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Kyungpook Mathematical Journal 2018; 58(3): 559-571

Published online September 30, 2018

### Stability and Constant Boundary-Value Problems of f-Harmonic Maps with Potential

Bouazza Kacimi, and Ahmed Mohammed Cherif*

Department of Mathematics, University Mustapha Stambouli, Mascara, 29000, Algeria
e-mail : kacimibouazza@yahoo.fr and med_cherif_ahmed@yahoo.fr

Received: October 3, 2017; Accepted: July 18, 2018

In this paper, we give some results on the stability of f-harmonic maps with potential from or into spheres and any Riemannian manifold. We study the constant boundary-value problems of such maps defined on a specific Cartan-Hadamard manifolds, and obtain a Liouville-type theorem. It can also be applied to the static Landau-Lifshitz equations. We also prove a Liouville theorem for f-harmonic maps with finite f-energy or slowly divergent f-energy.

Keywords: f-harmonic maps with potential, stability, boundary-value.

### 1. Preliminaries and Notations

We give some definitions.

(1) Let (M,g) be a Riemannian manifold. The divergence of (0, p)-tensor α on M is defined by

$(divM α)(X1,…,Xp-1)=(∇eiMα)(ei,X1,…,Xp-1),$

where ∇M is the Levi-Civita connection with respect to g, X1, …, Xp−1 ∈ Γ(TM), and {ei} is an orthonormal frame. Given a smooth function λ on M, the gradient of λ is defined by

$g(gradM λ,X)=X(λ),$

the Hessian of λ is defined by

$(HessM λ)(X,Y)=g(∇XM grad λ,Y),$

where X, Y ∈ Γ(TM), the Laplacian of γ is defined by

$ΔM(λ)=traceg HessM λ,$

(see [11]).

(2) Let ϕ : (M,g) → (N, h) be a smooth map between two Riemannian manifolds, τ (ϕ) the tension field of ϕ (see [1, 2, 6]), f a smooth positive function on M, and let H be a smooth function on N, the (f,H)-tension field of ϕ is given by

$τf,H(ϕ)=fτ(ϕ)+dϕ(gradM f)+(gradN H)∘ϕ,$

where gradM (resp. gradN) denotes the gradient operator with respect to g (resp. h). Then ϕ is called f-harmonic with potential H if the (f,H)-tension field vanishes, i.e. τf,H(ϕ) = 0 (for more details on the concept of f-harmonic maps with potential H see [7]). The notion of f-harmonic with potential H is a generalization of harmonic maps with potential H if f ≡ 1, f-harmonic maps if H = 0 and the usual harmonic maps if f ≡ 1 and H = 0. We define the index form for f-harmonic maps with potential H by

$If,Hϕ(v,w)=∫Mh(Jf,Hϕ(v),w)vM,$

for all v,w ∈ Γ(ϕ−1TN), where

$Jf,Hϕ(v)=-f traceg RN(v,dϕ)dϕ-traceg∇ϕ f∇ϕv-(∇vN gradN H)∘ϕ,$

RN is the curvature tensor of (N, h), ∇N is the Levi-Civita connection of (N, h), ∇ϕ denote the pull-back connection on ϕ−1TN, and vM is the volume form of (M,g) (see [1, 11]). If ϕ be a f-harmonic map with potential H and for any vector field v along ϕ, the index form satisfies $If,Hϕ(v,v)≥0$, then ϕ is called a stable f-harmonic map with potential H. Note that, the definition of stable f-harmonic maps with potential H is a generalization of stable harmonic maps if f = 1 on M and H = 0 on N (see [4, 16]).

For the smooth map ϕ : (M,g) → (N, h), S. Ouakkas et al. introduced in [12] the f-stress energy tensor Sf of ϕ associated to the f-energy functional

$Ef(ϕ)=∫Mef(ϕ)vg,$

is given by

$Sf(ϕ)=ef(ϕ)g-fϕ*h,$

where $ef(ϕ)=12f∣dϕ∣2$ is the f-energy density of ϕ. For any vector field X on M (see [12]), we have

$divM Sf(ϕ)(X)=-h(τf(ϕ),dϕ(X))+12X(f)∣dϕ∣2,$

where τf (ϕ) = (ϕ) + (gradM f). If ϕ is a f-harmonic map with potential H, it follows that

$divM Sf(ϕ)(X)=h((gradN H)∘ϕ,dϕ(X))+12X(f)∣dϕ∣2.$

### Theorem 2.1

Let ϕ be a stable f-harmonic map with potential H from sphere ( , g) (n > 2) to Riemannian manifold (N, h), where f is a smooth positive function onsatisfying, (·)) ≥ 0, and H is a smooth function on N. Then, ϕ is constant.

