ(g_t)=1/2∫_M ⟨ω,e(φ)g-φ^∗ h⟩ v_g.

Therefore g is a critical point of ℱ if and only if its stress-energy tensor S vanishes.


We mention here that, if φ:(M^m,g)→(N^n,h) is an arbitrary isometric immersion, then S=0.

A natural generalization of harmonic maps is given by biharmonic maps. A biharmonic map φ:(M^m,g)→(N^n,h) between two Riemannian manifolds is a critical point of the bienergy functional

    E_2:C^∞(M,N)→ℝ,   E_2(φ)=1/2∫_M|τ(φ)|^2 v_g,

and it is characterized by the vanishing of its bitension field

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

where

    Δ^φ=-_g(∇^φ∇^φ-∇^φ_∇)

is the rough Laplacian of φ^-1TN and the curvature tensor field is

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


We remark that the biharmonic equation τ_2(φ)=0 is a fourth-order non-linear elliptic equation and that any harmonic map is biharmonic. A non-harmonic biharmonic map is called proper biharmonic.

In <cit.>, G. Y. Jiang defined the stress-energy tensor S_2 of the bienergy (also called stress-bienergy tensor) by

    S_2(X,Y) =    1/2|τ(φ)|^2⟨ X,Y⟩ +⟨ dφ,∇τ(φ)⟩⟨ X,Y⟩
        - ⟨ dφ(X),∇_Y τ(φ)⟩ - ⟨ dφ(Y),∇_X τ(φ)⟩,

as it satisfies

    S_2=-⟨τ_2(φ),dφ⟩.


The tensor field S_2 has a variational meaning, as in the harmonic case. We fix a map φ:M^m→(N^n,h) and define a new functional

    ℱ_2:𝒢→ℝ,   ℱ_2(g)=E_2(φ).

Then we have the following result.

Let φ:M^m→(N^n,h) and assume that M is compact. Then

    .d/dt|_t=0ℱ_2(g_t)=-1/2∫_M ⟨ω,S_2⟩ v_g,

so g is a critical point of ℱ_2 if and only if S_2=0.


We mention that, if φ:(M^m,g)→(N^n,h) is an isometric immersion then S_2 does not necessarily vanish.

A submanifold of a given Riemannian manifold (N^n,h) is a pair (M^m,φ), where M^m is a manifold and φ:M→ N is an immersion. We always consider on M the induced metric g=φ^∗ h, thus φ:(M,g)→ (N,h) is an isometric immersion; for simplicity we will write φ:M→ N without mentioning the metrics. Also, we will write φ:M → N, or even M, instead of (M,φ).

A submanifold φ:M^m→ N^n is called  biharmonic if the isometric immersion φ is a biharmonic map from (M^m,g) to (N^n,h).

Even if the notion of biharmonicity may be more appropriate for maps than for submanifolds, as the domain and the codomain metrics are fixed and the variation is made only through the maps, the biharmonic submanifolds proved to be an interesting notion (see, for example, <cit.>).

In order to fix the notations, we recall here only the fundamental equations of first order of a submanifold in a Riemannian manifold. These equations define the second fundamental form, the shape operator and the connection in the normal bundle. Let φ:M^m→ N^n be an isometric immersion. For each p∈ M, T_φ(p)N splits as an orthogonal direct sum

    T_φ(p)N=dφ(T_pM)⊕ dφ(T_pM)^⊥,

and NM=⋃_p∈ Mdφ(T_pM)^⊥ is referred to as the normal bundle of φ, or of M, in N.

Denote by ∇ and ∇^N  the Levi-Civita connections on M and N, respectively, and by ∇^φ the induced connection in the pull-back bundle
φ^-1(TN)=⋃_p∈ MT_φ(p)N.
Taking into account the decomposition in (<ref>),
one has

    ∇^φ_X dφ(Y)=dφ(∇_X Y)+B(X,Y),    ∀ X, Y∈ C(TM),

where B∈ C(⊙^2 T^∗ M⊗ NM) is called the second fundamental form of M in N. Here T^∗ M denotes the cotangent bundle of M. The mean curvature vector field of M in N is
defined by H=( B)/m∈ C(NM), where the  is considered with respect to the metric g.

Furthermore, if η∈ C(NM), then

    ∇^φ_X η=-dφ(A_η(X))+∇^⊥_Xη,   ∀ X∈ C(TM),

where A_η∈ C(T^∗ M⊗ TM) is called the shape operator of M in N in the direction of η, and ∇^⊥ is the induced connection in the
normal bundle. Moreover, ⟨ B(X,Y),η⟩=⟨ A_η(X),Y⟩, for all X, Y∈ C(TM), η∈ C(NM). In the case of hypersurfaces, we denote f= A, where A=A_η and η is the unit normal vector field, and we have H=(f/m)η; f is the (m times) mean curvature function.

A submanifold M of N is called PMC if H is parallel in the normal bundle, and CMC if |H| is constant.

When confusion is unlikely we identify, locally, M with its image through φ, X with dφ(X) and ∇^φ_X dφ(Y) with ∇^N_X Y. With these identifications in mind, we write

    ∇^N_X Y=∇_X Y+B(X,Y),

and

    ∇^N_X η=-A_η(X)+∇^⊥_Xη.


If S_2=0 for a submanifold M in N, then M is called biconservative. Thus, M is biconservative if and only if the tangent part of its bitension field vanishes.

We have the following characterization theorem of biharmonic submanifolds, obtained by splitting the bitension field in the tangent and normal part.



A submanifold M^m of a Riemannian manifold N^n is biharmonic if and only if

    A_∇^⊥_· H(·)+∇ A_H +( R^N(·,H)·)^T=0

and

    Δ^⊥ H+ B(·,A_H(·)) +(R^N(·,H)·)^⊥=0,

where Δ^⊥ is the Laplacian in the normal bundle.


