# Existence and Stability ofα−Harmonic Maps

Authors: Seyed Mehdi Kazemi Torbaghan; Keyvan Salehi; Salman Babayi
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we first study theα−energy functional, Euler-Lagrange operator, and α-stress-energy tensor. Second, it is shown that the critical points of the α−energy functional are explicitly related to harmonic maps through conformal deformation. In addition, an α−harmonic map is constructed from any smooth map between Riemannian manifolds under certain assumptions. Next, we determine the conditions under which the fibers of horizontally conformal α−harmonic maps are minimal submanifolds. Then, the stability of any α−harmonic map on Riemannian manifold with nonpositive curvature is studied. Finally, the instability of α−harmonic maps from a compact manifold to a standard unit sphere is investigated.


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## Body

## 1. Introduction

Eells and Sampson introduced the concept of harmonic maps between Riemannian manifolds [1]. If ψ:M,g⟶N,h be a smooth map between Riemannian manifolds, then the energy functional of ψ is denoted by Eψ and defined as follows:(1)Eψ=12∫Mdψ2dVg,where dVg is the volume element of M,g. The Euler-Lagrange equation corresponding to Eψ is given by(2)τψ≔Trg∇dψ=0,where ∇ is the induced connection on the pull-back bundle ψ−1TN. The section τψ∈Γψ−1TN described in (2) is known as the tension field of ψ.The authors showed in [1] that any smooth map from a compact Riemannian manifold to a Riemannian manifold with nonpositive sectional curvature can be deformed into a harmonic map. This is well-known as the fundamental existence theorem for harmonic maps. Following that, several researchers conducted a study on these maps [2–6]. Harmonic maps have been examined in numerous theories in mathematical physics, such as liquid crystal, ferromagnetic material, superconductors [7, 8].Sacks and Uhlenbeck introduced perturbed energy functional that satisfies the Palais–Smale condition in their pioneering paper [9] in 1981 to prove the existence of harmonic maps from a closed surface and thus obtained so-called α-harmonic maps, as critical points of perturbed functional, to approximate harmonic maps. For α>1, the α−energy functional of the map ψ is denoted by Eαψ and defined as follows:(3)Eαψ≔∫M1+dψ2αdVg,where dψ2 denotes the Hilbert–Schmidt norm of the differential map dψ∈ΓT∗M⊗ψ−1TN with respect to g and h. Noting that, for α>1, Eα satisfies Morse theory and Ljusternik–Schnirelman theory [10]. Furthermore, the critical points of Eα are said to be the α-harmonic maps. Sacks and Uhlenbeck developed an existence theory for harmonic mappings of orientable surfaces into Riemannian manifolds using the critical maps of Eα. Furthermore, they demonstrated that the convergence of the critical points of Eα is sufficient to construct at least one harmonic map of the sphere into any Riemannian manifold [9]. Many studies have recently been conducted on these maps. For example, the authors of [11] investigated the convergence behaviour of a sequence of α−harmonic mappings uα with Eαuα<C from a compact surface M,g into a compact Riemannian manifold N,h without boundary. It is worth noticing that this sequence converges weakly to a harmonic map. Furthermore, in [10], it is studied the energy identity and necklessness for a sequence of Sacks–Uhlenbeck maps during blowing up.Following the concepts of [1, 4, 8–14], we study the stability, existence, and structure of α-harmonic maps, as well as their practical applications, in this work. The existence of α−harmonic maps in an arbitrary class of homotopy and the conditions under which the fibers of α−harmonic maps are minimal submanifolds are explored in particular. Furthermore, the stability of harmonic mappings from a Riemannian manifold to a nonpositive Riemannian curvature and unit standard sphere is explored. Sections 3–4.1 present our key findings. This manuscript is organized as follows: Section 1 investigates the concepts of ɑ–energy functional and ɑ–harmonic maps, and it provides an explicit relationship between ɑ–harmonic maps and harmonic maps via conformal deformation. Section 2 looks at the existence of ɑ–harmonic mappings. It is demonstrated that, under certain assumptions, every smooth map between Riemannian manifolds can be deformed into an ɑ–harmonic map. The ɑ–stress energy tensor and its practical applications are discussed in Section 3. It is also shown that the vanishing of the divergence of the ɑ–stress energy tensor of ψ is identical to the vanishing of the ɑ–hormonicity of ψ. Furthemore, the criteria that cause the fibres of horizontally conformal ɑ–harmonic mappings to be minimal submanifolds are then examined. In Section 4, the Jacobi operator and Green’s theorem are used to derive the second variation formula of the ɑ–energy functional. The stability of ɑ–harmonic maps is explored in the last part. Any ɑ–harmonic map from a Riemannian manifold to a Riemannian manifold with nonpositive curvature is proved to be stable in this sense. The instability of ɑ–harmonic mappings from a compact manifold to a unit standard sphere is also explored.