Proof

Choose a normal orthonormal frame {ei} at point x0 in . Set

$λ(x)=<α,x>ℝn+1,$

for all , where α ∈ ℝn+1 and let . Note that

$v=<α,ei>ℝn+1 ei, ∇XSnv=-λX, for all X∈Γ(TSn),traceg(∇Sn)2v=∇eiSn∇eiSnv-∇∇eiSneiSnv=-v,$

where is the Levi-Civita connection on with respect to the standard metric g of the sphere (see [16]). At point x0, we have

$∇eiϕf∇eiϕdϕ(v)=∇gradSn fϕdϕ(v)+f∇eiϕ∇eiϕdϕ(v),$

the first term of (2.1) is given by

$∇gradSn fϕdϕ(v)=∇vϕdϕ(gradSn f)+dϕ([gradSn f,v])=∇vϕdϕ(gradSn f)+dϕ(∇gradSn fSnv)-dϕ(∇vSngradSn f),$

the seconde term of (2.1) is given by

$f∇eiϕ∇eiϕdϕ(v)=f∇eiϕ∇vϕdϕ(ei)+f∇eiϕdϕ([ei,v])=fRN(dϕ(ei),dϕ(v))dϕ(ei)+f∇vϕ∇eiϕdϕ(ei)+fdϕ([ei,[ei,v]])+2f∇[ei,v]ϕdϕ(ei),$

from the definition of tension field, we get

$f∇eiϕ∇eiϕdϕ(v)=-fRN(dϕ(v),dϕ(ei))dϕ(ei)+f∇vϕτ(ϕ)+f∇vϕdϕ(∇eiSnei)+fdϕ(∇eiSn∇eiSnv)-fdϕ(∇eiSn∇vSnei)+2f∇[ei,v]ϕdϕ(ei)=-fRN(dϕ(v),dϕ(ei))dϕ(ei)+∇vϕfτ(ϕ)-v(f)τ(ϕ)+f∇vϕdϕ(∇eiSnei)+fdϕ(∇eiSn∇eiSnv)-fdϕ(∇eiSn∇vSnei)+2f∇[ei,v]ϕdϕ(ei),$

by equations (2.1), (2.2), (2.4), and the f-harmonicity with potential H condition of ϕ, we have (2.5)

$∇eiϕf∇eiϕdϕ(v)=dϕ(∇gradSn fSnv)-dϕ(∇vSngradSn f)-fRN(dϕ(v),dϕ(ei))dϕ(ei)-∇vϕ(gradN H)∘ϕ-v(f)τ(ϕ)+fdϕ(∇vSn∇eiSnei)+fdϕ(∇eiSn∇eiSnv)-fdϕ(∇eiSn∇vSnei)+2f∇∇eiSnvϕdϕ(ei),$

by the definition of Ricci tensor, we get

$∇eiϕf∇eiϕdϕ(v)=dϕ(∇gradSn fSnv)-dϕ(∇vSngradSn f)-fRN(dϕ(v),dϕ(ei))dϕ(ei)-∇vϕ(gradN H)∘ϕ-v(f)τ(ϕ)+fdϕ(RicciSn v)+fdϕ(trace(∇Sn)2v)+2f∇∇eiSnvϕdϕ(ei),$

from the property $∇XSnv=-λX$, we obtain

$∇eiϕf∇eiϕdϕ(v)=-λdϕ(gradSn f)-dϕ(∇vSngradSn f)-fRN(dϕ(v),dϕ(ei))dϕ(ei)-∇vϕ(gradN H)∘ϕ-v(f)τ(ϕ)+fdϕ(RicciSn v)+fdϕ(trace(∇Sn)2v)-2λfτ(ϕ).$