Various forms of the above result were obtained in <cit.>. From here we deduce some characterization formulas for the biconservativity.


Let M^m be a submanifold of a Riemannian manifold N^n. Then M is a biconservative submanifold if and only if:

    
  * A_∇^⊥_· H(·)+∇ A_H +( R^N(·,H)·)^T=0;
    
  * m/2(|H|^2)+2 A_∇^⊥_· H(·) + 2( R^N(·,H)·)^T=0;
    
  * 2∇ A_H-m/2(|H|^2)=0.



The following properties are immediate.


Let M^m be a submanifold of a Riemannian manifold N^n. If ∇ A_H=0 then M is biconservative.



Let M^m be a submanifold of a Riemannian manifold N^n. Assume that N is a space form, i.e., it has constant sectional curvature, and M is PMC. Then M is biconservative.



Let M^m be a submanifold of a Riemannian manifold N^n. Assume that M is pseudo-umbilical, i.e., A_H=|H|^2I, and m≠4. Then M is CMC.


If we consider the particular case of hypersurfaces, then Theorem <ref> becomes


If M^m is a hypersurface in a Riemannian manifold N^m+1, then M is biharmonic if and only if

    2A( f)+f f-2f(^N(η))^T=0,

and

    Δ f+f|A|^2-f^N(η,η)=0,

where η is the unit normal vector field of M in N.



A hypersurface M^m in a space form N^m+1(c) is biconservative if and only if

    A( f)=-f/2 f.




Any CMC hypersurface in N^m+1(c) is biconservative.


Therefore, the biconservative hypersurfaces may be seen as the next research topic after that of CMC surfaces.



§ INTRINSIC CHARACTERIZATION OF BICONSERVATIVE SURFACES


We are interested to study biconservative surfaces which are non-CMC. We will first look at them from a local, extrinsic point of view and then from a global point of view. While by “local” we will mean the biconservative surfaces φ:M^2→ N^3(c) with f>0 and f≠ 0 at any point of M, by “global” we will mean the complete biconservative surfaces φ:M^2→ N^3(c) with f>0 at any point of M and f≠ 0 at any point of an open and dense subset of M.

In this section, we consider the local problem, i.e.,  we take φ:M^2→ N^3(c) a biconservative surface and assume that f>0 and f≠ 0 at any point of M. Let X_1=( f)/| f| and X_2 two vector fields such that {X_1(p),X_2(p)} is a positively oriented orthonormal basis at any point p∈ M. In particular, we obtain that M is parallelizable. If we denote by λ_1≤λ_2 the eigenvalues functions of the shape operator A, since A(X_1)=-(f/2)X_1 and A=f, we get λ_1=-f/2 and λ_2=3f/2. Thus the matrix of A with respect to the (global) orthonormal frame field {X_1,X_2} is

    A=(
    [ -f/2    0; ;    0 3f/2 ]).

We denote by K the Gaussian curvature and, from the Gauss equation, K=c+ A, we obtain

    f^2=4/3(c-K).

Thus c-K>0 on M.

From the definitions of X_1 and X_2, we find that

    f=(X_1 f)X_1    and    X_2 f=0.

Using the connection 1-forms, the Codazzi equation and then the extrinsic and intrinsic expression for the Gaussian curvature, we obtain the next result which shows that the mean curvature function of a non-CMC biconservative surface must satisfy a second-order partial differential equation. More precisely, we have the following theorem.


Let φ:M^2→ N^3(c) a biconservative surface with f>0 and f≠ 0 at any point of M. Then we have

    fΔ f+| f|^2+4/3c f^2-f^4=0,

where Δ is the Laplace-Beltrami operator on M.


In fact, we can see that around any point of M there exists (U;u,v) local coordinates such that f=f(u,v)=f(u) and (<ref>) is equivalent to

    ff^''-7/4(f^')^2-4/3cf^2+f^4=0,

i.e., f must satisfy a second-order ordinary differential equation.

Indeed, let p_0∈ M be an arbitrary fixed point of M and let γ=γ(u) be an integral curve of X_1 with γ(0)=p_0. Let ϕ the flow of X_2 and (U;u,v) local coordinates with p_0∈ U such that

    X(u,v)=ϕ_γ(u)(v)=ϕ(γ(u),v).

We have

    X_u(u,0)=γ^'(u)=X_1(γ(u))=X_1(u,0)

and

    X_v(u,v)=ϕ^'_γ(u)(v)=X_2(ϕ_γ(u)(v))=X_2(u,v).

If we write the Riemannian metric g on M in local coordinates as

    g=g_11du^2+2g_12du dv+g_22dv^2,

we get g_22=|X_v|^2=|X_2|^2=1, and X_1 can be expressed with respect to X_u and X_v as

    X_1=1/σ(X_u-g_12X_v)=σ u,

where σ=√(g_11-g_12^2)>0, σ=σ(u,v).

Let f∘ X=f(u,v). Since X_2f=0, we find that

    f(u,v)=f(u,0)=f(u),   ∀ (u,v)∈ U.

It can be proved that

    [X_1,X_2]=3(X_1 f)/4fX_2,

and thus X_2 X_1 f=X_1 X_2 f-[X_1,X_2]f=0.

On the other hand we have

    [             X_2 X_1 f = X_v(1/σf^')=X_v(1/σ)f^';                       =                       0 ].

We recall that

    f=(X_1 f)X_1=(1/σf^')X_1≠ 0

at any point of U, and then f^'≠ 0 at any point of U. Therefore, from (<ref>), X_v(1/σ)=0, i.e., σ=σ(u). Since g_11(u,0)=1, and g_12(u,0)=0, we have σ=1, i.e.,

    X_1=X_u-g_12X_v= u.

In <cit.> it was found an equivalent expression for (<ref>), i.e.,

    (X_1X_1f)f=7/4(X_1 f)^2+4c/3f^2-f^4.

Therefore, using (<ref>), relation (<ref>) is equivalent to (<ref>).