### 1.1.α-Harmonic Maps

In this section, first, we study the notion ofα−harmonic maps. Then, the certain relation between α-harmonic maps and harmonic maps through conformal deformation is given.Letψ:M,g⟶N,h be a smooth map between Riemannian manifolds. Throughout this paper, it is considered that M,g is a compact Riemannian manifold. Furthermore, the Levi–Civita connections on M and N are denoted by ∇M and ∇N, respectively. Moreover, the induced connection on the pullback bundle ψ−1TN is denoted by ∇ and defined by ∇YV=∇dψYNV, for any smooth vector field Y∈χM and section V∈Γψ−1TN.Letα be a positive constant such that α>1. Then, the α−energy functional of ψ is defined as follows:(4)Eαψ≔∫M1+dψ2αdVg.The critical points ofEα are called α−harmonic maps. By Green’s theorem, the corresponding Euler-Lagrange equation of the α−energy functional, Eα, is given by(5)ταψ≔2α1+dψ2α−1τψ+dψgrad2α1+dψ2α−1=0.The sectionταψ∈Γψ−1TN is said to be the α−tension field of ψ.By (2) and (5), and [[12], Theorem 2.1], in which Ft=1+2tα, we have the following:Theorem 1.
Letψ:M,g⟶N,h be a smooth map between Riemannian manifolds. Then, ψ is α−harmonic if and only if it has a vanishing α−tension field.Definition 1.
Letψ:M,g⟶N,h be a smooth map between Riemannian manifolds. It is called nondegenerate if the induced tangent map ψ∗=dψ is nondegenerate, i.e., ker dψ=∅.Example 1.
Letα=3 and ψ:ℝ2⟶ℝ3 be a smooth map defined as follows:(6)ψx1,x2=2x1+x2,x1−x2,−4x1+3x2.
By equation (5) and Definition 1, it can be seen that ψ is a nondegenerate α−harmonic map.
Now, an explicit relation betweenα−harmonic maps and harmonic maps through conformal deformation is given, which provides examples of [[12], Proposition 2.2], where Ft=1+2tα.Proposition 1.
Letψ:Mm,g⟶Nn,h be a nondegenerate smooth map with m>2. Then, ψ is an α−harmonic map if and only if the map ψ is harmonic with respect to the conformally related metric g¯ defined by(7)g¯=2α2/m−21+∣dψ∣22α−2/m−2g.Proof 1.
Assume thatg¯ be a Riemannian metric for some smooth positive function μ on M conformally associated to g by g¯=μ2g. Denote the tension fields of the smooth map ψ with respect to g and g¯ by τψ and τ¯ψ, respectively. By (2), it can be obtained that(8)τ¯ψ=1μmμm−2τψ+dψgradμm−2.
Setting(9)μ=2α1/m−21+dψ2α−1/m−2.
By (5), (7), and (8), we get(10)μmτ¯ψ=μm−2τψ+dψgradμm−2=2α1+dψ2α−1+dψgrad2α1+dψ2α−1=ταψ.
The Proposition1 follows from (10).


## 1.1.α-Harmonic Maps

In this section, first, we study the notion ofα−harmonic maps. Then, the certain relation between α-harmonic maps and harmonic maps through conformal deformation is given.Letψ:M,g⟶N,h be a smooth map between Riemannian manifolds. Throughout this paper, it is considered that M,g is a compact Riemannian manifold. Furthermore, the Levi–Civita connections on M and N are denoted by ∇M and ∇N, respectively. Moreover, the induced connection on the pullback bundle ψ−1TN is denoted by ∇ and defined by ∇YV=∇dψYNV, for any smooth vector field Y∈χM and section V∈Γψ−1TN.Letα be a positive constant such that α>1. Then, the α−energy functional of ψ is defined as follows:(4)Eαψ≔∫M1+dψ2αdVg.The critical points ofEα are called α−harmonic maps. By Green’s theorem, the corresponding Euler-Lagrange equation of the α−energy functional, Eα, is given by(5)ταψ≔2α1+dψ2α−1τψ+dψgrad2α1+dψ2α−1=0.The sectionταψ∈Γψ−1TN is said to be the α−tension field of ψ.By (2) and (5), and [[12], Theorem 2.1], in which Ft=1+2tα, we have the following:Theorem 1.
Letψ:M,g⟶N,h be a smooth map between Riemannian manifolds. Then, ψ is α−harmonic if and only if it has a vanishing α−tension field.Definition 1.
Letψ:M,g⟶N,h be a smooth map between Riemannian manifolds. It is called nondegenerate if the induced tangent map ψ∗=dψ is nondegenerate, i.e., ker dψ=∅.Example 1.
Letα=3 and ψ:ℝ2⟶ℝ3 be a smooth map defined as follows:(6)ψx1,x2=2x1+x2,x1−x2,−4x1+3x2.
By equation (5) and Definition 1, it can be seen that ψ is a nondegenerate α−harmonic map.
Now, an explicit relation betweenα−harmonic maps and harmonic maps through conformal deformation is given, which provides examples of [[12], Proposition 2.2], where Ft=1+2tα.Proposition 1.
Letψ:Mm,g⟶Nn,h be a nondegenerate smooth map with m>2. Then, ψ is an α−harmonic map if and only if the map ψ is harmonic with respect to the conformally related metric g¯ defined by(7)g¯=2α2/m−21+∣dψ∣22α−2/m−2g.Proof 1.
Assume thatg¯ be a Riemannian metric for some smooth positive function μ on M conformally associated to g by g¯=μ2g. Denote the tension fields of the smooth map ψ with respect to g and g¯ by τψ and τ¯ψ, respectively. By (2), it can be obtained that(8)τ¯ψ=1μmμm−2τψ+dψgradμm−2.
Setting(9)μ=2α1/m−21+dψ2α−1/m−2.
By (5), (7), and (8), we get(10)μmτ¯ψ=μm−2τψ+dψgradμm−2=2α1+dψ2α−1+dψgrad2α1+dψ2α−1=ταψ.
The Proposition1 follows from (10).


