Willmore-Like Tori in Killing Submersions

The first variation formula and Euler-Lagrange equations for Willmore-like surfaces in Riemannian 3-spaces with potential are computed and, then, applied to the study of invariant Willmore-like tori with invariant potential in the total space of a Killing submersion. A connection with generalized elastica in the base surface of the Killing submersion is found, which is exploited to analyze Willmore tori in Killing submersions and to construct foliations of Killing submersions made up of Willmore tori with constant mean curvature.


Introduction
In 1811 S. Germain pointed out that the free energy controlling the physical system associated with an elastic plate is the total squared curvature or bending energy.For a surface  in R 3 the bending energy is given by where  denotes the mean curvature of , which, according to a popular model in Biophysics, can be seen also as part of the elastic energy of fluid membranes [1].Notice that, for curves  isometrically immersed in a Riemannian manifold,  is nothing but the geodesic curvature of , , and W() boils down to Following a classical model of D. Bernoulli, planar curves minimizing the bending energy E() can be thought of as the equilibrium positions of thin elastic rods (or elasticae).Elastica (and their generalizations) in semi-Riemannian manifolds is a classical subject which has many applications to Physics where they can be used, among other applications, to produce mathematical models of stiff rods, stiff polymers, vortices in fluids, superconductors, membranes, mechanical properties of DNA molecules, and also imaging and visual perception (for more details and references on this subject, see, for example, [2]).
The study of the variational problem associated with W() for surfaces in R 3 goes back to Blaschke's school in the 1920s.Thus, G. Thomsen in [3] obtained the first variation and the Euler-Lagrange equations corresponding to (1).Years later, T. J. Willmore reintroduced the problem in [4] and this is reason why W() is also known as the Willmore energy of the surface and its critical points are called Willmore surfaces.The Willmore energy is a conformal invariant [5,6], a property which has been constantly exploited in the vast literature on the subject (see, for example, [2,[7][8][9][10][11][12][13]).In the last years, there has been also an intensive investigation of Willmore surfaces in connection with Willmore problem, that is, the determination of the minima for the Willmore energy within a given topological class.In particular, T. Willmore proposed in 1965 the following conjecture: for every smooth immersed torus  in R 3 , W() ≥ 2 2 and the equality is obtained at the Clifford torus.The conjecture has been recently proved in [14], although prior to this proof the Willmore conjecture had already been shown for many special cases (see [14][15][16][17] and references therein).
A generalization of the Willmore energy is due to B-Y Chen [18].He extended the Willmore functional to any submanifold   of any Riemannian manifold   so that the conformal invariance property is preserved.Given a 2

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Riemannian submanifold   of   the Chen-Willmore functional is defined by where  and   denote the mean curvature and the extrinsic scalar curvature of , respectively.The energy CW() is conformally invariant and its critical points are known as Chen-Willmore submanifolds [18,19].In contrast to the case of Willmore surfaces in ambient spaces of constant curvature, there are not many results concerning Chen-Willmore submanifolds in background spaces with nonconstant sectional curvature.Nevertheless, Willmore surfaces and submanifolds have strong connections in Physics with applications to (just to mention a few) the analysis of elastic plates and biological membranes [1,20] and to bosonic string theories and sigma models (for more details, see [21,22] and references therein).Several of these applications are based on a beautiful link between Willmore surfaces and elastica which can be established by using a symmetry reduction procedure [12].
In this paper we will study the following generalization of the Willmore energy: where  is a surface of a 3-dimensional Riemannian manifold ,  denotes the mean curvature of , and Φ ∈  ∞ ().Here, W Φ () will be called Willmore energy with a potential Φ and in Section 3 the first variation formula and Euler-Lagrange equation associated with W Φ are computed.