If φ:M^2→ N^3(c) is a non-CMC biharmonic surface, then, there exists an open subset U such that f>0, f≠ 0 at any point of U, and f satisfies the following system

    {[ Δ f=f(2c-|A|^2); ;    A( f)=-f/2 f ]. .

As we have seen, this system implies

    {[            Δ f=f(2c-|A|^2); ; fΔ f+| f|^2+4/3c f^2-f^4=0 ]. .

which, in fact, is a ODE system. We get

    {[ ff^''-3/4(f^')^2+2cf^2-5/2f^4=0; ;  ff^''-7/4(f^')^2-4/3cf^2+f^4=0 ]. .

As an immediate consequence we obtain

    (f^')^2+10/3cf^2-7/2f^4=0,

and combining it with the first integral

    (f^')^2=2f^4-8cf^2+α f^3/2

of the first equation from (<ref>), where α∈ℝ is a constant, we obtain

    3/2f^5/2+14/3cf^1/2-α=0.

If we denote f̃=f^1/2, we get 3f̃^5/2+14 cf̃/3-α=0. Thus, f̃ satisfies a polynomial equation with constant coefficients, so f̃ has to be a constant and then, f is a constant, i.e., f=0 on U (in fact, f has to be zero). Therefore, we have a contradiction (see <cit.> for c=0 and <cit.>, for c=± 1).


We can also note that relation (<ref>), which is an extrinsic relation, together with (<ref>), allows us to find an intrinsic relation that (M,g) must satisfy. More precisely, the Gaussian curvature of M has to satisfy

    (c-K)Δ K-| K|^2-8/3K(c-K)^2=0,

and the conditions c-K>0 and K≠0.

Formula (<ref>) is very similar to the Ricci condition. Further, we will briefly recall the Ricci problem. Given an abstract surface (M^2,g), we want to find the conditions that have to be satisfied by M such that, locally, it admits a minimal embedding in N^3(c). It was proved (see <cit.>) that if (M^2,g) is an abstract surface such that c-K>0 at any point of M, where c∈ℝ is a constant, then, locally, it admits a minimal embedding in N^3(c) if and only if

    (c-K)Δ K-| K|^2-4K(c-K)^2=0.

Condition (<ref>) is called the Ricci condition with respect to c, or simply the Ricci condition. If (<ref>) holds, then, locally, M admits a one-parameter family of minimal embeddings in N^3(c).

We can see that relations (<ref>) and (<ref>) are very similar and, in <cit.>, the authors studied the link between them. Thus, for c=0, it was proved that if we consider a surface (M^2,g) which satisfies (<ref>) and K<0, then there exists a very simple conformal transformation of the metric g such that (M^2,√(-K)g) satisfies (<ref>). A similar result was also proved for c≠ 0, but in this case, the conformal factor has a complicated expression (and it is not enough to impose that (M^2,g) satisfy (<ref>), but we need the stronger hypothesis of it to admit a non-CMC biconservative immersion in N^3(c)).

Unfortunately, condition (<ref>) does not imply, locally, the existence of a biconservative immersion in N^3(c), as in the minimal case. We need a stronger condition. It was obtained the following local, intrinsic characterization theorem.


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

    κ=3| K|/8(c-K).




If the surface M in Theorem <ref> is simply connected, then the theorem holds globally, but, in this case, instead of a local isometric embedding we have a global isometric immersion.


We remark that unlike in the minimal immersions case, if M satisfies the hypotheses from Theorem <ref>, then there exists a unique biconservative immersion in N^3(c) (up to an isometry of N^3(c)), and not a one-parameter family.

The characterization theorem can be equivalently rewritten as below.


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 a constant. 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 of p∈ M. Then, the following conditions are equivalent:

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

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

  
  (c) locally, the metric g can be written as g=(c-K)^-3/4(du^2+dv^2), where (u,v) are local coordinates positively oriented, K=K(u), and K^'>0;
  
  (d) locally, the metric g can be written as g=e^2φ(du^2+dv^2), where (u,v) are local coordinates positively oriented, and φ=φ(u) satisfies the equation
      
    φ^''=e^-2φ/3-ce^2φ

      and the condition φ^'>0; moreover, the solutions of the above equation, u=u(φ), are
      
    u=_φ_0^φdτ/√(-3e^-2τ/3-ce^2τ+a)+u_0,

      where φ is in some open interval I and a,u_0∈ℝ are constants;
  
  (e) locally, the metric g can be written as g=e^2φ(du^2+dv^2), where (u,v) are local coordinates positively oriented, and φ=φ(u) satisfies the equation
      
    3φ^'''+2φ^'φ^''+8ce^2φφ^'=0

      and the conditions φ^'>0 and c+e^-2φφ^''>0; moreover, the solutions of the above equation, u=u(φ), are
      
    u=_φ_0^φdτ/√(-3be^-2τ/3-ce^2τ+a)+u_0,

      where φ is in some open interval I and a,b,u_0∈ℝ are constants, b>0.



The proof follows by direct computations and by using Remark 4.3 in <cit.> and Proposition 3.4 in <cit.>.

From the above theorem we have the following remarks.

  
  (i) If condition (a) is satisfied, i.e., the integral curves of X_2 are circles in M with a precise constant curvature, then the integral curves of X_1 are geodesics of M.

  
  (ii) If condition (c) is satisfied, then K has to be a solution of the equation
      
    3K^''(c-K)+3(K^')^2+8K(c-K)^5/4=0.


  
  (iii) If condition (c) is satisfied and c>0, then  (M^2, (c-K)^3/4g) is a flat surface and, trivially, a Ricci surface with respect to c.


  
  (iv) Let φ=φ(u) be a solution of equation (<ref>). We consider the change of coordinates
      
    (u,v)=(αũ+β,αṽ+β),

      where α∈ℝ is a positive constant and β∈ℝ, and define
      
    ϕ=φ(αũ+β)+logα.

