Biharmonic Maps from Tori into a 2-Sphere?

Zeping WANGYe-Lin OUHanchun YANG

Abstract Biharmonic maps are generalizations of harmonic maps.A well-known result on harmonic maps between surfaces shows that there exists no harmonic map from a torus into a sphere(whatever the metrics chosen)in the homotopy class of maps of Brower degree±1.It would be interesting to know if there exists any biharmonic map in that homotopy class of maps.The authors obtain some classifications on biharmonic maps from a torus into a sphere,where the torus is provided with a flat or a class of non- flat metrics whilst the sphere is provided with the standard metric.The results in this paper show that there exists no proper biharmonic maps of degree±1 in a large family of maps from a torus into a sphere.

Keywords Biharmonic maps,Biharmonic tori,Harmonic maps,Gauss maps,Maps into a sphere

1 Introduction

All objects including manifolds,tensor fields,and maps studied in this paper are supposed to be smooth.

where ? is a compact domain of M.Analytically,a harmonic map is a solution of a system of 2nd order PDEs

where τ(?)is called the tension field of the map ?.

Biharmonic maps are generalizations of harmonic maps,which are maps ? :(M,g) ?→(N,h)between Riemannian manifolds that are critical points of the bienergy functional defined by

where ? is a compact domain of M.Biharmonic map equation is a system of 4-th order nonlinear PDEs(see[24])

where RNdenotes the curvature operator of(N,h)defined by

As a harmonic map is always a biharmonic map,we call a biharmonic map that is not harmonic a proper biharmonic map.

Since 2000,the study of biharmonic maps has been attracting growing interest of many mathematicians and it has become a popular topic of research with many interesting results.For some recent geometric study of general biharmonic maps,we refer the readers to[3,5,8,28,32,35–36,39,43,46,50]and the references therein.For some recent progress on biharmonic submanifolds(i.e.,submanifolds whose defining isometric immersions are biharmonic maps),see a recent survey[1,7,9,11–19,22–23,25–26,30–31,34,38,41,44–45,48,51]and the references therein.For biharmonic conformal immersions and submersions,see[4,29,37,40,42,49]and the references therein.

For harmonic maps between surfaces,a very interesting result proved by Eells and Wood in[21]states that there exists no harmonic map from a torus T2into a sphere S2(whatever the metrics chosen)in the homotopy class of maps of Brower degree±1.It would be interesting to know if there exists any proper biharmonic map from a torus T2into a sphere S2(whatever the metrics chosen)in the homotopy class of maps of Brower degree±1.Motivated by this,we study biharmonic maps from a torus into a sphere in this paper.We are able to obtain some classifications of proper biharmonic maps in a large family of maps from T2into S2which include the Gauss map of the torus T2?→R3and the compositionsof some immersions of T2into S3followed by the Hopf fibration.Here,the torus is provided with a flat or a class of non- flat metrics whilst the sphere is provided with the standard metric(see Theorem 3.1 and Theorem 4.1).

2 Constructions of Maps from a Torus into a 2-Sphere

In order to study biharmonic maps from a torus T2into a sphere S2,we need to have a good source of maps from a torus into a sphere.In this section,we present three ways to construct maps from T2into S2.

(1)Construction via Hopf fibration.For any mapwe have a map from torus into 2-sphere,whereis the Hopf fibration.Here,we view the Hopf fibration as the restriction of the Hopf construction of the standard multiplication of complex numbers,i.e.,with

(2)Construction via radial projection.For any mapwe have a map from torus into 2-sphere,wherewithis the radial projection from R3{0}onto S2.

(3)Construction via Gauss map of a torus.It is well known that ifis an immersion of a torus into R3,then the Gauss map gives a map from the torus into a 2-sphere defined bywith n(x)being the unit normal vector at the point x∈T2.

