Stability of λ-Harmonic Maps

In this paper, λ-harmonic maps from a Finsler manifold to a Riemannian manifold are studied. Then, some properties of this kind of harmonic maps are presented and some examples are given. Finally, the stability of the λ-harmonic maps from a Finsler manifold to the standard unit sphere Sn(n > 2) is investigated.


Introduction
Harmonic maps between Riemannian manifolds were first introduced by Eells and Sampson in 1964.They showed that any map φ 0 : (M, g) −→ (N, h) from any compact Riemannian manifold (M, g) into a Riemannian manifold (N, h) with non-positive sectional curvature can be deformed into a harmonic maps.This is so-called the fundamental existence theorem for harmonic maps.In view of physics, harmonic maps have been studied in various fields of physics, such as super conductor, ferromagnetic material, liquid crystal, etc. [1][2][3][4][5][6][7][8][9].
Lichnerowicz first studied f -harmonic maps between Riemannian manifolds as a generalization of harmonic maps in 1970 [10].Recently, in Cherif et al. [11], the researchers proved that any stable f -harmonic map ψ from sphere S n (n > 2) to Riemannian manifold N is constant.Course [12] studied the f -harmonic flow on surfaces.Ou [13] analysed the f-harmonic morphisms as a subclass of harmonic maps which pull back harmonic functions to f-harmonic functions.Many scholars have studied and done research on f -harmonic maps, see for instance, [10,11,[13][14][15][16][17].
The concept of harmonic maps from a Finsler manifold to a Riemannian manifold was first introduced by Mo [18].On the workshop of Finsler Geometry in 2000, Professor S. S. Chern conjectured that the fundamental existence theorem for harmonic maps on Finsler spaces is true.In [19], Mo and Yang, the researchers have proved this conjecture and shown that any smooth map from a compact Finsler manifold to a compact Riemannian manifold of non-positive sectional curvature can be deformed into a harmonic map which has minimum energy in its homotopy class.Shen and Zhang [20] extended Mo's work to Finsler target manifold and obtained the first and second variation formulas.
As an application, He and Shen [21] proved that any harmonic map from an Einstein Riemannian manifold to a Finsler manifold with certain conditions is totally geodesic and there is no stable harmonic map from an Euclidean unit sphere S n to any Finsler manifolds.Harmonic maps between Finsler manifolds have been studied extensively by various researchers, see for instance, [18][19][20][21][22].
In [23], J. Lie introduced the notion of F -harmonic maps between Finsler manifolds.
, where e(ψ) is the energy density of ψ, c m−1 denotes the volume of the standard (m − 1)-dimensional sphere and dV SM is the canonical volume element of SM.
In view of physics, when (M, F) is a Riemannian manifold, p-harmonic maps have been extensively applied in image processing for denoising color images [26,27].Furthermore, exponential harmonic maps have been studied on gravity [28].The concept of F -harmonic maps, as an extension of harmonic, p-harmonic and exponential harmonic maps, have an important role in physics and physical cosmology.For instance, instead of the scalar field in the Lagrangian, some of the F -harmonic maps, such as the trigonometric functions, are studied in order to reproduce the inflation.Moreover, there are other F -harmonic maps, such as exponential harmonic maps, are investigated in order to depict the phenomenon of the quintessence [5,29,30].
Let ψ : (M, F) −→ (N, h) be a smooth map from a Finsler manifold (M, F) into a Riemannian manifold (N, h) and λ : where g is the fundamental tensor of (M, F), ψ * h is the pull-back of the metric h by the map ψ and λ ψ is a smooth function given by (x, y) ∈ SM −→ λ ψ (x, y) := λ(x, y, ψ(x)).By considering the Euler-Lagrange equation associated to the λ-energy functional, it can be seen that any F -harmonic map ψ : (M, F) −→ (N, h) from a Finsler manifold (M, F) into a Riemannian manifold (N, h) without critical points (i.e., | dψ x | = 0 for all x ∈ M), is a λ-harmonic map with λ = F (e(ψ)).
In particular, when grad h λ = 0 and (M, F) is a Riemannian manifold, λ-harmonic maps can be considered as the stationary solutions of inhomogeneous Heisenberg spin system, see for instance [13,14].Furthermore, the intersection of λ-harmonicity with curvature conditions justifies their application for gleaning valuable information on weighted manifolds and gradient Ricci solitons, see [15][16][17].
The current paper is organized as follows.In Section 2, a few concepts of Finsler geometry are reviewed.In Section 3, the λ-energy functional of a smooth map from a Finsler manifold to a Riemannian manifold is introduced and the corresponding Euler-Lagrange equation is obtained via calculating the first variation formula of the λ-energy functional and an example is given.In Section 4, the second variation formula of the λ-energy functional for a λ-harmonic map is derived.As an application, the stability theorems for λ-harmonic maps are given.