      Then g=e^2ϕ(dũ^2+dṽ^2) and ϕ also satisfies equation (<ref>). If φ=φ(u) satisfies the first integral
      
    φ^''=be^-2φ/3-ce^2φ,

      where b>0, then, for α=b^-3/8, ϕ=ϕ(ũ) satisfies
      
    ϕ^''=e^-2ϕ/3-ce^2ϕ.

      From here, as the classification is done up to isometries, we note that the parameter b in the solution of (<ref>) is not essential and only the parameter a counts. Thus we have a one-parameter family of solutions.

  
  (v) If φ is a solution of (<ref>), for some c, then φ+α, where α is a real constant, is a solution of (<ref>) for ce^2α.

  
  (vi) If c=0, we note that if φ is a solution of (<ref>), then also φ+ is a solution of the same equation, i.e, condition (a) from Theorem <ref> is invariant under the homothetic tranformations of the metric g. Then, we see that equation (<ref>) is invariant under the affine change of parameter u=αũ+β, where α>0. Therefore, we must solve equation (<ref>) up to this change of parameter and an additive constant of the solution φ. The additive constant will be the parameter that counts.




In the c=0 case, the solutions of equation (<ref>), are explicitly determined in the next proposition.


The solutions of the equation

    3φ^'''+2φ^'φ^''=0

which satisfy the conditions φ^'>0 and φ^''>0, up to affine transformations of the parameter with α>0, are given by

    φ(u)=3log(cosh u)+,      u>0.



We note that, when c=0, we have a one-parameter family of solutions of equation (<ref>), i.e., g_C_0=C_0(cosh u)^6(du^2+dv^2), C_0 being a positive constant.

If c≠0, then we can not determine explicitly φ=φ(u). Another way to see that in the c≠ 0 case we have only a one-parameter family of solutions of equation (<ref>) is to rewrite the metric g in certain non-isothermal coordinates.

Further, we will consider only the c=1 case and we have the next result.


Let (M^2,g) be an abstract surface with g=e^2φ(u)(du^2+dv^2), where u=u(φ) satisfies

    u=_φ_0^φdτ/√(-3be^-2τ/3-e^2τ+a)+u_0,

where φ is in some open interval I, a,b∈ℝ are positive constants, and u_0∈ℝ is a constant. Then (M^2,g) is isometric to

    (D_C_1,g_C_1=3/ξ^2(-ξ^8/3+3C_1ξ^2-3)dξ^2 +1/ξ^2dθ^2),

where D_C_1=(ξ_01,ξ_02)×ℝ, C_1∈(4/(3^3/2),∞) is a positive constant, and ξ_01 and ξ_02 are the positive vanishing points of -ξ^8/3+3C_1ξ^2-3, with 0<ξ_01<ξ_02.



Let us consider

    (D_C_1,g_C_1=3/ξ^2(-ξ^8/3+3C_1ξ^2-3)dξ^2+ 1/ξ^2dθ^2)

and

    (D_C_1^',g_C_1^'=3/ξ̃^2(-ξ̃^8/3+3C_1^'ξ̃^2-3)dξ̃^2+1/ξ̃^2dθ̃^2).

The surfaces (D_C_1,g_C_1) and (D_C_1^',g_C_1^') are isometric if and only if C_1=C_1^' and the isometry is Θ(ξ,θ)=(ξ,±θ+). Therefore, we have a one-parameter family of surfaces.



We note that the expression of the Gaussian curvature of (D_C_1,g_C_1) does not depend on C_1. More precisely,

    K_C_1(ξ,θ)=-1/9ξ^8/3+1.

But, if we change further the coordinates (ξ,θ)=(ξ_01+ξ̃(ξ_02-ξ_01),θ̃), then we “fix” the domain, i.e., (D_C_1,g_C_1) is isometric to ((0,1),g̃_C_1) and C_1 appears in the expression of K_C_1(ξ̃,θ̃).




§ COMPLETE BICONSERVATIVE SURFACES IN ℝ^3


In this section we consider the global problem and construct complete biconservative surfaces in ℝ^3 with f>0 everywhere and f≠ 0 at any point of an open dense subset. Or, from intrinsic point of view, we construct a complete abstract surface (M^2,g) with K<0 everywhere and K≠ 0 at any point of an open dense subset of M, that admits a biconservative immersion in ℝ^3, defined on the whole M, with f>0 on M and | f|>0 on the open dense subset.

First, we recall a local extrinsic result which provides a characterization of biconservative surfaces in ℝ^3.


Let M^2 be a surface in ℝ^3 with f(p)>0 and ( f)(p)≠0 for any p∈ M. Then, M is biconservative if and only if, locally, it is a surface of revolution, and the curvature κ=κ(u) of the profile curve σ=σ(u), |σ^'(u)|=1, is a positive solution of the following ODE

    κ^''κ=7/4(κ^')^2-4κ^4.



In <cit.> there was found the local explicit parametric equation of a biconservative surface in ℝ^3.


Let M^2 be a biconservative surface in ℝ^3 with f(p)>0 and ( f)(p)≠0 for any p∈ M. Then, locally, the surface can be parametrized by

    X_C̃_0(ρ,v)=(ρcos v,ρsin v, u_C̃_0(ρ)),

where

    u_C̃_0(ρ)=3/2C̃_0(ρ^1/3√(C̃_0ρ^2/3-1)+1/√(C̃_0)log(√(C̃_0)ρ^1/3+√(C̃_0ρ^2/3-1)))

with C̃_0 a positive constant and ρ∈(C̃_0^-3/2,∞).


We denote by S_C̃_0 the image X_C̃_0((C̃_0^-3/2,∞)×ℝ). We note that any two such surfaces are not locally isometric, so we have a one-parameter family of biconservative surfaces in ℝ^3. These surfaces are not complete.