Example 2.1Letwithbe a family of immersions studied by Lawson[27].Then,the compositiongives a family of maps from a torus into a 2-sphere defined by

If we use geodesic polar coordinates(ρ,φ)on the 2-sphere,then this family of maps can be represented as

with

Example 2.2Let f:S1×S1?→R4be a family of immersions of flat tori into R4defined bywith constants A,B satisfyingPostcomposing this map with the radial projectionwe have a family of map F=withIf we denotethen=sins,then the family of the maps can be written aswithApplying construction via Hopf fibration,we have a map from T2into S2:

with

where(ρ,φ)are the geodesic polar coordinates on S2.

Remark 2.1As it was observed in[6,Example 3.3.18]that exceptthe family of maps Fsare embeddings of tori into S3and all these maps are harmonic maps with constant energy density called eigenmaps.

with

Example 2.4Letbe the standard embedding X(r,θ)=(asinr,(b+acosr)cosθ,(b+acosr)sinθ)of a torus into R3.A straightforward computation gives the induced metric and the unit normal vector field of the torus to be

and

respectively.If we use the geodesic polar coordinates(ρ,φ)on the unit sphere so that a generic point(x,y,z) ∈ S2is represented as(x,y,z)=(sinρ,eiφcosρ),then,the Gauss map of the torus can be written as

with

Example 2.5Letbe the standard embedding X(r,θ)=(asinr,(b+acosr)cosθ,(b+acosr)sinθ)of a torus into R3.Using the construction via radial projectionwithwe have a family of maps from tori into a 2-sphere given by ? =P ?X:T2?→ S2with

where

Again,with respect to the geodesic polar coordinates(ρ,φ)on the unit sphere so that a generic point(x,y,z)∈ S2is represented as(x,y,z)=(cosρ,eiφsinρ),then,this family of maps from tori into S2can be written as

with

which are rotationally symmetric maps.

The 6 families of maps given in Examples 1–6 lead us to study the biharmonicity of the following family of maps

Clearly,the family of maps defined by(2.8)includes families of maps defined in(2.2)–(2.3),(2.5),and parts of the families given in(2.4)and(2.7).Our main results in this paper include a complete classifications of proper biharmonic maps in the family of maps defined by(2.8),where the torus is provided with a flat or a non- flat metric whilst the sphere is provided with the standard metric(see Theorem 3.1 and Theorem 4.1).

3 Biharmonic Maps from a Flat Torus into a 2-Sphere

Lemma 3.1(see[43])Letbe a map between Riemannian manifolds with ?(x1,···,xm)=(?1(x),···,?n(x))with respect to local coordinates(xi)in M and(yα)in N.Then,? is biharmonic if and only if it is a solution of the following system of PDE’s:

where τ1,···,τnare components of the tension field of the map ?, ?, ? denote the gradient and the Laplace operators defined by the metric g,andare the components of the connection and the curvature of the target manifold.

In order to prove our classification theorem,we need the following lemma.

Lemma 3.2Letwith ?(r,θ)=(ar+bθ+c,mr+nθ+l).Then,? is biharmonic if and only if it solves the system

ProofOne can easily compute the connection coefficients of the domain and the target surfaces to get

and

A further computation gives the components of the Riemannian curvature of the target surface as

We compute the components of tension field of the map ? to have

and

Substituting(3.8)–(3.19)into(3.1),we conclude that ? is biharmonic if and only if it solves the system

which is equivalent to the system(3.2).Thus,the lemma follows.

Now we are ready to prove the following theorem that gives a classification of all biharmonic maps in a large family that includes most examples mentioned in Section 2.

Theorem 3.1The mapfrom a flat torus into a 2-sphere with ?(r,θ)=(ar+bθ +c,mr+nθ +l)is biharmonic if and only if one of the following cases happens:

(B)m=n=0.In this case,the map ?(r,θ)=(ar+bθ+c,l)is actually a harmonic map,or

ProofApplying Lemma 3.2 with σ(r)=1, λ(ρ)=sinρ,we conclude that the mapfrom a flat torus into a 2-sphere with ?(r,θ)=(ar+bθ+c,mr+nθ+l)is biharmonic if and only if

Substituting the last two equations into the first two equations of(3.21),we see that the biharmonicity of the map ? becomes

Noting that 0< ρ=ar+bθ+c< π (and hence)sin,we conclude that Equation(3.22)is equivalent to

We solve Equation(3.23)by considering the following cases.