Preliminaries
Throughout this paper, let (M, F) be an m-dimensional smooth, oriented, compact Finsler manifold without boundary.In the local coordinates (x i , y i ) on TM \ {0}, the fundamental tensor of (M, F) is defined as follows: In this paper, the following conventions of index ranges are used Let ρ : SM −→ M be the natural projection on the projective sphere bundle SM.The Finsler structure F determines two important quantities on the pull-back bundle ρ * T * M as follows: which are called the Hilbert form and Cartan tensor, respectively.The dual of the Hilbert form ω is the distinguished section := y i F ∂ ∂x i of the pull-back bundle ρ * TM.Note that, all indices related to ρ * TM are raised and lowered with the metric g.
On the pull-back bundle ρ * TM, there exists uniquely the Chern connection ∇ c whose connection 1-forms {ω j i } are satisfied the following equations where δy i := dy i + N i j dx j [31].Here N i j := γ i jk y k − A i jk y p y q and γ i jk are the formal Christoffel symbols of the second kind for g ij .The curvature 2-forms of the Chern connection, Ω i j := dω i j − ω k j ∧ ω i k , have the following structure See [22] for an expository proof.The Landsberg curvature of (M, F) is defined as follows: By [22], we have where "." denotes the covariant derivative along the Hilbert form.Consider a g-orthonormal frame {ω i = v i j dx j } for any fibre of ρ * T * M where ω m is the Hilbert form ω, and let {e i = u j i ∂ ∂x j } be its dual frame where e m is the distinguished section dual to the Hilbert form ω(= ω m ).Set where } is a local basis for TSM.By (2), it can be seen that Tangent vectors on SM which are annihilated by all {ω a m }'s form the horizontal sub-bundle HSM of TSM.The fibres of HSM are m-dimensional.On the other hand, let VSM := ∪ x∈M TS x M be the vertical sub-bundle of TSM, its fibres are m − 1 dimensional.The decomposition TSM = HSM ⊕ VSM holds because HSM and VSM are direct summands, (see [32], p. 7).The inner product g = g ij dx i dx j on ρ * TM induces a Riemannian metric ĝ on SM as follows: Furthermore, the volume element dV SM of SM with respect to ĝ is defined as follows where "|" denotes the horizontal covariant derivative with respect to the Chern connection and e H i is defined in (2).

The First Variation Formula
Let ψ : (M m , F) −→ (N n , h) be a smooth map from a Finsler manifold (M, F) into a Riemannian manifold (N, h), ρ : SM −→ M a natural projection on SM and ψ = ψ • ρ.In the sequel, we denote the Chern connection on ρ * TM by c ∇, the connection induced by the Chern connection of (M, F) on the pulled-back bundle ψ * TN over SM by ∇ and the Levi-Civita connection on (N, h) by ∇ N .
Let λ : SM × N −→ (0, ∞) be a smooth positive function.The λ-energy density of ψ is the function e λ (ψ) : SM −→ R, defined by where ψ * h is the pull-back of h by the map ψ and Tr g stands for taking the trace with respect to g (the fundamental tensor of F) at (x, y) ∈ SM.By making use of ( 5), the λ-energy functional is defined as follows: where c m−1 denotes the volume of the standard (m − 1)-dimensional sphere and dV SM is the canonical volume element of SM.A map ψ : (M, F) −→ (N, h) is said to be λ-harmonic if it is a critical point of the λ-energy functional.Let {ψ t } t∈I be a smooth variation of ψ 0 = ψ with the variational vector field For any t ∈ I, in local coordinate (x i , U) on M and ( xα , V) on N, the λ-energy density of ψ t can be written as follows: where x = ψ t (x) and dψ t ( Due to the fact that {V α } is independent of y and using (7), we have where λ ψ is a smooth function given by (x, y) ∈ SM −→ λ ψ (x, y) := λ(x, y, ψ(x)).By using the definition of the gradient operator, we get where e(ψ) := 1 2 Tr g h(dψ, dψ) and λ (x,y) is the smooth function z ∈ N −→ λ (x,y) (z) = λ(x, y, z).The function e(ψ) is called the energy density of ψ.Let Ψ := λ ψ h(V, dψ(e i ))ω i , which is a section of p * T * M .By Lemma 4, we have where is a section of ρ * TM.By (10), we get By substituting ( 9) and ( 12) in ( 8) and considering the Green's theorem, the first variation formula of the λ-energy functional is obtained as follows: where Here grad H λ denotes the horizontal part of grad ĝ λ ∈ Γ(TSM) and K is defined by (11).The field τ λ (ψ) is said to be the λ-tension field of ψ.Theorem 1.Let ψ : (M, F) −→ (N, h) be a smooth map from a Finsler manifold (M, F) to a Riemannian manifold (N, h) and λ ∈ C ∞ (SM × N).Then, ψ is the λ-harmonic map if and only if τ λ (ψ) ≡ 0.
Example 1. Assume that (R 2 , F) is a locally Minkowski manifold and (R 3 , , ) be the three-dimensional Euclidean space.Consider the map ψ : (R 2 , F) −→ (R 3 , , ) defined by . Due to the fact that the Landsberg curvature of locally Minkowski manifold vanishes and considering Theorem 1 and Equation ( 13), one can see that ψ is λ-harmonic.Now, we discuss the relation between λ-harmonic maps and F -harmonic maps from a Finsler manifolds to a Riemannian manifolds.Let F : [0, ∞) −→ [0, ∞) be a C 2 function such that F > 0 on (0, ∞).The smooth map ψ : (M, F) −→ (N, h) is said to be F -harmonic if it is an extermal point of the F -energy functional The concept of F -harmonic maps from a Finsler manifold was first introduced by J. Li [23] in 2010.The F -energy functional is the energy , the p-energy, the α-energy of Sacks-Uhlenbeck and exponential energy when F (t) is equal to t, (2t) ) and e t , respectively.The Euler-Lagrange equation associated to F -energy functional is given by For more details, see [23].The field τ F (ψ) is called the F -tension field of ψ.