If φ:M^2→ℝ^3 is a biconservative surface with f>0 and f≠ 0 at any point, then there exists a unique C̃_0 such that φ(M)⊂ S_C̃_0. Indeed, any point admits an open neighborhood which is an open subset of some S_C̃_0. Let us consider p_0∈ M. Then, there exists a unique C̃_0 such that φ(U)⊂ S_C̃_0, where U is an open neighborhood of p_0. If A denotes the set of all points of M such that they admit open neighborhoods which are open subsets of that S_C̃_0, then the set A is non-empty, open and closed in M. Thus, as M is connected, it follows that A=M.


The “boundary” of S_C̃_0, i.e., S_C̃_0∖ S_C̃_0, is the circle (C̃_0^-3/2cos v,C̃_0^-3/2sin v,0), which lies in the Oxy plane. At a boundary point, the tangent plane to the closure S_C̃_0 of S_C̃_0 is parallel to Oz. Moreover, along the boundary, the mean curvature function is constant f_C̃_0=(2C̃_0^3/2)/3 and f_C̃_0=0.

Thus, in order to obtain a complete biconservative surface in ℝ^3, we can expect to “glue” along the boundary two biconservative surfaces of type S_C̃_0 corresponding to the same C̃_0 (the two constants have to be the same) and symmetric to each other, at the level of C^∞ smoothness.

In fact, it was proved that we can glue two biconservative surfaces S_C̃_0 and S_C̃_0^', at the level of C^∞ smoothness, only along the boundary and, in this case, C̃_0=C̃_0^'.


If we consider the symmetry of u_C, with respect to the Oρ(=Ox) axis, we get a smooth, complete, biconservative surface S̃_C̃_0 in ℝ^3. Moreover, its mean curvature function f̃_C̃_0 is positive and f̃_C̃_0 is different from zero at any point of an open dense subset of S̃_C̃_0.



The profile curve σ_C̃_0=(ρ,0,u_C̃_0(ρ))≡(ρ,u_C̃_0(ρ)) can be re­para­metrized as

    [                                                      σ_C̃_0(θ)=                                      (σ_C̃_0^1(θ), σ_C̃_0^2(θ)); ;                                                               = C̃_0^-3/2((θ+1)^3/2,3/2(√(θ^2+θ) + log(√(θ)+√(θ+1)))),     θ>0, ]

and now X_C̃_0=X_C̃_0(θ,v).



The homothety of ℝ^3, (x,y,z)→C̃_0(x,y,z), renders S̃_1 onto S̃_C̃_0^-2/3.


In <cit.>, there were also found the complete biconservative surfaces in ℝ^3 with f>0 at any point and f≠ 0 at any point of an open dense subset, but there, the idea was to use the intrinsic characterization of the biconservative surfaces. More precisely, we have the next global result.


Let (ℝ^2,g_C_0=C_0 (cosh u)^6(du^2+dv^2)) be a surface, where C_0∈ℝ is a positive constant. Then we have:


  (a) the metric on ℝ^2 is complete;

  (b) the Gaussian curvature is given by
           
    K_C_0(u,v)=K_C_0(u)=-3/C_0(cosh u)^8<0,    K^'_C_0(u)=24 sinh u/C_0(cosh u)^9,

           and therefore K_C_0≠ 0 at any point of ℝ^2∖ Ov;


  (c) the immersion φ_C_0:(ℝ^2,g_C_0)→ℝ^3 given by
    
    φ_C_0(u,v)=(σ_C_0^1(u)cos (3v), σ_C_0^1(u)sin (3v), σ_C_0^2(u))

    is biconservative in ℝ^3, where
    
    σ_C_0^1(u)=√(C_0)/3(cosh u)^3,   σ_C_0^2(u)=√(C_0)/2(1/2sinh (2u)+u),      u ∈ℝ.





The first two items follow by standard arguments. For the last part, we note that choosing C̃_0=(9/C_0)^1/3 in  (<ref>) and using the change of coordinates (θ,v)=((sinh u)^2,3v), where u>0, the metric induced by X_(9/C_0)^1/3 coincides with g_C_0. Then, we define φ_C_0 as: for u>0, φ_C_0(u,v) is obtained by rotating the profile curve

    σ^+_(9/C_0)^1/3(u)=σ_(9/C_0)^1/3(u)=(σ_(9/C_0)^1/3^1(u), σ_(9/C_0)^1/3^2(u)),

and for u<0, φ_C_0(u,v) is obtained by rotating the profile curve

    σ^-_(9/C_0)^1/3(u)=(σ_(9/C_0)^1/3^1(-u), -σ_(9/C_0)^1/3^2(-u)).



By simple transformations of the metric, (ℝ^2,g_C_0) becomes a Ricci surface or a surface with constant Gaussian curvature.


Consider the surface (ℝ^2,g_C_0). Then (ℝ^2,√(-K_C_0)g_C_0) is complete, satisfies the Ricci condition and can be minimally immersed in ℝ^3 as a helicoid or a catenoid.



Consider the surface (ℝ^2,g_C_0). Then (ℝ^2,-K_C_0g_C_0) has constant Gaussian curvature 1/3 and it is not complete. Moreover, (ℝ^2,-K_C_0g_C_0) is the universal cover of the surface of revolution in ℝ^3 given by

    Z(u,v)=(α(u)cosh(√(3)/av),α(u)sinh(√(3)/av),β(u)),      (u,v)∈ℝ^2,

where a∈ (0,√(3)] and

    α(u)=a/cosh u,  β(u)=_0^u√((3-a^2)cosh^2τ+a^2)/cosh^2τ dτ.




When a=√(3), the immersion Z has only umbilical points and the image Z(ℝ^2) is the round sphere of radius √(3), without the North and the South poles. Moreover, if a∈ (0,√(3)), then Z has no umbilical points.





Concerning the biharmonic surfaces in ℝ^3 we have the following non-existence result.