Case Icosρ =0.This means that cos(ar+bθ+c)=0 for any r,θ∈ R.This,together with 0< ρ =ar+bθ+c< π,implies that a=b=0,Noting that the components of the tension field of the map ? are given by the last two equations of(3.21),we conclude that the solutions?a=b=0,,m,n,l∈ R?given in this case are actually harmonic maps.From this we obtain the case(A).

Case IIIn this case,the biharmonicity of the map ? is equivalent to

If m=n=0,then,m=n=0,a,b,c,l∈R are solutions of(3.24)and we see from the last two equations of(3.21)that the maps given by these solutions are actually harmonic maps.This gives the case(B).

Since the first equation of(3.26)means thatfor any r,θ∈ R,we conclude that a=b=0 and hence cos(2c)=0.Recalling that 2ρ=2c∈ (0,2π),we obtain solutionsIn these cases,It follows from the third equation of(3.21)that the first component of the tension fieldThus,the biharmonic mapsare proper biharmonic maps.From this we obtain the case(C).

Summarizing the above results we obtain the theorem.

As immediate consequences of Theorem 3.1,we have the following corollaries.

Corollary 3.1Forthe mapwithis a proper biharmonic map.In particular,the compositions of the family of harmonic embeddingsfollowed by the Hopf fibration

with

are proper biharmonic maps if and only if

Remark 3.1Note that Corollary 3.1 provides many new examples of proper biharmonic maps from a flat torus into a sphere,including special caseswithThese special cases,up to an isometry of the target sphere,are the same maps obtained in[39]by construction of orthogonal multiplication of complex numbers(see[39,Theorem 2.2]for details).These special cases of proper biharmonic map were also known as special solutions to the biharmonic equation for rotationally symmetric maps from a flat torus into a 2-sphere(see[33,50]).

Corollary 3.2The map

with

is neither a harmonic map nor a biharmonic map.

Corollary 3.3The Gauss map of the torus,X(r,θ)=(asinr,(b+acosr)cosθ,(b+acosr)sinθ),viewed as a mapfrom a flat torus into a sphere,is neither harmonic nor biharmonic.

Corollary 3.4Letwithbe the family of immersions studied in[27].Then,the family of maps from a torus into a 2-sphere defined by the construction via Hopf fibration

with

contains no proper biharmonic map.

4 Biharmonic Maps from a Non-Flat Torus into a 2-Sphere

In this section,we will study biharmonic maps from a non- flat torus into a 2-sphere.The non- flat metric on the torus we consider is dr2+(k+cosr)2dθ2which is homothetic to the induced metric gT=a2dr2+(b+acosr)2dθ2(see Example 2.4 for details)from the standard embedding,X(r,θ)=(asinr,(b+acosr)cosθ,(b+acosr)sinθ).

Theorem 4.1The map from a non- flat torus into a 2-sphere,(k>1)with ?(r,θ)=(ar+bθ+c,mr+nθ+l)is biharmonic if and only if it is harmonic.

ProofLet(k>1)with ?(r,θ)=(ar+bθ+c,mr+nθ+l).Using Lemma 3.2 with σ(r)=k+cosr, λ(ρ)=cosρ,we see that ? is biharmonic if and only if it solves the system

A straightforward computation using the last two equations of(4.1)yields

Substituting(4.2)into the first equation of(4.1),we have

We will solve Equation(4.3)by the following cases.