Stability of λ-Harmonic Maps
In this section, the second variation formula of the λ-energy functional for a λ-harmonic map from a Finsler manifold to a Riemannian manifold is obtained.As an application, it is shown that any stable λ-harmonic map ψ : (M, F) −→ S n from a Finsler manifold (M, F) to the standard sphere S n (n > 2) is constant.
Theorem 2. (The second variation formula).Let (M m , F) be a Finsler manifold and (N, h) be a Riemannian manifold and let ψ : (M, F) −→ (N, h) be a λ-harmonic map.Assume that {ψ t } t∈I is a smooth variation of ψ 0 = ψ with the variation vector field V = ∂ψ t ∂t | t=0 .Then, the second variation of λ-energy functional is where R N is the curvature tensor on (N, h) and K is defined by (11).
Proof.Let p : SM −→ M be a natural projection and ψ = ψ • p, and let c ∇ and ∇ be the Chern connection on p * TM and the pull-back Chern connection on ψ * TN, respectively.For any t ∈ I, we shall use the same notation of ∇ for the pull-back Chern connection on ψ * TN.By (5), it can be shown that Now, we calculate each term of the right hand side (RHS) of (17).By (9) and considering the definition of gradient operator, we get ). ( Thus, the first term of the RHS of ( 17) is obtained as follows: By calculating the second term of the RHS of ( 17), we get Now, we calculate the last term of the RHS of (17).By definition of the function λ ψ , we have Therefore, Let Ψ := λ ψ t h(∇ e H i dψ t ( ∂ ∂t ), dψ t ( ∂ ∂t ))ω i , which is a section of (p * T * M).By Lemma 4, we get )) By Green's theorem and Equation ( 23), the first term of the RHS of ( 22) is obtained as follows: Similarly, let Ψ := h(∇ ∂ ∂t dψ t ( ∂ ∂t ), λ ψ t dψ t (e i ))ω i , which is a section of p * T * M. By Lemma (4), we also have By (25), we get the second term of the RHS of ( 22) as follows: By making use of ( 24) and ( 26), we have By substituting (19), ( 20) and ( 27) in (17), the Equation ( 16) is obtained and hence completes the proof.Definition 1.By considering the assumptions of Theorem 2, set A λ-harmonic map ψ is said to be stable λ-harmonic map if Q ψ λ (V) ≥ 0 for any vector field V along ψ.
where (Υ 1 ) x is the tangential part of Υ 1 to S n at the point x.By (35), at any point (z, x) ∈ SM × S 3 , the left hand side of (32) for the function λ which is defined in (34), can be calculated as follows: where we use | x | 2 = 1, in the last equality.Due to the fact that (z, x) is an arbitrary point on SM × S 3 and considering Equation (36), it can be seen that the function λ satisfies Definition 2. Thus, λ is an an adopted function.
Remark 1.Any smooth function λ ∈ C ∞ (SM) is an adopted function on S n .
Based on the above notations, we prove the following result.Theorem 3. Let (M m , F) be a Finsler manifold and (S n , h) the n-dimensional Euclidean sphere (n > 2), and let λ : SM × S n −→ R be an adopted function on S n .Then, any non-constant λ-harmonic map ψ : (M, F) −→ S n is unstable.
Proof.Choose an arbitrary point z 0 ∈ SM.Then, set ψ = ψ • ρ and x 0 = ψ(z 0 ), where ρ : SM −→ M is the natural projection on SM.By (2), one can see that [22] Let {Υ 1 , • • • , Υ n+1 } be a λ-frame field in R n+1 at x 0 .By (2), we have where ∇ and R S denote the induced connection on the pull-back bundle (ψ • ρ) −1 TS n and the curvature tensor of S n , respectively.Now we discuss at the point x 0 .Since {Υ α } is a parallel frame field in R n+1 and considering the definition of shape operator, we obtain According to the definition of the function µ α in (33), one can easily check that By ( 2), (37), (39) and ( 40) and considering the definition of the energy density, we have where we use in the third equality.Here {e A } = {e H i , êm+a } is a local basis for TSM which is defined in (2).By similar calculation, we get By considering (39) and (40), it can be concluded that dψ(e i ).On the other hand, due to the fact that λ is an adopted function and e(ψ) is a non-negative function and considering Equations ( 32) and (33), it can be seen that  44) and ( 47)-( 49) in (38) and considering the assumptions of this theorem, it follows that Thus, the map ψ is unstable and hence completes the proof.
By considering the Theorem 1 and the Remark 3, we obtain the following result.