There exists no proper biharmonic surface in ℝ^3.




§ COMPLETE BICONSERVATIVE SURFACES IN 𝕊^3


As in the previous section, we consider the global problem for biconservative surfaces in 𝕊^3, i.e., our aim is to construct complete biconservative surfaces in 𝕊^3 with f>0 everywhere and f≠ 0 at any point of an open and dense subset.

We start with the following local, extrinsic result.


Let M^2 be a biconservative surface in 𝕊^3 with f(p)>0 and ( f)(p)≠ 0 at any point p∈ M. Then, locally, the surface, viewed in ℝ^4, can be parametrized by

    Y_C̃_1(u,v)=σ(u)+4κ(u)^-3/4/3√(C̃_1)( f_1 (cos v -1)+f_2 sin v),

where C̃_1∈(64/(3^5/4),∞) is a positive constant; f_1, f_2∈ℝ^4 are two constant orthonormal vectors; σ(u) is a curve parametrized by arclength that satisfies

    ⟨σ(u),f_1⟩ = 4κ(u)^-3/4/3√(C̃_1),     ⟨σ(u),f_2⟩=0,

and, as a curve in 𝕊^2, its curvature κ=κ(u) is a positive non constant solution of the following ODE

    κ^''κ=7/4(κ^')^2+4/3κ^2-4κ^4

such that

    (κ^')^2=-16/9κ^2-16κ^4+C̃_1κ^7/2.




The constant C̃_1 determines uniquely the curvature κ, up to a translation of u, and then κ, f_1 and f_2 determines uniquely the curve σ.


We consider f_1=e_3 and f_2=e_4 and change the coordinates (u,v) in (κ,v). Then, we get

    [                                                         Y_C̃_1(κ,v)= (√(1-(4/3√(C̃_1)κ^-3/4)^2)cosμ(κ), √(1-(4/3√(C̃_1)κ^-3/4)^2)sinμ(κ),;                                                                                            4/3√(C̃_1)κ^-3/4cos v, 4/3√(C̃_1)κ^-3/4sin v), ]

where (κ,v)∈(κ_01,κ_02)×ℝ, κ_01 and κ_02 are positive solutions of

    -16/9κ^2-16κ^4+C̃_1κ^7/2=0

and

    μ(κ)=± 108_κ_0^κ√(C̃_1)τ^3/4/(-16+9C̃_1τ^3/2) √( 9C̃_1τ^3/2-16(1+9τ^2)) dτ +c_0,

with c_0∈ℝ a constant and κ_0∈(κ_01,κ_02). We note that an alternative expression for Y_C̃_1 was given in <cit.>.


The limits lim_κ↘κ_01μ(κ)=μ(κ_01) and  lim_κ↗κ_02μ(κ)=μ(κ_02) are finite.



For simplicity, we choose κ_0=(3C̃_1/64)^2.


If we denote S_C̃_1 the image of Y_C̃_1, then we note that the boundary of S_C̃_1 is made up from two circles and along the boundary, the mean curvature function is constant (two different constants) and its gradient vanishes. More precisely, the boundary of S_C̃_1 is given by the curves

    [ (√(1-(4/3√(C̃_1)κ_01^-3/4)^2)cosμ(κ_01), √(1-(4/3√(C̃_1)κ_01^-3/4)^2)sinμ(κ_01),;                             4/3√(C̃_1)κ_01^-3/4cos v, 4/3√(C̃_1)κ_01^-3/4sin v ) ]

and

    [ (√(1-(4/3√(C̃_1)κ_02^-3/4)^2)cosμ(κ_02), √(1-(4/3√(C̃_1)κ_02^-3/4)^2)sinμ(κ_02),;                            4/3√(C̃_1)κ_02^-3/4cos v, 4/3√(C̃_1)κ_02^-3/4sin v ). ]

These curves are circles in affine planes in ℝ^4 parallel to the Ox^3x^4 plane and
their radii are (4κ_01^-3/4)/(3√(C̃_1)) and (4κ_02^-3/4)/(3√(C̃_1)), respectively.

At a boundary point, using the coordinates (μ, v), we get that the tangent plane to the closure of S_C̃_1 is spanned by a vector which is tangent to the corresponding circle and by

    [ (-√(1-(4/3√(C̃_1)κ_0i^-3/4)^2)sinμ(κ_0i), √(1-(4/3√(C̃_1)κ_0i^-3/4)^2)cosμ(κ_0i),0,0 ), ]

where i=1 or i=2.

Thus, in order to construct a complete biconservative surface in 𝕊^3, we can expect to glue along the boundary two biconservative surfaces of type S_C̃_1, corresponding to the same C̃_1. In fact, if we want to glue two surfaces corresponding to C̃_1 and C̃_1^' along the boundary, then these constants have to coincide and there is no ambiguity concerning along which circle of the boundary we should glue the two pieces. But this process is not as clear as in ℝ^3 since we should repeat it infinitely many times.

Further, as in the ℝ^3 case, we change the point of view and use the intrinsic characterization of the biconservative surfaces in 𝕊^3.

The surface (D_C_1,g_C_1) defined in Section <ref> is not complete but it has the following properties.


Consider (D_C_1, g_C_1). Then, we have

  
  (a) K_C_1(ξ,θ)=K(ξ,θ),
  
    1-K(ξ,θ)=1/9ξ^8/3>0,   K^'(ξ)=-8/27ξ^5/3

  and K≠ 0 at any point of D_C_1;
  
  (b) the immersion ϕ_C_1:(D_C_1, g_C_1)→𝕊^3 given by
  
    ϕ_C_1(ξ,θ)=(√(1-1/C_1ξ^2)cosζ(ξ),√(1-1/C_1ξ^2)sinζ(ξ),cos(√(C_1)θ)/√(C_1)ξ, sin(√(C_1)θ)/√(C_1)ξ),

  is biconservative in 𝕊^3, where
  
    ζ(ξ)=±_ξ_00^ξ√(C_1)τ^4/3/(-1+C_1τ^2) √(-τ^8/3+3C_1τ^2-3) dτ+c_1,

  with c_1∈ℝ a constant and ξ_00∈(ξ_01,ξ_02).