## 2. The Existence ofα−Harmonic Map

In this section, it is assumed thatM,g and N,h are compact Riemannian manifolds, and ℋ is a homotopy class of a smooth given map ψ:Mm,g⟶Nn,h. The following theorem is due to Hong.Theorem 2 (see [4]).
Suppose thatψ:M,g⟶N,h is a harmonic map. Then, for any ε^>0, there exists a smooth metric g^ conformally equivalent to g such that ψ:M,g^⟶N,h is harmonic and dψg^≤ε^, where dψg^ is the Hilbert–Schmidt norm with respect to g^ and h.By Theorem2, the following theorem follows and provides examples of [[12], Theorem 3.5], in which Ft=1+2tα and F−harmonic become α−harmonic.Theorem 3.
Letψ∈ℋ and ψ:Mm,g⟶Nn,h be a harmonic map. Then, there is a smooth metric g¯ on M conformally equivalent to g such that ψ:M,g¯⟶N,h is α−harmonic if m>2α.Proof 2.
Assume thatε be a positive constant and ht=1+2tα. Let kt=1/2t/2α1/α−1−1 be the inverse function of h′t on 0,ε. Then, we have(11)h′kt=t,(12)k′th″kt=1,on 0,ε. Setting(13)yu=kum−2u2.
The derivative ofy with respect to u can be obtained as follows:(14)dydu=1u3m−2um−24αα−1um−22α2−α/α−1−um−22α1/α−1−1.
Due to the fact thatk is the inverse function of h′ and m>2α, we get(15)dyduh′t1/m−2=2m−4αt+m−22α−12α1+2t3α−1/m−2≠0,for t∈0,ε. Then, dy/du≠0, on u between h′01/m−2 and h′ε1/m−2. Thus, there exists a smooth function θy on 0,ε′ for some positive constant ε′, such that(16)y=kθym−2θy2.
Moreover, by (11), we have(17)h′kθym−2=θym−2.
By using (16) and (17), we get(18)h′yθy2=θym−2.
Sinceh′t is a positive function on 0,ε′, by (18), it can be seen that θ is a positive function on 0,ε′.
Now, by Theorem5, for ε^≔ε′, there exists a smooth metric g^ conformally equivalent to g such that ψ:M,g^⟶N,h is harmonic and dψg^2/2<ε′. Due to the harmonicity of ψ:M,g^⟶N,h, the tension field of ψ associated to the metrics g^ and h vanishes, τ^ψ=Trg^dψ=0. Assume that g¯=μ−2g^ for a smooth positive function μ:M⟶ℝ, by (8), we get(19)0=τ^ψ=1μmμm−2τ¯ψ+dψgradg¯μm−2.
Due to the fact thatdψg^2/2<ε′, the positive function μ can be defined as follows:(20)μ≔θdψg^22.
By (18) and (20), it yields that(21)μm−2=θdψg^22m−2=h′dψg^22θdψg^222=h′dψg^22μ2=h′dψg¯22=2α1+dψg¯α−1.
By substituting (21) in (19) and using (5), we get(22)τ¯αψ=2αx1+dψg¯2αx−1τ¯ψ+dψgradg¯2αx1+dψg¯2αx−1=0.
And this completes the proof.


### 2.1. Theα−Stress-Energy Tensor

Letψ:Mm,g⟶Nn,h be a smooth map between Riemannian manifolds. The stress-energy tensor of ψ associated with the α−energy functional (in the sequel, we call the α-stress-energy tensor of ψ, in short) is denoted by Sαψ and defined as follows:(23)Sαψ=1+∣dψ2αg−2α1+dψ2α−1ψ∗h.Remark 1.
The stress-energy tensor sometimes called the stress-energy-momentum tensor or the energy-momentum tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and nongravitational force fields. This density and flux of energy and momentum are the sources of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity [3].
In general relativity, the symmetric stress-energy tensor acts as the source of spacetime curvature and is the current density associated with gauge transformations of gravity which are general curvilinear coordinate transformations [14].
The following Proposition2 and Remark 2 provide examples of [[12], Proposition 4.1], in which SF=Sα and τF=τα.Proposition 2.
Based on the above notations, we get(24)divSαψY=−hταψ,dψY,for all Y∈χM.Proof 3.
Choose a local orthonormal frameeii=1m on M with ∇eiej|p=0, at a point p∈M. Let Y be a smooth vector field on M. According to the definition of divergence operator on a Riemannian manifold, at point p, we have(25)divSαψY=∑i=1m∇eiMSαψei,Y=−hdψgrad2α1+dψ2α−1,dψY−2α1+dψ2α−1∑j=1mhdψej,∇Ydψej−2α1+dψ2α−1∑i=1mh∇eidψei,dψY+hdψei,∇eidψY−dψ∇eiMY.
At pointp, the last term of the right-hand side of 24 can be obtained as follows:(26)∇eidψY−dψ∇eiMY=∇Ydψei+dψei,Y−dψ∇eiMY=∇Ydψei+dψ∇eiMY−∇YMei−dψ∇eiMY=∇Ydψei,where we use(27)∇YMei=∑j=1m∇gej,YejMei=∑j=1mgej,Y∇ejMei=0,at the point p, for the third equality. Furthermore,(28)∇eidψei=∇eidψei−dψ∇eiMei=τψ.
By substituting (26) and (28) in the last equation of (25), we get(29)divSαψY=−2α1+dψ2α−1hτψ,dψY−hdψgrad2α1+dψ2α−1,dψY=−hταψ,dψY,and hence completes the proof.Remark 2.
Based on Proposition2, it can be seen that if ψ is a submersion and divSαψ=0, then ψ is an α−harmonic map. Conversely, if ψ is an α−harmonic map, then divSαψ=0.