On the other hand, our analysis will be focused mainly on surfaces living in the total space of a Killing submersion.A Riemannian submersion  :  →  of a 3-dimensional Riemannian manifold  over a surface  will be called a Killing submersion if its fibers are the trajectories of a complete unit Killing vector field .Killing submersions are determined by two functions on , its Gaussian curvature and the so-called bundle curvature (for more details, see [23,24] and Section 2).A remarkable family of Killing submersions are the homogeneous 3-spaces.With the exception of the hyperbolic 3space, H 3 (−1), simply connected homogeneous Riemannian 3-manifolds with isometry group of dimension 4 or 6, can be represented by a 2-parameter family E(, ), where ,  ∈ R.These E(, )-spaces are 3-manifolds admitting a global unit Killing vector field whose integral curves are the fibers of a certain Riemannian submersion over the simply connected constant Gaussian curvature surface () and, therefore, determine Killing submersions  : E(, ) → () (for more details, see [25]).A local description of these examples can be given by using the so-called Bianchi-Cartan-Vranceanu spaces (see Section 2).
In Section 4, we establish a connection between invariant Willmore-like tori with -invariant potentials in the total space  of Killing submersions and elastic curves with related potentials in the base surface .We show also how one can apply this connection to study -invariant Willmore tori in the total space  of a Killing submersion and, in particular, to the case of -invariant Willmore tori in orthonormal frame bundles on Riemannian surfaces.Finally, the previous results are applied in Section 5 to the construction of foliations, of certain Killing submersions, consisting of Willmore tori with constant mean curvature.

Killing Submersions
A Riemannian submersion  :  →  of a 3-dimensional Riemannian manifold  over a surface  will be called a Killing submersion if its fibers are the trajectories of a complete unit Killing vector field  (for more details, see [23,24]).Fibers of Killing submersions are geodesics in  and form a foliation called the vertical foliation.Most of the geometry of a Killing submersion is encoded in a pair of functions: , .
Here  represents the Gaussian curvature function of the base surface , while  denotes the so-called bundle curvature which is defined as follows.Since  is a (vertical) unit Killing vector field, then it is clear that for any vector field, , on , there exists a function   (which, a priori, depends on the vector field ) such that ∇   =    ∧ , where ∇ denotes the Levi-Civita connection of .Actually, it is not difficult to see that   does not depend on the vector field  (see [23] for details) so we get a function  ∈  ∞ (), the bundle curvature, satisfying where ∧ denotes the vector product.The bundle curvature is obviously constant along the fibers and, consequently, it can be seen as a function on the base surface,  ∈  ∞ ().In product spaces  × R the projection over the first factor is a Killing submersion, so its bundle curvature is  ≡ 0.More generally, it is easy to deduce from (5) that  ≡ 0 in a Killing submersion, if and only if, the horizontal distribution in the total space, , is integrable.It is customary to denote the total space of a Killing submersion by (, ).Existence of Killing submersions over a given simply connected surface  for a prescribed bundle curvature,  ∈  ∞ (), has been proved in [24].Uniqueness of these submersions, up to isomorphisms, is guaranteed under the assumption that the total space is also simply connected.This fact leads to the classification of Killing submersions on a simply connected surface as quotients of simply connected total spaces under vertical translations.In particular, we can suppose that fibers have finite length since, if necessary, we can always take a suitable quotient under a vertical translation in order to get a circle bundle over .The following result provides existence of Killing submersions with prescribed bundle curvature over arbitrary Riemannian surfaces Theorem 1.Let  be a Riemannian surface with Gaussian curvature  and choose any function  ∈  ∞ ().Then there exists a Killing submersion over  with bundle curvature , ((, ), ).Moreover, ((, ), ) can be chosen having compact fibers.
Proof.If  is simply connected, then the result was proved in [24].Therefore we assume that  is not simply connected and, thus, it has a nontrivial fundamental group  1 ().
Denote by   its universal covering and by  :   →  the universal covering map.Now, if  is the metric on , then we consider   endowed with the metric   =  * () so that  is a Riemannian covering map.Denote by Γ the group of deck transformations of (  , ) which is a subgroup of the group of isometries (  ,   ) isomorphic to  1 ().