Case I

In this case,using the assumption thatand the fact that the functions 1,sin2ρ,cos2ρ and sin4ρ are linearly independent as functions of variable θ,we conclude from(4.3)that

From the fourth and the first equation of(4.4),we have a=m=n=0.In this case,we use the last two equations in(4.1)to conclude that the components of the tension field x= τ1=0,y= τ2=0.This implies that the map ? is actually harmonic.

Case IIb=0 and a=0.

In this case,substituting a=b=0 and(4.2)into the second equation of(4.1),we obtain

If c=0,then sinρ=0 and(4.5)reduces to m(k2?1)=0 which implies m=0 since k>1 by assumption.In this case,the last two equations in(4.1)imply that τ(?)=0 and hence ? is harmonic.

Note that k>1 and the functions 1,cosr,cos2r are linearly independent,then Equation(4.7)implies that sin2ρ=0 and reduces to k2?1=0,a contradiction.

If otherwise,i.e.,m=0,substituting a=b=m=0 and(4.2)into the first equation of(4.1),we have

The above equation implies that n2sin2ρ =0,i.e.,n=0.It follows that x= τ1=0,y= τ2=0,meaning that the map ? is harmonic in this case.

Case IIIb=0,We will show that Equation(4.3)has no solution in this case.

Multiplying(k+cosr)4to both sides of Equation(4.3)and simplifying the resulting equation by using the product-to-sum formulas,we have

or,equivalently

where

We observe that the 40 trigonometric functions appearing in the linear combination on the left hand side of Equation(4.9)are linearly independent for the values of a that produce no like terms among them.Note also that even in the case the values of a produce like terms,we can collect the like terms so that the resulting set of functions are linearly independent.

Case AFor those values of a that do not turn any of sin(2ar),sin(2a±i)r,sin(4ar),sin(4a±i)r(i=1,2,3,4)into a like term of sinr,sin2r,sin3r,sin4r.In this case,we have all coefficients vanish,including d1=d4=0,which imply a=0,a contradiction.

Case BFor those values of a that turn one of sin(2ar),sin(2a±i)r,sin(4ar),sin(4a±i)r(i=1,2,3,4)into a like term to one of sinr,sin2r,sin3r,sin4r.We can check that the only values of a that turn one of sin(2ar),sin(2a±i)r,sin(4ar),sin(4a±i)r(i=1,2,3,4)into a like term to one of sinr,sin2r,sin3r,sin4r are the following:

It is not difficult to check that none of positive values of a given in(4.22)can produce like term of sin(4a+4)r and none of the negative values of a given in(4.22)can produce like term of sin(4a?4)r.It follows that for each value of a given in(4.22),we can,after a possible collecting of like terms in(4.9),use the linear independence of the resulting set of functions to conclude thatThis implies that m=0 for any values of a given in(4.22).Substituting m=0 into Equation(4.9),we have

where

We can check that the only values of a that turn sin(2ar)into a like term to one of sin(2a±i)r(i=1,2),sin(4ar),sinr,sin2r are

Summarizing the results in Cases A and B we conclude that for the case b=0,a 6=0,the biharmonic map equations have no solution.

Combining the results proved in Cases I–III,we obtain the theorem.

From the proof of Theorem 4.1 we have seen the following corollary.

Corollary 4.1Forthere exists no biharmonic map in the family of the maps from a non- flat torus into a 2-sphere,(k>1)with ?(r,θ)=(ar+bθ+c,mr+nθ+l).

Corollary 4.2The composition of the family of immersions

followed by the Hopf fiberation

with

is neither a harmonic map nor a biharmonic map.

Corollary 4.3The composition of the mapwith

given in[39]followed by the construction via Hopf fibration

with

is neither a harmonic map nor a biharmonic map.

Corollary 4.4Letwith

be the family of immersions studied in[27].Then,the family of maps from a torus into a 2-sphere defined by the construction via Hopf fibration

with

is neither a harmonic map nor a biharmonic map.

Corollary 4.5Letbe an embedding with

with b>a>0.Then,the Gauss map

with ?(r,θ)=(r,θ)is neither harmonic nor biharmonic.