## 2.1. Theα−Stress-Energy Tensor

Letψ:Mm,g⟶Nn,h be a smooth map between Riemannian manifolds. The stress-energy tensor of ψ associated with the α−energy functional (in the sequel, we call the α-stress-energy tensor of ψ, in short) is denoted by Sαψ and defined as follows:(23)Sαψ=1+∣dψ2αg−2α1+dψ2α−1ψ∗h.Remark 1.
The stress-energy tensor sometimes called the stress-energy-momentum tensor or the energy-momentum tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and nongravitational force fields. This density and flux of energy and momentum are the sources of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity [3].
In general relativity, the symmetric stress-energy tensor acts as the source of spacetime curvature and is the current density associated with gauge transformations of gravity which are general curvilinear coordinate transformations [14].
The following Proposition2 and Remark 2 provide examples of [[12], Proposition 4.1], in which SF=Sα and τF=τα.Proposition 2.
Based on the above notations, we get(24)divSαψY=−hταψ,dψY,for all Y∈χM.Proof 3.
Choose a local orthonormal frameeii=1m on M with ∇eiej|p=0, at a point p∈M. Let Y be a smooth vector field on M. According to the definition of divergence operator on a Riemannian manifold, at point p, we have(25)divSαψY=∑i=1m∇eiMSαψei,Y=−hdψgrad2α1+dψ2α−1,dψY−2α1+dψ2α−1∑j=1mhdψej,∇Ydψej−2α1+dψ2α−1∑i=1mh∇eidψei,dψY+hdψei,∇eidψY−dψ∇eiMY.
At pointp, the last term of the right-hand side of 24 can be obtained as follows:(26)∇eidψY−dψ∇eiMY=∇Ydψei+dψei,Y−dψ∇eiMY=∇Ydψei+dψ∇eiMY−∇YMei−dψ∇eiMY=∇Ydψei,where we use(27)∇YMei=∑j=1m∇gej,YejMei=∑j=1mgej,Y∇ejMei=0,at the point p, for the third equality. Furthermore,(28)∇eidψei=∇eidψei−dψ∇eiMei=τψ.
By substituting (26) and (28) in the last equation of (25), we get(29)divSαψY=−2α1+dψ2α−1hτψ,dψY−hdψgrad2α1+dψ2α−1,dψY=−hταψ,dψY,and hence completes the proof.Remark 2.
Based on Proposition2, it can be seen that if ψ is a submersion and divSαψ=0, then ψ is an α−harmonic map. Conversely, if ψ is an α−harmonic map, then divSαψ=0.


## 3. Horizontal Conformalα−Harmonic Maps

Letψ:Mm,g⟶Nn,h be a smooth map between Riemannian manifolds and let dψp≠0 for each p∈M. The vertical space at p with respect to ψ is denoted by Vp and defined by Vp=kerdψp. The orthogonal complement of Vp in TpM is denoted by Hp and called the horizontal space at p. For any X∈TpM, the tangent vector X can be decomposed as follows:(30)X=XH+XV,where XV∈Vp and XH∈Hp. The map ψ is called a horizontally conformal map if there exists a positive function μ∈C∞M such that(31)hdψX,dψY=μ2gX,Y,for any X,Y∈Hp and p∈M. The function μ is said to be the dilation of ψ.Remark 3.
In Relativistic Astrophysics, conformal mappings are used as a mathematical mechanism to obtain exact solutions to the Einstein field equations in general relativity. The behaviour of the space-time geometry quantities is given under a conformal transformation, and the Einstein field equations are exhibited for a perfect fluid distribution matter configuration [3].
The following Theorem4 and Corollary 1 provide examples of [[12], Theorem 5.1].Theorem 4.
Letψ:Mm,g⟶Nn,h be a horizontally conformal α−harmonic map with dilation μ where m>n and n>2α. Then, the fibers of ψ are minimal submanifold if and only if gradμ2 is vertical.Proof 4.
Forp∈M, choose a local orthonormal frame field eii=1m near p, such that e1,…,en are horizontal vectors and en+1,…,em are vertical tangent vectors. Due to the fact that ψ is horizontally conformal with dilation μ, we get dψ2=nμ2. By (23), it is obtained that(32)Sαψ=1+nμ2αg−2α1+nμ2α−1ψ∗h.
By Proposition2 and considering the fact that the map ψ is α−harmonic, we get(33)0=divSαψej=∑i=1m∇eiSαψei,ej=nα1+nμ2α−1ejμ2−∑i=1nei2αμ21+nμ2α−1hdψei,dψej+2α1+nμ2α−1∑i=1mhdψ∇eiMei,dψej+hdψei,dψ∇eiMej.
Forj1≤j≤n, we have(34)0=∑i=1neigei,ej=∑i=1ng∇eiMei,ej+gei,∇eiMej=∑i=1ng∇eiMeiH,ej+gei,∇eiMejH=1μ2∑i=1nhdψ∇eiMeiH,dψej+hdψei,dψ∇eiMejH=1μ2∑i=1nhdψ∇eiMei,dψej+hdψei,dψ∇eiMej.
By making use of (33) and (34), for 1≤j≤n, we get(35)0=nα1+nμ2α−1ejμ2−ej2αμ21+nμ2α−1+2α1+nμ2α−1∑i=n+1mhdψ∇eiMei,dψej+hdψei,dψ∇eiMej=nα1+nμ2α−1ejμ2−2α1+nμ2α−1ejμ2−2nμ2αα−11+nμ2α−2ejμ2+2α1+nμ2α−1∑i=n+1mhdψ∇eiMeiH,dψej=α1+nμ2α−2n−2+nμ2n−2αejμ2+2αμ21+nμ2α−1∑i=n+1mg∇eiMei,ej.
By (35) and the definition of the mean curvature vector H, of the fiber of ψ, given as follows:(36)H=1m−n∑j=1n∑i=n+1mg∇eiMei,ejej,
it is obtained that(37)α1+nμ2α−2n−2+nμ2n−2αgradμ2H+2m−nα1+nμ2α−1H=0.
From (37) and the assumptions of this theorem, it can be seen that(38)α1+nμ2α−2n−2+nμ2n−2α>0.
From (37) and (38), Theorem 4 follows.Corollary 1.
Under the assumptions of Theorem4, the fibers of ψ are minimal submanifold if and only if the horizontal distribution has mean curvature gradμ2/2μ2.Proof 5.
By (33), for jn+1≤j≤m, we get(39)0=α1+nμ2α−2nejμ2+2∑i=1nhdψei,dψ∇eiMej.
Fori1≤i≤n, it is obtained that(40)hdψei,dψ∇eiMej=hdψei,dψ∇eiMejH=μ2gei,∇eiMejH=μ2gei,∇eiMej=−μ2g∇eiMei,ej.
By (39) and (40), we get(41)nejμ2−2μ2∑i=1ng∇eiMei,ej=0.
Thus, the mean curvatureH¯, of the horizontal distribution, is obtained as follows:(42)H¯=1n∑j=n+1m∑i=1ng∇eiMei,ejej=12μ2∑j=n+1mejμ2ej=12μ2gradμ2V.
From Theorem4 and equation (42), Corollary 1 follows.