From the definition of Jacobi operator (1.7) and equation (2.7) we have

$Jϕf(dϕ(v))=λdϕ(gradSn f)+dϕ(∇vSngradSn f)+v(f)τ(ϕ)-fdϕ(RicciSn v)-fdϕ(trace(∇Sn)2v)+2λfτ(ϕ),$

since and (see [1, 16]), we conclude

$h(Jϕf(dϕ(v)),dϕ(v))=λh(dϕ(gradSn f),dϕ(v))+h(dϕ(∇vSngradSn f),dϕ(v))+v(f)h(τ(ϕ),dϕ(v))-(n-2)fh(dϕ(v),dϕ(v))+2λfh(τ(ϕ),dϕ(v)),$

by (2.9) and the f-harmonicity with potential H condition of ϕ, it follows that

$traceα h(Jϕf(dϕ(v)),dϕ(v))=h(dϕ(∇ejSngradSn f),dϕ(ej))+h(τ(ϕ),dϕ(gradSn f))-(n-2)f∣dϕ∣2,$

note that

$h(τ(ϕ),dϕ(gradSn f))=h(∇eiϕdϕ(ei),dϕ(gradSn f))=divSn η-h(dϕ(ei),∇eiϕdϕ(gradSn f)),$

with η(X) = h((X), . We obtain

$traceα h(Jϕf(dϕ(v)),dϕ(v))=-h((∇dϕ)(ej,gradSn f),dϕ(ej))+divSn η-(n-2)f∣dϕ∣2,$

since , from the stable f-harmonic with potential H condition, and equation (2.11), we get

$0≤traceαIfϕ(dϕ(v),dϕ(v))+∫Snh((∇dϕ)(ej,gradSn f),dϕ(ej))vSn=-(n-2)∫Snf∣dϕ∣2vSn≤0.$

Consequently, || = 0, that is ϕ is constant, because n > 2.

If f = 1 on , we get the following result:

### Corollary 2.2

([14]) Any stable harmonic map ϕ from sphereto Riemannian manifold (N, h) is constant.

### Corollary 2.3

([3]) Any stable harmonic map with potential from sphereto Riemannian manifold (N, h) is constant.

Using the similar technique we have:

### Theorem 2.4

Let (M,g) be a compact Riemannian manifold, anda stable f-harmonic map with potential H, where f is a smooth positive function on M, and H is a smooth function onsatisfying. Then, ϕ is constant.

Proof

Choose a normal orthonormal frame {ei} at point x0 in M. When the same data of previous proof, we have

$∇eiϕf∇eiϕ(v∘ϕ)=∇gradM fϕ(v∘ϕ)+f∇eiϕ∇eiϕ(v∘ϕ),$

the first term of (2.12) is given by

$∇gradM fϕ(v∘ϕ)=-(λ∘ϕ)dϕ(gradM f),$

the seconde term of (2.12) is given by

$f∇eiϕ∇eiϕ(v∘ϕ)=-f∇eiϕ(λ∘ϕ)dϕ(ei)=-fdϕ(gradM(λ∘ϕ))-(λ∘ϕ)fτ(ϕ),$

by the definition of gradient operator, we get

$-fdϕ(gradM (λ∘ϕ))=-fdϕ(ei),$

substituting the formulas (2.13), (2.14), (2.15) into (2.12) gives

$∇eiϕf∇eiϕ(v∘ϕ)=-(λ∘ϕ)dϕ(gradM f)-fdϕ(ei)-(λ∘ϕ)fτ(ϕ),$

from the f-harmonicity with potential H condition of ϕ, and equation (2.16), we have

$<∇eiϕf∇eiϕ(v∘ϕ),v∘ϕ>=-f+(λ∘ϕ)<(gradSn H)∘ϕ,v∘ϕ>,$

since the sphere has constant curvature, we obtain

$=f∣dϕ∣2-f,$

by the definition of Jacobi operator and equations (2.17), (2.18), we get

$=2f-f∣dϕ∣2-(λ∘ϕ)<(gradSn H)∘ϕ,v∘ϕ>-<(∇v∘ϕSngradSn H)∘ϕ,v∘ϕ>,$

so that

$traceα=(2-n)f∣dϕ∣2-(ΔSnH)∘ϕ,$

and then

$traceαIf,Hϕ(v∘ϕ,v∘ϕ)=(2-n)∫Mf∣dϕ∣2vM-∫M[(ΔSnH)∘ϕ]vM$

Hence Theorem 2.4 follows from (2.20) and the stable f-harmonicity with potential H condition of ϕ with n > 2 and .

From Theorem 2.4, we deduce:

### Corollary 2.5

([14]) Let (M,g) be a compact Riemannian manifold. When n > 2, any stable harmonic mapmust be constant.

### Corollary 2.6

([5]) Let (M,g) be a compact Riemannian manifold. When n > 2, any stable f-harmonicmust be constant, where f is a smooth positive function on M.

