HAN Ying-bo,FENG Shu-xiang

(College of Mathematics and Information Science,Xinyang Normal University,Xinyang 464000,China)

SOME RESULTS OF WEAKLY f-STATIONARY MAPS WITH POTENTIAL

HAN Ying-bo,FENG Shu-xiang

(College of Mathematics and Information Science,Xinyang Normal University,Xinyang 464000,China)

In this paper,we investigate a generalized functional Φf,Hrelated to the pullback metric.By using the stress-energy tensor,we obtain some Liouville type theorems for weakly fstationary maps with potential under some conditions on H.

weakly f-stationary map with potential;stress-energy tensor;Liouville type theorems

1 Introduction

Let u:(Mm,g) → (Nn,h)be a smooth map between Riemannian manifolds(Mm,g) and(Nn,h).Recently,Kawai and Nakauchi[1]introduced a functionalrelated to the pullback metric u∗h as follows:

(see[2–5]),where u∗h is the symmetric 2-tensor defined by

for any vector fields X,Y on M and||u∗h||is given by

with respect to a local orthonormal frame(e1,···,em)on(M,g).The map u is stationary for Φ if it is a critical point of Φ(u)with respect to any compact supported variation of u. Asserda[6]introduced the following functionalΦFbywhere F:[0,∞) → [0,∞)is a C2function such that F(0)=0 and F′(t) ＞ 0 on[0,∞).The map u is F-stationary for Φ ifit is a criticalpoint of Φ(u)with respect to any compact supported variation of u.Following[6],Han and Feng in[5]introduced the following functional Φfby

where f:(M,g) → (0,+∞)is a smooth function.They derived the first variation formula of Φfand introduced the f-stress energy tensor SΦfassociated to Φf.Then,by using the f-stress energy tensor,they obtained the monotonicity formula and vanishing theorems for stationary map for the functional Φf(u)under some conditions on f.

The theory of harmonic maps was developed by many researchers so far,and a lot of results were obtained(see[7,8]).Lichnerowicz in[9](also see[7])introduced the fharmonic maps,generalizing harmonic maps.Since then,there were many results for fharmonic maps such as[10–14].Ara[15]introduced the notion of F-harmonic map,which is a special f-harmonic map and also is a generalization of harmonic maps,p-harmonic maps or exponentially harmonic maps.Since then,there were many results for F-harmonic maps such as[16–19].

On the other hand,Fardon and Ratto in[20]introduced generalized harmonic maps of a certain kind,harmonic maps with potential,which had its own mathematical and physical background,for example,the static Landu-Lifschitz equation.They discovered some properties quite different from those ofordinary harmonic maps due to the presence of the potential.After this,there were many results for harmonic map with potential such as [21,22],p-harmonic map with potential such as[23],F-harmonic map with potential such as[24],f-harmonic map with potential such as[25]and F-stationary maps with potential such as[4].

In this paper,we generalize and unify the concept of critical point of the functionalΦ. For this,we define the functional Φf,Hby

where H is a smooth function on Nn.If H=0,then we have Φf,H= Φf.If H=0 and f=1,then we have Φf,H= Φ.Let

be a variation of u,i.e.,ut= Ψ(t,.)with u0=u,where Ψ :(−∈,∈) × M → N is a smooth map.Let Γ0(u−1T N)be a subset of Γ(u−1T N)consisting of all elements with compact supports contained in the interior of M.For each ψ ∈ Γ0(u−1T N),there exists a variation ut(x)=expu(x)(tψ)(for t small enough)of u,which has the variational field ψ.Such a variation is said to have a compact support.Let

Defi nition 1.1A smooth map u is called f-stationary map with potential H for the functional Φf,H(u),if

for V ∈ Γ0(u−1T N).

It is known that du(X) ∈ Γ(u−1T N)for any vector field X of M.If X has a compact support which is contained in the interior of M,then du(X) ∈ Γ0(u−1T N).

Defi nition 1.2A smooth map u is called weakly f-stationary map with potential H for the functional Φf,H(u)if Ddu(X)Φf,H(u)=0 for all X ∈ Γ0(T M).

Remark 1.1From Definition 1.1 and Definition 1.2,we know that f-stationary maps with potential H must be weakly f-stationary maps with potential H,that is,the weakly f-stationary maps with potential H are the generalization of the f-stationary maps with potential H.

In this paper,we investigate weakly f-stationary maps with potential H.By using the stress-energy tensor,we obtain some Liouville type theorems for weakly f-stationary maps with potentialunder some conditions on H.