## 4. Second Variational Formula

In this section, the second variation formula of theα−energy functional is calculated by using the Jacobi operator and Stokes’ theorem, illuminating [[12], Theorem 6.1] in which EF=Eα.Remark 4.
In physics, the Jacobi operator plays a key role to solve the nonlinear boundary value of Troesch’s equation. This equation arises in the investigation of the confinement of a plasma column by radiation pressure and also in the theory of gas porous electrodes [14, 15].Theorem 5.
(The second variation formula) Letψ:M,g⟶N,h be an α−harmonic map and ψt,s:M⟶N−ε<s,t<ε be a 2-parameter smooth variation of ψ such that ψ0,0=ψ. Then,(43)∂2∂t∂sEαψ|t=s=0=−∫MhJαυ,ω,where(44)υ=∂ψt,s∂t|s=t=0,ω=∂ψt,s∂s|s=t=0,and Jαυ∈Γψ−1TN is given by(45)Jαυ=2α1+dψ2α−1TrgRNυ,dψdψ+4αα−1Trg∇∇υ,dψ1+dψ2α−2dψ+2αTrg∇1+dψ2α−1∇υ,where RN is the curvature tensor on N,h, and , denotes the inner product on T∗M⊗ψ−1TN.Proof 6.
Let−ε,ε×−ε,ε×M be a product manifold that is equipped with the product metric, and let the natural extension of ∂/∂t on −ε,ε, ∂/∂s on −ε,ε, and ∂/∂x on M to the product manifold −ε,ε×−ε,ε×M is denoted by ∂/∂t,∂/∂s, and ∂/∂x again, respectively. Assume that Ψ:−ε,ε×−ε,ε×M⟶N be a smooth map defined by Ψt,s,x=ψt,sx. The same notation ∇ shall be used for the induced connection on Ψ−1TN. Consider an orthonormal frame ei with respect to g on M. Then, by (4), we have(46)∂2∂t∂sEαψt,s|s=t=0=∫M∂2∂t∂s1+dψt,s2α|s=t=0dVg.
By computing the second derivative, we get(47)∂2∂t∂s1+dψt,s2α=∂A∂sh∇∂∂tdψt,sei,dψt,sei+Ah∇∂/∂sdψt,sei,∇∂/∂sdψt,sei+h∇∂/∂s∇∂/∂tdψt,sei,dψt,sei,where A=2α1+dψt,s2α−1. Due to the fact that(48)∂A∂s|s=t=0=Bh∇eiω,dψei,where B=4αα−11+dψ2α−2, and the first term of the right-hand side of (47) can be obtained as follows:(49)∂A∂sh∇∂/∂tdψt,sei,dψt,sei|s=t=0=B∇υ,dψh∇eiω,dψei=eihω,B∇υ,dψdψei−hω,∇eiB∇υ,dψdψei.
Furthermore, by calculating the second term of the right-hand side of (47), we get(50)Ah∇∂/∂tdψt,sei,∇∂/∂sdψt,sei|s=t=0=eih2α1+dψ2α−1∇eiυ,ω−h∇ei2α∇ei1+dψ2α−1∇eiυ,ω.
By the definition and properties of the curvature tensor ofN,h and the compatibility of ∇ with the metric h, the last term of the right-hand side of (47) can be obtained as follows:(51)Ah∇/∂∂s∇∂/∂tdψt,sei,dψt,sei|s=t=0=−2α1+dψ2α−1hRNυ,dψeidψei,ω+eih∇∂/∂sdψt,s∂∂t,Adψt,sei|s=t=0−h∇∂/∂sdψt,s∂∂t,∇eiAdψt,sei|s=t=0.
From equations (46), (47), (49), (50), and (44) and considering Green’s theorem and α−harmonicity of ψ, Theorem 5 follows.