### 3. Liouville Theorems

Let ϕ: (M,g) → (N, h) be a smooth map. For any fixed x0M, by r(x) we denote the distance function from x0 to x, and by BR(x0) the geodesic ball with radius R and center x0. We say that the f-energy of ϕ is divergent slowly if there exists a positive function ψ(t) with $∫R0∞dttψ(t)=∞ (R0>0)$, such that

$limR→∞∫BR(x0)ef(ϕ)(x)ψ(r(x))<∞,$

(see [7]). The next lemma is very useful in the sequel.

### Lemma 3.1

Let ϕ: (Mm, g) → (Nn, h) be a smooth map, DM a compact domain such thatD is a smooth hypersurface in M. Let n denotes the unit normal vector ofD. Let X be any vector field in M with compact support. Then

$∫∂Def(ϕ)g(X,n)=∫∂Dfh(dϕ(X),dϕ(n))+∫DdivMSf(ϕ)(X)+∫D〈Sf(ϕ),∇X〉.$

Here 〈,〉 denote the inner product on T*MT*M.

Proof

Choosing a local orthonormal frame field {ei} on M, and define $∇X(ei,ej)=g(∇XMei,ej)$, then

$divM(ef(ϕ)X)=g(∇eiM(ef(ϕ)X),ei)=g(∇eiM(ef(ϕ)X),ei)+ef(ϕ)g(∇eiMX,ei)=∇XMef(ϕ)+ef(ϕ)g(∇eiMX,ei),$

and

$∇XMef(ϕ)=12∇XM(fh(dϕ(ei),dϕ(ei)))=12X(f)∣dϕ∣2+fh((∇Xdϕ)ei,dϕ(ei))=12X(f)∣dϕ∣2+fh((∇eidϕ)X,dϕ(ei))=12X(f)∣dϕ∣2+h(∇eiϕdϕ(X),fdϕ(ei))-fh(dϕ(∇eiMX),dϕ(ei))=12X(f)∣dϕ∣2+∇eiMh(dϕ(X),fdϕ(ei))-h(dϕ(X),(∇ei(fdϕ))ei)-fh(dϕ(∇eiMX),dϕ(ei))=12X(f)∣dϕ2∣+divM(fh(dϕ(X),dϕ(ei))ei)-h(dϕ(X),τf(ϕ))-f〈∇X,ϕ*h〉.$

Hence we obtain

$divM(ef(ϕ)X)=12X(f)∣dϕ∣2+divM(fh(dϕ(X),dϕ(ei))ei)-h(dϕ(X),τf(ϕ)〉-f〈∇X,ϕ*h〉+ef(ϕ)g(∇eiMX,ei))=12X(f)∣dϕ∣2+divM(fh(dϕ(X),dϕ(ei))ei)-h(dϕ(X),τf(ϕ))+〈Sf(ϕ),∇X〉.$

Now, for compact domain D in M with its smooth hypersurface ∂D, taking local orthonormal frame field {ei} on M along ∂D, such that {e1, …, em−1} ∈ Γ(TD), and em = n be the unit normal vector of ∂D. Since SuppX is compact, integrating the formula (3.2) on D, by means of Green’s theorem and using (1.9), we have the desired formula.

### Theorem 3.2

Let M be an m–dimensional complete, simply connected Riemannian manifold with non-positive sectional curvature KM, m > 2. Assuming that KM satisfies

• a2 < KM < −b2, where a > 0, b>0 and$(m-1)b2≥a$; or

• $-A1+r2≤KM≤0$, where$0,

assume that ϕ is a f-harmonic map with potential H from BR(x0) to any Riemannian manifold N with ϕ |BR(x0)= P, where PN satisfies$H(P)=maxy∈N H(y)$, and X(f) ≥ 0 such that$X=r∂∂r$. Then ϕ must be constant in BR(x0).