Consider   =  ∘  ∈  ∞ (  ).Since   is simply connected, we know there exists a unique (up to isomorphisms) Killing submersion   :   →   with simply connected total space   and bundle curvature   (see [24] for details).Now, if ℎ ∈ Γ, then it is obvious that   ∘ ℎ =   .Consequently, once an initial condition has been chosen, there exists a unique isometry ℎ ∈ (  ,   ) preserving the Killing fiber flow (ℎ *  = ) and satisfying   ∘ ℎ = ℎ ∘   .Hence, we have a monomorphism from Γ into (  ,   ) whose image Γ is isomorphic to Γ.It is not difficult to see that this determines a properly discontinuous action of Γ on   , defining a Riemannian covering map  : (  ,   ) → ( =   /Γ, ).Finally we define  :  →  by (()) = (  ()) which satisfies  * () = .It can be checked that it provides a Killing submersion over  with bundle curvature .[24], which include, as we will see later, the so-called Bianchi-Cartan-Vranceanu spaces for suitable choices of the functions , , .
Regardless of the values of the functions ,  ∈ C ∞ (Ω), the Riemannian metric given by ( 7) satisfies that  is a Killing submersion over (Ω, d 2  ) with  =   as unit vertical Killing field.The bundle curvature and the Gaussian curvature are given by where Δ  represents the Laplacian with respect to the standard metric in the plane.
Example 3 (Bianchi-Cartan-Vranceanu spaces).Particularizing the above construction one can get models for all Killing submersions over R 2 , H 2 () and the punctured sphere S 2 * (). where Then, the metric  2  (d 2 + d 2 ) in Ω  has constant Gaussian curvature  = .If, in addition, we choose  = − and  =  for some real constant , then one obtains the metrics of the Bianchi-Cartan-Vranceanu spaces E(, ) ≡ Ω  × R ([25, Section 2.3]): A simple computation, using (9), gives  = .Thus the Bianchi-Cartan-Vranceanu spaces (BCV spaces) can be seen as the canonical models of Killing submersions with constant bundle curvature and constant Gaussian curvature.
Cartan in [26] showed that the examples above cover, in fact, all possible 3-dimensional homogeneous spaces with 4dimensional isometry group.The BCV family also includes two real space forms, which have 6-dimensional isometry group.
Example 4 (bundle-like metrics).Let  be the threedimensional total space of a principal fiber bundle on a surface, , and denote by  :  →  the natural projection.Consider a principal connection  and denote by  2 the metric on the fiber.Now, given a positive smooth function  ∈  ∞ () and any Riemannian metric, , on , we can define the following generalized Kaluza-Klein Riemannian metric on : In particular, if the function  is constant, then  is called a Kaluza-Klein metric or a bundle-like metric.As it is well known, the following properties are satisfied by the above class of metrics: (i) The natural action of the structure group, G, on  is carried out by isometries of (, ).Consequently, the fiber flow is associated with a Killing vector field of (, ).
(ii) The natural projection  : (, ) → (, ) is a Riemannian submersion.Furthermore, its leaves are geodesics in (, ), if and only if,  is a constant function, i.e.,  is a bundle-like metric.
(iii) Setting one sees that generalized Kaluza-Klein metrics are conformal to bundle-like metrics.
Hence, once a suitable conformal change has been made (if necessary), every bundle-like metric provides a Riemannian submersion whose fiber flow is associated with a vertical unit Killing vector field.In other words, we have a Killing submersion.Moreover, we see that homogeneous 3-spaces with 4-dimensional isometry group pose bundle-like metrics and their classes of congruence are completely determined, up to topology, for a pair of constants: the curvature of the base surface , and the bundle curvature .

Willmore-Like Energies
Let (,  = ⟨, ⟩) be a 3-dimensional Riemannian manifold, Φ ∈  ∞ () a smooth function, and  a compact surface with no boundary.In the space of isometric immersions of  in , I(, ), we define the following Willmore-like energy: where  ∈ I(, ),   denotes the mean curvature function of  and   is the induced element of area via the immersion .
The main purpose of this section is to compute the field equations associated with this kind of functionals.To this end, we consider a variation of  ∈ I(, ).That is, a smooth map Ψ :  × (−, ) →  satisfies the following conditions: (i) For any  ∈ (−, ) the map   :  →  defined by   () = Ψ(, ) belongs to I(, ).