### 4.1. Stability ofα-Harmonic Maps

In this section, the stability ofα−harmonic maps is studied. First, we showed that any α−harmonic map from a Riemannian manifold to a Riemannian manifold with nonpositive curvature is stable. Then, the stability of α−harmonic maps from a compact manifold to a unit standard sphere is investigated.Remark 5.
The stability of harmonic maps plays a key role in mathematical physics and mechanics, [5]. For instance, the linearized Vlasov–Maxwell equations are used to investigate harmonic stability properties for planar wiggler free-electron laser (FEL). It is worth noting that the analysis is carried out in the Compton regime for a tenuous electron beam propagating in the z direction through the constant amplitude planar wiggler magnetic field B0=−Bωcosk0ze^x [16].Definition 2.
Under the assumptions of Theorem5, setting(52)Iυ,ω=∂2∂t∂sEαψt,s|s=t=0.
Then,ψ is said to be stable α−harmonic if Iυ,υ≥0 for any compactly supported vector field υ along ψ.
From Theorem5, we obtain the following corollary and provide examples of [[12], Theorem 6.2], in which F−harmonic map is α−harmonic map.Corollary 2.
LetN be a Riemannian manifold with nonpositive Riemannian curvature. Then, any α−harmonic map ψ:M,g⟶N,h is stable.Proof 7.
Setting(53)δX=2α1+dψ2α−1h∇Xυ,ω,(54)ηX=4αα−11+dψ2α−2∇υ,dψhdψX,ω,for any X∈χM. Then, we have(55)−h2αTrg∇1+dψ2α−1∇υ,ω=−∑i=1mh2α∇ei1+dψ2α−1∇eiυ,ω=−∑i=1meih2α1+dψ2α−1∇eiυ,ω+h2α1+dψ2α−1∇eiυ,∇eiω=−divδ+2α1+dψ2α−1∇υ,∇ω.
On the other hand, by (54), we get(56)−hTrg∇∇υ,dψBdψ,ω=∑i=1m−h∇ei∇υ,dψBdψei,ω=∑i=1m−eih∇υ,dψBdψei,ω+h∇υ,dψBdψei,∇eiω=−divη+B∇υ,dψ∇ω,dψ,where B=4αα−11+dψ2α−2. By substituting (55) and (56) in (45) and using the Divergence theorem and Definition 2, we get(57)Iυ,υ=∫M4αα−11+dψ2α−2∇υ,dψ2−2α1+dψ2α−1hTrgRNυ,dψdψ,υ+2α1+dψ2α−1∇υ2dVg.
By (57) and the assumptions, Corollary 2 follows.
Now, the stability ofα−harmonic maps from a compact without boundary Riemannian manifold to a standard unit sphere is studied. As in [12], an extrinsic average variational method of Wei ([17, 18]) is employed.
Consider a unit standard sphereSn as a submanifold in n+1−dimensional Euclidean space ℝn+1. Denote the Levi-Civita connections on Sn and ℝn+1 by ∇S and ∇R, respectively. At p∈Sn, any vector V in ℝn+1 can be decomposed as follows:(58)V=V⊤+V⊥,where V⊥=V,pp is the normal part to Sn, and V⊤ is the tangent part to Sn. The second fundamental form of Sn in ℝn+1 is denoted by B and defined as follows:(59)BX,Y=−X,Yp,where X,Y are tangent vectors of Sn at p and , is the Euclidean metric on ℝn+1. Furthermore, the shape operator AW corresponding to a normal vector field W on Sn is defined by(60)AWX=−∇XRW⊤,where X is a tangent vector field on Sn. Noting that, the second fundamental form and the shape operator of Sn are satisfied by the following equation:(61)BX,Y,W=AWX,Y=−X,Yp,W,for any tangent vectors X and Y of Sn at p.
From the sectional curvature ofSn and using (59), (60), and (61), the following lemma is obtained.Lemma 1.
Letψ:M,g⟶Sn be a smooth map, and let Λ is a parallel vector field in ℝn+1. Then, at p∈Sn, we have(1)
∇XΛ⊤=AΛ⊥dψX,(2)
∇XΛ⊤,dψX=−dψX2p,Λ,(3)
∇XΛ⊤2=dψX2p,Λ2,(4)
RSΛ⊤,dψXdψX,Λ⊤=dψX2Λ⊤2−dX,Λ2,where ∇ and RS denote the induced connection on ψ−1TSn and the curvature tensor of  Sn, respectively.Proof 8.
By (58) and (60), we have(62)∇XΛ⊤=∇dψXSΛ⊤=∇dψXRΛ⊤⊤=∇dψXRΛ−Λ⊥⊤=−∇dψXRΛ⊥⊤=AΛ⊥dψX.
Making use of (61) and (62), we get(63)∇XΛ⊤,dψX=AΛ⊥dψX,dψX=−dψX2p,Λ⊥=−dψX2p,Λ∇XΛ⊤2=∇XΛ⊤,∇XΛ⊤=AΛ⊥dψX,AΛ⊥dψX=−AΛ⊥dψX,dψXp,Λ⊥=dψX,dψXp,Λ⊥2=dψX2p,Λ2.
Due to the fact that the sectional curvature ofSn is constant, we get(64)RSΛ⊤,dψXdψX,Λ⊤=dψX2Λ⊤2−dψX,Λ⊤2=dψX2Λ⊤2−dψX,Λ2,and hence completes the proof.Theorem 6.
LetM be a compact without boundary manifold and let ψ:M,g⟶Sn be an α−harmonic map such that dψ2<n−2/2α−n. Then, ψ is unstable.Proof 9.
LetΛkk=1n+1 be a parallel orthonormal frame field in ℝn+1 and eii=1m be a local orthonormal frame field on M, and let RS and ∇ denote the curvature tensor of Sn and the induced connection on ψ−1TSn, respectively. By (57), we have(65)∑k=1n+1IΛk⊤,Λk⊤=∫MB∑k=1n+1∑i=1m∇eiΛk⊤,dψei2+2α1+dψ2α−1∑k=1n+1∑i=1m∇eiΛk⊤2−RSΛk⊤,dψeidψei,Λk⊤dVg,where B=4αα−11+dψ2α−2. According to Lemma 1’s second assertion, the first term of the right-hand side of (65) can be obtained as follows:(66)∑k=1n+1∑i=1m∇eiΛk⊤,dψei2=∑k=1n+1∑i=1m−dψei2p,Λk2=∑k=1n+1dψ4p,Λk2=dψ4p2,=dψ4.
By virtue of Lemma1’s third claim, the second term of the right-hand side of (65) can be calculated as follows:(67)∑k=1n+1∑i=1m∇eiΛk⊤2=∑k=1n+1∑i=1mdψei2p,Λk2=∑k=1n+1dψ2p,Λk2=dψ2p2=dψ2.
Since the sectional curvature ofSn+1 is constant and using Lemma 1’s fourth assertion, the last term of the right-hand side of (65) can be calculated as follows:(68)∑k=1n+1∑i=1mRSΛk⊤,dψeidψei,Λk⊤=∑k=1n+1∑i=1mdψei2Λk⊤2−dψei,Λk2=−dψ2+dψ2∑k=1n+1Λk−Λk⊥2=−dψ2+dψ2∑k=1n+1Λk−Λk,pp2=−dψ2+dψ2∑k=1n+1Λk2−Λk,p2=−dψ2+dψ2n+1−p2=n−1dψ2.
By substituting (66), (67), and (68) in (65), we get(69)∑k=1n+1IΛk⊤,Λk⊤=∫M4αα−11+dψ2α−2dψ4+2α1+dψ2α−12−ndψ2dVg=∫MC2−n+2α−ndψ2dVg,where C=2α1+dψ2α−1dψ2. By (69) and the assumptions, it is obtained that(70)∑k=1n+1IΛk⊤,Λk⊤<0.
Then,ψ is unstable and hence completes the proof.