The first item follows by standard arguments. For the second item, we note that choosing C̃_1=3^1/4·16C_1 in (<ref>) and using the change of coordinates (κ,v)=(3^-3/2ξ^4/3,(3^-1/8√(C_1)θ)/4), the metric induced by Y_3^1/4· 16C_1 coincides with g_C_1.

Then, we define ϕ_C_1 as

    ϕ_C_1(ξ,θ)=Y_3^1/4· 16C_1(3^-3/2ξ^4/3,3^-1/8√(C_1)θ/4).




The limits lim_ξ↘ξ_01ζ(ξ)=ζ(ξ_01) and  lim_ξ↗ξ_02ζ(ξ)=ζ(ξ_02) are finite.



For simplicity, we choose ξ_00=(9C_1/4)^3/2.



The immersion ϕ_C_1 depends on the sign ± and on the constant c_1 in the expression of ζ. As the classification is up to isometries of 𝕊^3, the sign and the constant are not important, but they will play an important role in the gluing process.





The construction of complete biconservative surfaces in 𝕊^3 consists in two steps, and the key idea is to notice that (D_C_1,g_C_1) is, locally and intrinsically, isometric to a surface of revolution in ℝ^3.

The first step is to construct a complete surface of revolution in ℝ^3 which on an open dense subset is locally isometric to (D_C_1,g_C_1). We start with the next result.



Let us consider (D_C_1,g_C_1) as above. Then (D_C_1,g_C_1) is the universal cover of the surface of revolution in ℝ^3 given by

    ψ_C_1,C_1^∗(ξ,θ)=(χ(ξ)cosθ/C_1^∗, χ(ξ)sinθ/C_1^∗,ν(ξ)),

where χ(ξ)=C_1^∗/ξ,

    ν(ξ)=±_ξ_00^ξ√(3τ^2-(C_1^∗)^2(-τ^8/3+3C_1τ^2-3)/τ^4(-τ^8/3 +3C_1τ^2-3)) dτ+c_1^∗,

C_1^∗∈(0,(C_1-4/3^3/2)^-1/2) is a positive constant and c_1^∗∈ℝ is constant.



The immersion ψ_C_1, C_1^∗ depends on the sign ± and on the constant c_1^∗ in the expression of ν. We denote by S^±_C_1,C_1^∗,c_1^∗ the image of ψ_C_1, C_1^∗.




The limits lim_ξ↘ξ_01ν(ξ)=ν(ξ_01) and  lim_ξ↗ξ_02ν(ξ)=ν(ξ_02) are finite.


We note that the boundary of S^±_C_1,C_1^∗,c_1^∗ is given by the curves

    (C_1^∗/ξ_01cosθ/C_1^∗,C_1^∗/ξ_01sinθ/C_1^∗, ν(ξ_01))

and

    (C_1^∗/ξ_02cosθ/C_1^∗,C_1^∗/ξ_02sinθ/C_1^∗, ν(ξ_02))

These curves are circles in affine planes in ℝ^3 parallel to the Oxy plane and their radii are C_1^∗/ξ_01 and C_1^∗/ξ_02, respectively.

At a boundary point, using the coordinates (ν, θ), we get that the tangent plane to the closure of S^±_C_1,C_1^∗,c_1^∗ is spanned by a vector which is tangent to the corresponding circle and by the vector (0,0,1). Thus, the tangent plane is parallel to the rotational axis Oz.

Geometrically, we start with a piece of type S^±_C_1,C_1^∗,c_1^∗ and by symmetry to the planes where the boundary lie, we get our complete surface S̃_C_1,C_1^∗; the process is periodic and we perform it along the whole Oz axis.

Analytically, we fix C_1 and C_1^∗, and alternating the sign and with appropriate choices of the constant c_1^∗, we can construct a complete surface of revolution S̃_C_1,C_1^∗ in ℝ^3 which on an open subset is locally isometric to (D_C_1,g_C_1). In fact, these choices of + and -, and of the constants c_1^∗ are uniquely determined by the “first” choice of +, or of -, and of the constant c_1^∗. We start with + and c_1^∗=0.

The profile curve of S^±_C_1,C_1^∗,c_1^∗ can be seen as the graph of a function depending on ν and this allows us to obtain a function F such that the profile curve of S̃_C_1,C_1^∗ to be the graph of the function χ∘ F depending on ν and defined on the whole Oz (or Oν). The function F:ℝ→[ξ_01,ξ_02] is periodic and at least of class C^3.



The surface of revolution given by

    Ψ_C_1,C^∗_1(ν,θ)=((χ∘ F)(ν)cosθ/C^∗_1, (χ∘ F)(ν)sinθ/C^∗_1,ν),      (ν,θ)∈ℝ^2,

is complete and, on an open dense subset, it is locally isometric to (D_C_1, g_C_1). The induced metric is given by

    g_C_1, C^∗_1(ν,θ)=3F^2(ν)/3F^2(ν)- (C^∗_1)^2(-F^8/3(ν)+3C_1F^2(ν)-3)dν^2 +1/F^2(ν)dθ^2,

(ν,θ)∈ℝ^2. Moreover, K≠ 0 at any point of that open dense subset, and 1-K>0 everywhere.


From Theorem <ref> we easily get the following result.


The universal cover of the surface of revolution given by Ψ_C_1,C^∗_1 is ℝ^2 endowed with the metric g_C_1,C^∗_1. It is complete, 1-K>0 on ℝ^2 and, on an open dense subset, it is locally isometric to (D_C_1,g_C_1) and K≠ 0 at any point. Moreover any two surfaces (ℝ^2,g_C_1,C_1^∗) and (ℝ^2,g_C_1,C_1^∗') are isometric.