Proof

First of all, from the definition of Sf (ϕ), we obtain

$〈Sf(ϕ),∇X〉=(ef(ϕ)g(eα,eβ)-fh(dϕ(eα),dϕ(eβ)))g(∇eαMX,eβ)=ef(ϕ)g(∇eαMX,eα)-fh(dϕ(eα),dϕ(eβ))g(∇eαMX,eβ).$

Let $eα={es,∂∂r}$ be the orthonormal frame field of BR(x0) and $X=r∂∂r$, then

$∇∂∂rMX=∂∂r,$$∇esMX=r∇esM∂∂r=r HessM(r) (es,et)et,$$divM X=g(∇eαMX,eα)=1+r HessM(r) (es,es).$

Substituting (3.4), (3.5) and (3.6) into (3.3), we get

$〈Sf(ϕ),∇X〉=ef(ϕ)(1+r HessM(r)(es,es))-fh(dϕ(es),dϕ(et))g(∇esMX,et)-fh(dϕ(∂∂r),dϕ(∂∂r))g(∇∂∂rMX,∂∂r)-fh(dϕ(∂∂r),dϕ(et))g(∇∂∂rMX,et)-fh(dϕ(es),dϕ(∂∂r))g(∇esMX,∂∂r)=ef(ϕ)(1+r HessM(r)(es,es))-fh(dϕ(es),dϕ(et))r HessM(r)(es,et)-fh(dϕ(∂∂r),dϕ(∂∂r)).$

Under the assumption (1) in Theorem 3.2, from Hessian comparison theorem (see [8]) we have

$b coth(br)(g-dr⊗dr)≤HessM(r)≤a coth(ar)(g-dr⊗dr).$

Therefore, (3.7) becomes

$〈Sf(ϕ),∇X〉≥ef(ϕ)(1+(m-1)(br) coth(br))-f(ar) coth(ar)h(dϕ(es),dϕ(es))-fh(dϕ(∂∂r),dϕ(∂∂r))=f(m-12(br) coth(br)-12)h(dϕ(∂∂r),dϕ(∂∂r))+f(12+m-12(br) coth(br)-(ar) coth(ar))h(dϕ(es),dϕ(es))≥m-22fh(dϕ(∂∂r),dϕ(∂∂r))+f(12+r coth(br)(m-12b-a))h(dϕ(es),dϕ(es)).$

Hence, when $(m-1)b2≥a$, it follows from (3.9)

$〈Sf(ϕ),∇X〉≥δef(ϕ),$

where δ > 0.

Under the assumption (2), also by Hessian comparison theorem (see [8]) we have

$1r(g-dr⊗dr)≤HessM(r)≤βr(g-dr⊗dr),$

where $β=12+12(1+4A)12$. By (3.7), it follows that

$〈Sf(ϕ),∇X〉≥mef(ϕ)-fβh(dϕ(es),dϕ(es))-fh(dϕ(∂∂r),dϕ(∂∂r))=m-22fh(dϕ(∂∂r),dϕ(∂∂r))+m-2β2fh(dϕ(es),dϕ(es))≥δef(u).$

Then, under the two assumptions of Theorem 3.2 we obtain

$〈Sf(ϕ),∇X〉≥δef(ϕ),$

where δ > 0. Now choosing the geodesic polar coordinates (θ, r) in BR(x0) and a local orthonormal frame field {e1, …, em−1, $∂∂r$} on M. After applying D = BR(x0), $X=r∂∂r$ and $n=∂∂r$ to (3.1), we get

$R∫∂BR(x0)ef(ϕ)=R∫∂BR(x0)f∣dϕ(∂∂r)∣2+∫BR(x0)divM Sf(ϕ)(r∂∂r)+∫BR(x0)〈Sf(ϕ),∇X〉.$

Noting that ϕ is f-harmonic map with potential H, and using (1.10), we have

$divM Sf(ϕ)(r∂∂r)=∫BR(x0)r∂(H∘ϕ)∂r+12∫BR(x0)r∂f∂r∣dϕ∣2,$

so, (3.11) becomes

$R∫∂BR(x0)ef(ϕ)=R∫∂BR(x0)f∣dϕ(∂∂r)∣2+∫BR(x0)r∂(H∘ϕ)∂r+12∫BR(x0)r∂f∂r∣dϕ∣2+∫BR(x0)〈Sf(ϕ),∇X〉.$

Since ϕ is constant at ∂BR(x0), by (3.10) and (3.12), we have

$∫BR(x0)r∂(H∘ϕ)∂r+12∫BR(x0)r∂f∂r∣dϕ∣2+δ∫BR(x0)ef(ϕ)≤0.$

Denote J(θ, r)dθdr the volume element of BR(x0) in polar coordinates around x0. Since $∂∂r(rJ(θ,r))>0$ (see [3]), we obtain

$∫0Rr∂(H∘ϕ)∂rJ(θ,r)dr=RJ(θ,R)H(P)-∫0RH∘ϕ(θ,r)∂(rJ(θ,r))∂rdr≥RJ(θ,R)H(P)-H(P)∫0R∂(rJ(θ,r))∂rdr≡0.$

Therefore

$∫BR(x0)r∂(H∘ϕ)∂r=∫∂BR(x0)(∫0Rr∂(H∘ϕ)∂rJ(θ,r) dr) dθ≥0.$

By (3.13) and (3.14) and X(f) ≥ 0, we immediately conclude that ef (ϕ) ≡ 0 in BR(x0), namely, ϕ is constant in BR(x0), which completes the proof of Theorem 3.2.