## 4.1. Stability ofα-Harmonic Maps

In this section, the stability ofα−harmonic maps is studied. First, we showed that any α−harmonic map from a Riemannian manifold to a Riemannian manifold with nonpositive curvature is stable. Then, the stability of α−harmonic maps from a compact manifold to a unit standard sphere is investigated.Remark 5.
The stability of harmonic maps plays a key role in mathematical physics and mechanics, [5]. For instance, the linearized Vlasov–Maxwell equations are used to investigate harmonic stability properties for planar wiggler free-electron laser (FEL). It is worth noting that the analysis is carried out in the Compton regime for a tenuous electron beam propagating in the z direction through the constant amplitude planar wiggler magnetic field B0=−Bωcosk0ze^x [16].Definition 2.
Under the assumptions of Theorem5, setting(52)Iυ,ω=∂2∂t∂sEαψt,s|s=t=0.
Then,ψ is said to be stable α−harmonic if Iυ,υ≥0 for any compactly supported vector field υ along ψ.
From Theorem5, we obtain the following corollary and provide examples of [[12], Theorem 6.2], in which F−harmonic map is α−harmonic map.Corollary 2.
LetN be a Riemannian manifold with nonpositive Riemannian curvature. Then, any α−harmonic map ψ:M,g⟶N,h is stable.Proof 7.
Setting(53)δX=2α1+dψ2α−1h∇Xυ,ω,(54)ηX=4αα−11+dψ2α−2∇υ,dψhdψX,ω,for any X∈χM. Then, we have(55)−h2αTrg∇1+dψ2α−1∇υ,ω=−∑i=1mh2α∇ei1+dψ2α−1∇eiυ,ω=−∑i=1meih2α1+dψ2α−1∇eiυ,ω+h2α1+dψ2α−1∇eiυ,∇eiω=−divδ+2α1+dψ2α−1∇υ,∇ω.
On the other hand, by (54), we get(56)−hTrg∇∇υ,dψBdψ,ω=∑i=1m−h∇ei∇υ,dψBdψei,ω=∑i=1m−eih∇υ,dψBdψei,ω+h∇υ,dψBdψei,∇eiω=−divη+B∇υ,dψ∇ω,dψ,where B=4αα−11+dψ2α−2. By substituting (55) and (56) in (45) and using the Divergence theorem and Definition 2, we get(57)Iυ,υ=∫M4αα−11+dψ2α−2∇υ,dψ2−2α1+dψ2α−1hTrgRNυ,dψdψ,υ+2α1+dψ2α−1∇υ2dVg.
By (57) and the assumptions, Corollary 2 follows.
Now, the stability ofα−harmonic maps from a compact without boundary Riemannian manifold to a standard unit sphere is studied. As in [12], an extrinsic average variational method of Wei ([17, 18]) is employed.
Consider a unit standard sphereSn as a submanifold in n+1−dimensional Euclidean space ℝn+1. Denote the Levi-Civita connections on Sn and ℝn+1 by ∇S and ∇R, respectively. At p∈Sn, any vector V in ℝn+1 can be decomposed as follows:(58)V=V⊤+V⊥,where V⊥=V,pp is the normal part to Sn, and V⊤ is the tangent part to Sn. The second fundamental form of Sn in ℝn+1 is denoted by B and defined as follows:(59)BX,Y=−X,Yp,where X,Y are tangent vectors of Sn at p and , is the Euclidean metric on ℝn+1. Furthermore, the shape operator AW corresponding to a normal vector field W on Sn is defined by(60)AWX=−∇XRW⊤,where X is a tangent vector field on Sn. Noting that, the second fundamental form and the shape operator of Sn are satisfied by the following equation:(61)BX,Y,W=AWX,Y=−X,Yp,W,for any tangent vectors X and Y of Sn at p.