The second step is to construct effectively the biconservative immersion from (ℝ^2,g_C_1,C_1^∗) in 𝕊^3, or from S̃_C_1,C_1^∗ in 𝕊^3. The geometric ideea of the construction is the following: from each piece S^±_C_1,C_1^∗,c_1^∗ of S̃_C_1,C_1^∗ we “go back” to (D_C_1,g_C_1) and then, using ϕ_C_1 and a specific choice of + or - and of the constant c_1, we get our biconservative immersion Φ_C_1,C_1^∗. Again, the choices of + and -, and of the constant c_1 are uniquely determined  (modulo 2π, for c_1) by the “first” choice of +, or of -, and of the constant c_1 (see <cit.> for all details).




Some numerical experiments suggest that Φ_C_1,C_1^∗ is not periodic and it has self-intersections along circles parallel to Ox^3x^4.

The projection of Φ_C_1,C_1^∗ on the Ox^1x^2 plane is a curve which lies in the annulus of radii √(1-1/(C_1ξ_01^2)) and √(1-1/(C_1ξ_02^2)). It has self-intersections and is dense in the annulus.




Concerning the biharmonic surfaces in 𝕊^3 we have the following classification result.


Let φ:M^2→𝕊^3 be a proper biharmonic surface. Then φ(M) is an open part of the small hypersphere 𝕊^2(1/√(2)).




§ 

In the c=0 case, the idea was to construct, by symmetry, a complete biconservative surface in ℝ^3 starting with a piece of a biconservative surface. We illustrate this in the following figure obtained for C_0=1.


[inner sep=0pt] (local) at (0,0) 
    < g r a p h i c s >
;

[inner sep=0pt] (global) at (10,0) 
    < g r a p h i c s >
;
[->,thin](local)–(global);





In the c=1 case, the construction of a complete biconservative surface in 𝕊^3 can be summarized in the next diagram, obtained for C_1=C_1^∗=1, c_1^∗=0 and we started with + in the expression of ν.


[inner sep=0pt] (manifold) at (3.6,0) 
    < g r a p h i c s >
;
(4.3,-2) node[below] (M^2,g);
[inner sep=0pt] (band) at (12,0) 
    < g r a p h i c s >
;

[->,>=triangle 45] (10,0) – (14,0);
[->,>=triangle 45] (10.5,-2.5) – (10.5,2.5);
(11.2,0) node[below] ξ_01;
(12.9,0) node[below] ξ_02;
(14.1,0) node[below] ξ;
(10.5,2.6) node[left] θ;
(12,-2.5) node[below] (D_C_1,g_C_1);
[<->,thick](manifold)– node[midway,fill=none,above] ISOMETRY(9.5,0);
[-,thick](11,-3.3)-| (11,-4);
[-,thick](11,-4)-| (13,-4);
[->,thick](13,-4) – (13,-8) node[midway,sloped,fill=none,below]  ϕ_C_1=ϕ^±_C_1,c_1 node[midway,sloped,fill=none,above] BICONSERVATIVE;
[ellipse, fill=brown, inner sep=0.2in] at (13,-9.3)𝕊^3;
[<-,thick](10.7,-3.3)-| (10.7,-4);
[->,thick](10.7,-4) –(3.5,-4) node[midway,sloped,fill=none,below] ψ_C_1,C_1^∗=ψ^±_C_1,C_1^∗,c_1^∗ node[midway,sloped,fill=none,above] ISOMETRY ;
[inner sep=0pt] at (2,-4)
    < g r a p h i c s >
;
(2,-5.5) node[below] S^±_C_1,C_1^∗,c_1^∗⊂ℝ^3;
[inner sep=0pt] (graph6) at (5.3,-9.3) 
    < g r a p h i c s >
;
(5.5,-10.2) node[below] S̃_C_1,C_1^∗⊂ℝ^3 complete;
[->,thick](2,-6.2)–(5,-8)  node[pos=0.5,sloped, fill=none,above] playing with the node[pos=0.5,sloped,fill=none,below] constant c_1^∗ and ±;





[scale=2]
        (A) at (4,-4.2) ;
        (B) at (6.2,-4.1);
        [blue] plot [smooth,tension=1]
        coordinates (A) (2.4,-3.2) (2.8,-2.7) (5.2,-2.6) (5.7,-3.6) (B)
        [arrow inside=end=stealth,opt=red,scale=20.2,0.4,0.75,0.999];

(6.5,-6.4) node[right] [ playing with the constant;                c_1  and ± ];





The projection of Φ_1,1 on the Ox^1x^2 plane is represented in the next figure (c_1=0).



[xscale=1,yscale=1]

[inner sep=0pt]  at (.2,0)
    < g r a p h i c s >
;
[->,>=triangle 45] (-3,0) – (3.5,0);
[->,>=triangle 45] (0,-3) – (0,3);
(3.5,0) node[below] x^1;
(0,3) node[left] x^2;







The last two figures represent the signed curvature of the profile curve of S̃_C_1,C_1^∗ and the signed curvature of the curve obtained projecting Φ_1,1 on the Ox^1x^2 plane.



[xscale=1,yscale=1]

[inner sep=0pt]  at (.4,0)
    < g r a p h i c s >
;
[->,>=triangle 45] (-4.5,0.2) – (5,0.2);
[->,>=triangle 45] (0.85,-1.2) – (0.85,1.5);
(5,0.2) node[below] ν;
(0.85,1.5) node[left] κ;






[xscale=1,yscale=1]

[inner sep=0pt]  at (.2,0.4)
    < g r a p h i c s >
;
[->,>=triangle 45] (-4.5,-0.5) – (5,-0.5);
[->,>=triangle 45] (0.7,-1) – (0.7,2);
(5,-0.5) node[below] ν;
(0.7,2) node[left] κ;










99

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