Then, the vector field on Ψ given by determines a vector field along , () := (, 0), which is called the variational vector field associated with the variation Ψ.Thus, we can identify the tangent space   (I(, )) with the space of vector fields along  and, consequently, we have Now, the following lemma collects some formulae that will be needed in the sequel.They can be obtained using similar computations to those included in [13,22].For simplicity, we have eliminated the symbol  in most of our notation.

Lemma 5.
With the previous notation, the following statements hold: (1) Let  → (, ) be the mean curvature vector field of   at  ∈ , then where Δ is the Laplacian associated with the connection  in the normal bundle of (), Ã stands for the Simons operator [13], Ric is the Ricci curvature of , and N denotes the unit normal vector field of .Here () ⊺ and () ⊥ denote tangential and normal components, respectively.
(2) The variation of the area element is given by the following formula: where  is the one-form defined by () =   ( ⊺ , ).
Combining the above two formulae and an argument involving integration by parts, we obtain the first variation formula of the first term in (15): where   denotes the Gaussian curvature of  endowed with the induced metric associated with the initial immersion , and  stands for its extrinsic Gaussian curvature (that is, the sectional curvature of  on the tangent plane of ).Now, Lemma 5 can be used once more to compute the first variation formula of the second term in (15): where ∇Φ denotes the gradient of the potential Φ. Now, it is easy to see that Φ ∧  =  ⊺ (Φ) and then Finally, we combine ( 20) and ( 22) to characterize the extremals of W Φ () as the solutions of the following Euler-Lagrange equation: Along this note, solution surfaces of (23), that is extremals of W Φ (), will be called Willmore-like surfaces.Hence, we have Theorem 6.Let (, ) be a 3-dimensional Riemannian manifold, Φ ∈  ∞ () a smooth function, and  a compact surface isometrically immersed in .Then  is a Willmore-like surface, if and only if, ( 23) is satisfied.
Notice that if the potential Φ is a constant function, say  (in particular, the case in which W Φ () is the Willmore energy in a space of constant curvature ), then every minimal surface is automatically a Willmore-like surface.However, if the potential is not constant, it follows from ( 23) that a minimal surface, , is Willmore-like if and only if ∇Φ is tangent to  anywhere.

Vertical Willmore Tori in Killing Submersions
Assume we have a Killing submersion  : (, ) →  of a 3-dimensional Riemannian manifold  over a surface .
Let  be an immersed curve in (, ), then   =  −1 () is a surface isometrically immersed in  (by the natural inclusion  = ) which is invariant under G := {  :  ∈ R}, the one-parameter group of isometries associated with the Killing vector field .In fact, any G-invariant surface in , , is obtained in this way:  =  −1 (), for some curve  of .It is usual to call   the vertical tube (or vertical cylinder) shaped on the curve .Notice that   is embedded if  is a simple curve, and it is a torus when  is closed and G = S 1 is a circle group.
If  is parametrized by its arc-length, then any horizontal lift of , , is also arc-length parametrized.Now, using as coordinate curves the horizontal lifts of  and the fibers of the submersion, these vertical tubes   can be parametrized by Ψ(, ) =   (()) and, as a consequence,   are flat.It is also known that the mean curvature of these surfaces  is related to the curvature function of the corresponding cross sections by the formula (for details, see [27]): denoting the geodesic curvature of  in .Proof.Let   be a torus over .The S 1 -action on (, ) can be naturally extended to I(, ) by W Φ () = W Φ (  ∘ ), for all  ∈ R,  ∈ I(, ).On the other hand the space, Σ, of the S 1 -invariant immersions, can be identified with In this setting, we can apply the symmetric criticality principle of Palais [28] to reduce symmetry and then   =  −1 () is a Willmore torus, if and only if, it is an extremal of W Φ restricted to Σ. Finally, we use (24) to conclude that this happens, precisely, when  is a critical curve for E  .
Remark 8.If the potential  is constant, then extremals of the above energy E  are known as elastic curves (see [29]).Thus, critical curves  of E  will be called elasticae with potential 4.