From the sectional curvature ofSn and using (59), (60), and (61), the following lemma is obtained.Lemma 1.
Letψ:M,g⟶Sn be a smooth map, and let Λ is a parallel vector field in ℝn+1. Then, at p∈Sn, we have(1)
∇XΛ⊤=AΛ⊥dψX,(2)
∇XΛ⊤,dψX=−dψX2p,Λ,(3)
∇XΛ⊤2=dψX2p,Λ2,(4)
RSΛ⊤,dψXdψX,Λ⊤=dψX2Λ⊤2−dX,Λ2,where ∇ and RS denote the induced connection on ψ−1TSn and the curvature tensor of  Sn, respectively.Proof 8.
By (58) and (60), we have(62)∇XΛ⊤=∇dψXSΛ⊤=∇dψXRΛ⊤⊤=∇dψXRΛ−Λ⊥⊤=−∇dψXRΛ⊥⊤=AΛ⊥dψX.
Making use of (61) and (62), we get(63)∇XΛ⊤,dψX=AΛ⊥dψX,dψX=−dψX2p,Λ⊥=−dψX2p,Λ∇XΛ⊤2=∇XΛ⊤,∇XΛ⊤=AΛ⊥dψX,AΛ⊥dψX=−AΛ⊥dψX,dψXp,Λ⊥=dψX,dψXp,Λ⊥2=dψX2p,Λ2.
Due to the fact that the sectional curvature ofSn is constant, we get(64)RSΛ⊤,dψXdψX,Λ⊤=dψX2Λ⊤2−dψX,Λ⊤2=dψX2Λ⊤2−dψX,Λ2,and hence completes the proof.Theorem 6.
LetM be a compact without boundary manifold and let ψ:M,g⟶Sn be an α−harmonic map such that dψ2<n−2/2α−n. Then, ψ is unstable.Proof 9.
LetΛkk=1n+1 be a parallel orthonormal frame field in ℝn+1 and eii=1m be a local orthonormal frame field on M, and let RS and ∇ denote the curvature tensor of Sn and the induced connection on ψ−1TSn, respectively. By (57), we have(65)∑k=1n+1IΛk⊤,Λk⊤=∫MB∑k=1n+1∑i=1m∇eiΛk⊤,dψei2+2α1+dψ2α−1∑k=1n+1∑i=1m∇eiΛk⊤2−RSΛk⊤,dψeidψei,Λk⊤dVg,where B=4αα−11+dψ2α−2. According to Lemma 1’s second assertion, the first term of the right-hand side of (65) can be obtained as follows:(66)∑k=1n+1∑i=1m∇eiΛk⊤,dψei2=∑k=1n+1∑i=1m−dψei2p,Λk2=∑k=1n+1dψ4p,Λk2=dψ4p2,=dψ4.
By virtue of Lemma1’s third claim, the second term of the right-hand side of (65) can be calculated as follows:(67)∑k=1n+1∑i=1m∇eiΛk⊤2=∑k=1n+1∑i=1mdψei2p,Λk2=∑k=1n+1dψ2p,Λk2=dψ2p2=dψ2.
Since the sectional curvature ofSn+1 is constant and using Lemma 1’s fourth assertion, the last term of the right-hand side of (65) can be calculated as follows:(68)∑k=1n+1∑i=1mRSΛk⊤,dψeidψei,Λk⊤=∑k=1n+1∑i=1mdψei2Λk⊤2−dψei,Λk2=−dψ2+dψ2∑k=1n+1Λk−Λk⊥2=−dψ2+dψ2∑k=1n+1Λk−Λk,pp2=−dψ2+dψ2∑k=1n+1Λk2−Λk,p2=−dψ2+dψ2n+1−p2=n−1dψ2.
By substituting (66), (67), and (68) in (65), we get(69)∑k=1n+1IΛk⊤,Λk⊤=∫M4αα−11+dψ2α−2dψ4+2α1+dψ2α−12−ndψ2dVg=∫MC2−n+2α−ndψ2dVg,where C=2α1+dψ2α−1dψ2. By (69) and the assumptions, it is obtained that(70)∑k=1n+1IΛk⊤,Λk⊤<0.
Then,ψ is unstable and hence completes the proof.



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*Source: 1906905-2022-10-28.xml*