2 Preliminaries

Let ▽ andN▽ always denote the Levi-Civita connections of M and N respectively.Let ~▽ be the induced connection on u−1T N defined by~▽XW=N▽du(X)W,where X ∈ Γ(T M) and W ∈ Γ(u−1T N).We choose a local orthonormal frame field{ei}on M.We define the tension field τΦf,H(u)of u by

where σu= ∑jh(du(.),du(ej))du(ej),which was defined in[1].

Under the notation above we have the following:

Lemma 2.1[5](The first variation formula)Let u:M → N be a C2map.Then

where V=ddtut|t=0.

Let u:M → N be a weakly f-stationary map with potential H and X ∈ Γ0(T M). Then from Lemma 2.1 and the definition of weakly f-stationary maps with potential H,we have

Recall that for a 2-tensor field T ∈ Γ (T∗M ⊗ T∗M),its divergence div T ∈ Γ (T∗M)is defined by

where X is any smooth vector field on M.For two 2-tensors T1,T2∈ Γ(T∗M ⊗ T∗M),their inner product is defined as follows:

where{ei}is an orthonormal basis with respect to g.For a vector field X ∈ Γ(T M),we denote by θXits dual one form,i.e., θX(Y)=g(X,Y),where Y ∈ Γ(T M).The covariant derivative of θXgives a 2-tensor field ▽θX:

If X= ▽ϕ is the gradient field ofsome C2function ϕ on M,then θX=dϕ and ▽θX=Hessϕ.

Lemma 2.2(see[26,27])Let T be a symmetric(0,2)-type tensor fi eld and let X be a vector field,then

where LXis the Lie derivative ofthe metric g in the direction of X.Indeed,let{e1,···,em} be a local orthonormalframe field on M.Then

Let D be any bounded domain of M with C1boundary.By using the Stokes’theorem, we immediately have the following integralformula

where ν is the unit outward normalvector field along ∂D.

From equation(2.8),we have

Lemma 2.3If X is a smooth vector field with a compact support contained in the interior of M,then Z

which is called the f-stress-energy tensor.

Han and Feng in[5]introduced a symmetric 2-tensor SΦfto the functionalΦf(u)by

Lemma 2.4[5]Let u:(M,g) → (N,h)be a smooth map,then for all x ∈ M and for each vector X ∈ TxM,

where

By using equations(2.3),(2.9)and(2.11),we know that if u:M → N is a weakly f-stationary map with potential H,then we have

for any X ∈ Γ0(T M).

On the other hand,we may introduce the stress-energy tensor with potential SΦf,Hby the following

Then

By using equations(2.3),(2.9)and(2.14),we know that if u:M → N is a weakly fstationary map with potential H,then we have

for any X ∈ Γ0(T M).

3 Liouville Type Theorems

Let(M,g0)be a complete Riemannian manifold with a pole x0.Denote by r(x)the g0-distance function relative to the pole x0,that is r(x)=distg0(x,x0).Set

It is known that∂∂ris always an eigenvector of Hessg0(r2)associated to eigenvalue 2.Denote by λmax(resp. λmin)the maximum(resp.minimal)eigenvalues of Hessg0(r2)− 2dr ⊗ dr at each point of M −{x0}.Let(Nn,h)be a Riemannian manifold,and H be a smooth function on N.

From now on,we suppose that u:(Mm,g) → (N,h)is an f-stationary map with potential H,where

Clearly the vector field ν = ϕ−1∂is an outer normal vector field along ∂B(r) ⊂ (M,g).

∂rThe following conditions that we willassume for ϕ are as follows:

(ϕ2)There is a constant C0＞ 0 such that

RemarkIfϕ(r)=r14,conditions(ϕ1)and(ϕ2)turn into the following

Now we set

Theorem 3.1Let u:(M,ϕ2g0) → (N,h)be a weakly f-stationary map with potential H where 0 ＜ ϕ ∈ C∞(M).If ϕ satisfies(ϕ1)(ϕ2),H ≤ 0(or Hu(M)≤ 0),C0− µ ＞ 0 and

then u is constant.