As an application, we obtain the following known characterization of closed elasticae with potential which will be used later Corollary 9. A closed unit speed curve () of a Riemannian surface  is an extremal of the elastic-like energy E  () = ∫  ( 2 + 4) ( being its geodesic curvature in ), if and only if, where ( ) Since   is flat    = 0, moreover, a straightforward computation shows that on these vertical flat tori 2 + Ric(N, N) = , N being the unit normal along   .Hence (23) gives that   must provide a solution of the following differential equation: where  is the mean curvature function of   in (, ).
Finally we combine (24) and (27) to conclude that the curvature of the elastica with 4-potential must satisfy

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→  denoting the unit normal of  in .
On the other hand, as it has been mentioned previously in the introduction, in the context of Killing submersions the Willmore problem can be described as the variational problem associated with the so-called Chen-Willmore energy: where  ∈ I(, (, )), and  denotes the extrinsic Gaussian curvature associated with the immersion , that is the sectional curvature of (, ) restricted to the tangent bundle () of the corresponding surface.Recall that extremals of W are known as Willmore surfaces.It should be noticed that, in general, the energy (29) does not coincide with those given in (15), because the potential in ( 29) is defined on the Grassmannian of two planes, but if ((, ), ) has constant sectional curvature, , then the Willmore energy (29) would correspond to a Willmore-like energy with constant potential Φ = .Nevertheless, the situation for vertical cylinders shaped on closed curves of the base surface changes and we have the following.
Proposition 10.Assume that  : (, ) →  is a Killing submersion with compact fiber, i.e., G = S 1 .Then, a vertical torus over a closed curve  in ,   =  −1 (), is a Willmore surface in (, ), if and only if,   is an extremal of the following Willmore-like energy: And this happens in turn, if and only if,  is an elastica with potential 4 2 in .
Proof.The extrinsic Gaussian curvature of vertical cylinders   is related to the bundle curvature by where  denotes the horizontal lift of the unit tangent vector field  =   .Now, combining ( 29) and ( 31) and applying similar arguments to those used in Proposition 7 and the symmetric criticality principle of Palais [28], we draw the conclusion.
It is well known that compact minimal surfaces in spaces of constant curvature are examples of Willmore surfaces.The Euclidean 3-space does not contain compact minimal surfaces.However, the round three-sphere has plenty of compact minimal surfaces.These Willmore surfaces can be conformally projected by the stereographic projection to the Euclidean space, obtaining so examples of Willmore surfaces in the Euclidean 3-spaces because the Willmore energy is invariant under conformal changes in the surrounding metric.Both the Euclidean 3-space and the round 3-sphere provide examples of total 3-spaces associated with Killing submersions.In the former case  =  = 0 and  is just the projection over a plane.In the spherical case  = 4 2 is a constant and  is the well-known Hopf map from S 3 on S 2 .As it has been said previously, besides these spaces (whose group of isometries has dimension 6) the class of homogeneous three-space with 4-dimensional group of isometries provides also nice examples of Killing submersions with both  and  being constant.In this case vertical minimal tori are automatically Willmore (see [30] and references therein).In contrast, for general Killing submersions we have the following direct consequence of Proposition 10 and (28) Corollary 11.Let  : (, ) →  be a Killing submersion and () be a closed geodesic in , then   =  −1 () is a Willmore torus in (, ) if and only if () is a tangent line of  2 .
As an application, let us consider, for example, the unit round 2-sphere, S 2 ⊂ R 3 , which we assume to be centered at the point   = (0, 0, 1).Now, choose the nonnegative function ℎ ∈  ∞ (S 2 ) obtained as the restriction of the height function, in R 3 , to the plane  = 0.It is clear that the gradient flow associated with ∇ℎ is made up of the great circles through the origin, which obviously are geodesics.On the other hand, we know that there exists a Killing submersion ( = 1, ℎ) → S 2 with bundle curvature ℎ [24].This Killing submersion is unique up to isomorphism and in this case, it coincides with the Hopf map  : S 3 → S 2 where the 3-sphere is endowed with a suitable metric.Therefore, in the corresponding conformal class, we get a class of minimal Willmore tori having a pair of great circles in common.