### Remark 3.3

Consider the following static Landau-Lifshitz equation

$Δϕ+ϕ∣dϕ∣2-ℝ3ϕ+H0=0,$

where |ϕ(x)|2 = 1, x ∈ Ω ⊂ ℝm, H0 ≠ 0 is a constant vector in ℝ3. Then the solution ϕ of (3.15) can be seen as a harmonic map with potential: with the potential H(y) =< H0, y >3, (see [3]). Moreover, Hong [9] asserted that the static Landau-Lifshitz equation (3.15) with constant boundary-value problem $ϕ∣∂Ω=H0∣H0∣$, has only constant solution, if Ω = B3, where B3 denote the unit ball in ℝ3. On the other hand, if we choose M = ℝm(m > 2), , H(y) =< H0, y >3, , then Theorem 3.2 for f ≡ 1 leads to a conclusion for the static Landau-Lifshitz equation, in particular, when m = 3, it is just the result of Hong. Theorem 3.2 also generalizes the result of [10] for the usual harmonic maps and Theorem 3 in [3] for the harmonic maps with potential.

For f-harmonic maps, we have

### Theorem 3.4

Let M be as in Theorem 3.2. If ϕ is a f-harmonic map from M whose f-energy is finite or divergent slowly. Then ϕ must be a constant map when X(f) ≥ 0.

Proof

By setting D = BR(x0), $X=r∂∂r$ and $n=∂∂r$ in (3.1), we obtain

$∫BR(x0)(divM Sf(ϕ))(X)+∫BR(x0)〈Sf(ϕ),∇X〉=R∫∂BR(x0)ef(ϕ)-R∫∂BR(x0)f∣dϕ(∂∂r)∣2≤R∫∂BR(x0)ef(ϕ).$

According to (1.9), (3.10) and (3.16), for a f-harmonic map ϕ, we get

$R∫∂BR(x0)ef(ϕ)≥δ∫BR(x0)ef(ϕ)+12∫BR(x0)X(f)∣dϕ∣2≥δ∫BR(x0)ef(ϕ).$

Now suppose that ϕ is a nonconstant map, i.e. the f-energy density ef (ϕ) does not vanish everywhere, so there exists R0 > 0 such that for R > R0,

$∫BR(x0)ef(ϕ)≥C0,$

where C0 be a positive constant. So when R > R0, we have from (3.17) and (3.18)

$∫∂BR(x0)ef(ϕ)≥δC0R,$

therefore, (3.19) will imply

$Ef(ϕ)>∫BR(x0)ef(ϕ)=∫0R(∫∂BR(x0)ef(ϕ))dr≥∫R0R(∫∂BR(x0)ef(ϕ)) dr≥∫R0RδC0rdr=δC0 lnRR0.$

Let R → ∞, this contradicts the assumption of the finite f-energy, then ϕ is constant. If the f-energy of ϕ divergent slowly, therefore (3.19) leads to

$limR→∞∫BR(x0)ef(ϕ)ψ(r(x))=∫0∞drψ(r)∫∂BR(x0)ef(ϕ)≥δC0∫0∞drrψ(r)≥δC0∫R0∞drrψ(r)=∞.$

Which is in contradiction with f-energy of ϕ being slowly divergent. So ϕ must be a constant map.

### Remark 3.5

When f ≡ 1, it is clear that Theorem 3.4 recovers the results due to Sealey [13] and Xin [16] as special cases. If the manifold M in the Theorems 3.2 and 3.4 satisfies −a2 < KM < 0 and RicM < −b2 < 0 with b > 2a, then this two theorems remain true. Note that this kind of manifolds includes the bounded symmetric domains and complex hyperbolic spaces see ([15]).

### Acknowledgements

The authors would like to thank the reviewers for their useful remarks and suggestions. The authors are supported by National Agency Scientific Research of Algeria and Laboratory of Geometry, Analysis, Controle and Applications, Algeria.

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