ProofWe takewhere ▽0denotes the covariant derivative determined by g0and φ(r)is a nonnegative function determined later.By a direct computation,we have

Let{ei}mi=1be an orthonormalbasis with respect to g0and em=∂∂r.We may assume that Hessg0(r2)becomes a diagonal matrix with respect to{ei}mi=1.Then{e~i= ϕ−1ei}is an orthonormalbasis with respect to g. Now we compute

From(3.2),(2.14),(3.3),(ϕ1)and(ϕ2),we have

From(3.4),we have

For any fixed R ＞ 0,we take a smooth function φ(r)which takes value 1 on B(R2),0 outside B(R)and 0 ≤ φ(r) ≤ 1 on T(R)=B(R) − B(R2).And φ(r)also satisfies the condition |φ′(r)|≤Cr1on M,where C1is a positive constant.

From(2.15)and(3.5),we have

From(3.6)and(3.7),we have we have

So we know that u is a constant.

RemarkLetbe a complete Riemannian manifold with a pole x0.Assume that the radial curvature Krof M satisfi es the following conditions:withand.From the equation(3.1)and Lemma 4.4 in[5], we have.Letsmooth function on

Theorem 3.2Let u:be a weakly f-stationary map with potential H whereIfϕ satisfiesand,then u is constant.

ProofBy using the similar method in the proof in Theorem 3.1,we can obtain the following

From∂H

∂r◦u≥ 0 and(3.8),we have

For any fixed R ＞ 0,we take a smooth function φ(r)which takes value 1 on B(R2),0 outside B(R)and 0 ≤ φ(r) ≤ 1 on T(R)=B(R) − B(R2).And φ(r)also satisfi es the condition: |φ′(r)|≤Cr1on M,where C1is a positive constant.

From(2.12)and(3.9),we have

From(3.10)and(3.11),we have

So we know that u is a constant.

Theorem 3.3Suppose u:(M,ϕ2g0) → (N,h)is a smooth map which satisfi es the following

for any X ∈ Γ(T M).If ϕ satisfies(ϕ1)(ϕ2),H ≤ 0(or Hu(M)≤ 0),C0− µ ＞ 0 and Φf,H(u) of u is slowly divergent,then u is constant.

ProofFrom the inequality(3.5)for φ(r)=1,we have

On the other hand,taking D=B(r)and T=SΦf,Hin(2.8),we have

Now suppose that u is a nonconstant map,so there exists a constant R1＞ 0 such that for R ≥ R1,

where C3is a positive constant. From(3.13),we have

so we know that there exists a positive constant R2＞ R1such that for R ≥ R2,we have

From(3.14)(3.15)and(3.18),we have for R ＞ R2,

From(3.19)and|▽r|= ϕ−1,we have

This contradicts(3.12),therefore u is a constant.

Theorem 3.4Suppose u:(M,ϕ2g0) → (N,h)is a smooth map which satisfies the following

for any X ∈ Γ(T M).If ϕ satisfies(ϕ1)(ϕ2),∂H

∂r◦u≥ 0 C0− µ ＞ 0 and Φf(u)of u is slowly divergent(see(3.12)),then u is constant.

ProofFrom inequality(3.9)for φ(r)=1,we have

On the other hand,taking D=B(r)and T=SΦfin(2.8),we have

Now suppose that u is a nonconstant map,so there exists a constant R3＞ 0 such that for R ≥ R3,

where C4is a positive constant.

From(3.21),we have

so we know that there exists a positive constant R4＞ R3such that for R ≥ R4,we have

From(3.22),(3.23)and(3.26),we have for R ＞ R4,

From(3.27)and|▽r|= ϕ−1,we have

This contradicts(3.12),therefore u is a constant.

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具有势函数的弱f-稳态映射的若干结果

韩英波,冯书香
(信阳师范学院数学与信息科学学院,河南 信阳 464000)

本文研究了与拉回度量有 关广义泛函Φf,H. 利用应力能 量张量的方法, 得到具有势函数的弱f-稳态映射的一些刘维尔型定理.

具有势函数的弱f-稳态映射;应力能量张量;刘维尔型定理

:58E20;53C21

O186.15

tion:58E20;53C21

A < class="emphasis_bold">Article ID:0255-7797(2017)02-0301-14

0255-7797(2017)02-0301-14

∗Received date:2014-11-16 Accepted date:2015-03-19

Foundation item:Supported by National Natural Science Foundation of China(11201400; 10971029);Basic and Frontier Technology Reseach Project of Henan Province(142300410433);Project for youth teacher of Xinyang Normal University(2014-QN-061).

Biography:Han Yingbo(1978–),male,born at Heze,Shandong,PH.D.,associate professor,major in diff erential geometry.