To end this section, we consider frame bundles of surfaces as a particular case of Killing submersions and study Willmore surfaces in this context.Let (, ) be a Riemannian surface with metric  and Gaussian curvature .Denote by FM its orthonormal frame bundle.Then, the natural projection,  : FM →  gives a principal bundle with structure group O(2).Certainly the geometry of FM is more closely related to  than that of any other principal bundle.Remember, for example, the popular definitions of the Levi-Civita connection and the curvature operator of (, ) in terms of the connection form  and the curvature form Ω of FM.Now, FM can be endowed with bundle-like metrics as follows: providing a large class of Killing submersions.The corresponding bundle curvature  can be computed using (31) and [31]: where  is the third O'Neill invariant [32] defined in terms of the connection form  = (  ) by and we obtain where || = 1.Hence, from (33) we see that the bundle curvature of the frame bundle Killing submersion is given by Thus, combining Proposition 10 and (37) we have the following.
Corollary 12. Let  be a closed curve in a Riemannian surface, , with Gaussian curvature .Then   =  −1 () is a Willmore torus in the orthonormal frame bundle FM if and only if  is an extremal of the following elastic energy with potential acting on the space of closed curves in .To this end, we consider the following construction.

Killing Submersions and Willmore Tori Foliations
Let () be a smooth function defined on a real interval  = (, ) and consider the surface  × S 1 endowed with the warped product metric  2 + () 2 .The corresponding Riemannian surface is called a warped product surface and is simply denoted by   := ×  S 1 .The group S 1 acts trivially on   by isometries and the orbits of this action,   () := {}×S 1 , are called fibers of   .We first want to determine the warped surfaces for which all fibers are extremals of being the Gaussian curvature of   .According to the results in previous section, they will give rise to orthonormal frame bundles admitting foliations by Willmore tori with constant mean curvature.Then, we have where  2 ,  ∈ R + and Lam denotes the Lambert function [33].
Proof.The Euler-Lagrange equation for closed extremals of (39) in a surface are obtained using Corollary 9: 2  () +  () 3 + 2 ()  () −  () 2  () where  is the arc-length parameter of the curve, the derivative with respect to  is denoted by ( )  = /, () is the Gaussian curvature of the surface along the curve, and  →  is the normal vector to the curve in the surface.Assume that every orbit of   is critical for (39).In   the geodesic curvature of any orbit, (, ) = − ḟ ()/(), is constant on it (where now we are using overdots for /) while the Gaussian curvature is given by (, ) = − f ()/(s).Observe that  →  ≡ / is the normal vector field to the orbits on the surface.Since all orbits are critical, (41) tells us that 2 () ḟ () f () − ( ḟ ()) 3 + 2 () f () ...

Proposition 14 .
Let   := ×  S 1 be a warped product surface all whose fibers are critical for the energy functional (39).Then, locally, either  is constant and   is a Riemannian product surface, or () is determined by +  2 )  − 2Lam (− (1/)   1 −(2/) ) − Lam 2 (− (1/)   1 −(2/) )) 1/2 ,(40) = /,  →  denotes the unit normal of  in  and  is the Gaussian curvature of .Proof.If  is the Gaussian curvature of , choose any  ∈  ∞ () and a Killing submersion  : (, ) →  with compact fibers, as guaranteed by Theorem 1. Define Φ = ∘ on (, ).Now, from Proposition 7, we have that  is an extremal of the elastic-like energy E  () = ∫  ( 2 + 4), if and only if,   is a critical point of W Φ 5.1.Willmore Foliations of Orthonormal Frame Bundles onSurfaces.We first show how one can get minimal Willmore tori families foliating the frame bundle of compact rotation surfaces in R 3 .Let   be the rotation torus swept out by revolving around the -axis a closed curve  contained in the half plane  = 0,  > 0. Denote by   () the meridian obtained by rotating an angle  the profile curve .It is obvious that  −1 (  ()) is a minimal torus in FS  , where  : FM →   denotes the natural projection.Moreover, meridians are not only geodesics, but also they are extremals of the elastic energy A because  →  () = 0 (seeCorollary 12 and (28)).Consequently, { −1 (  ()) :  ∈ R} constitutes a foliation of FS  by minimal Willmore tori.Now we want to obtain orthonormal frame bundles of surfaces admitting foliations by nonminimal Willmore tori.