We generalize wave maps to biwave maps. We prove that the composition of a biwave map and a totally geodesic map is a biwave map. We give examples of biwave nonwave maps. We show that if f is a biwave map into a Riemannian manifold under certain circumstance, then f is a wave map. We verify that if f is a stable biwave
map into a Riemannian manifold with positive constant sectional curvature satisfying the conservation law, then f is a wave map. We finally obtain a theorem involving an unstable biwave map.

1. Introduction

Harmonic maps between Riemannian manifolds were first introduced and established by Eells and Sampson [1] in 1964. Afterwards, there were two reports on harmonic maps by Eells and Lemaire [2, 3] in 1978 and 1988. Biharmonic maps, which generalized harmonic maps, were first studied by Jiang [4, 5] in 1986. In this decade, there has been progress in biharmonic maps made by Caddeo et al. [6, 7], Loubeau and Oniciuc [8], Montaldo and Oniciuc [9], Chiang and Wolak [10], Chiang and Sun [11, 12], Chang et al. [13], Wang [14, 15], and so forth.

Wave maps are harmonic maps on Minkowski spaces, and their equations are the second-order hyperbolic systems of partial differential equations, which are related to Einstein’s equations and Yang-Mills fields. In recent years, there have been many new developments involving local well-posedness and global-well posedness of wave maps into Riemannian manifolds achieved by Klainerman and Machedon [16, 17], Shatah and Struwe [18, 19], Tao [20, 21], Tataru [22, 23], and so forth. Furthermore, Nahmod et al. [24] also studied wave maps from R×Rm into (compact) Lie groups or Riemannian symmetric spaces, that is, gauged wave maps when m≥4, and established global existence and uniqueness, provided that the initial data are small. Moreover, Chiang and Yang [25] , Chiang and Wolak [26] have investigated exponential wave maps and transversal wave maps.

Bi-Yang-Mills fields, which generalize Yang-Mills fields, have been introduced by Ichiyama et al. [27] recently. The following connection between bi-Yang Mills fields and biwave equations motivates one to study biwave maps.

Let P be a principal fiber bundle over a manifold M with structure group G and canonical projection π, and let 𝒢 be the Lie algebra of G. A connection A can be considered as a 𝒢-valued 1-form A=Aμ(x)dxμ locally. The curvature of the connection A is given by the 2-form F=Fμνdxμdxν with
Fμν=∂μAν-∂νAμ+[Aμ,Aν].
The bi-Yang-Mills Lagrangian is defined
L2(A)=12∫M∥δF∥2dvM,
where δ is the adjoint operator of the exterior differentiation d on the space of E-valued smooth forms on M (E=End(P), the endormorphisms of P). Then the Euler-Lagrange equation describing the critical point of (1.2) has the form
(δd+F)δF=0,
which is the bi-Yang-Mills system. In particular, letting M=R×R2 and G=SO(2), the group of orthogonal transformations on R2, we have that Aμ(x) is a 2×2 skew symmetric matrix Aμij. The appropriate equivariant ansatz has the form
Aμij(x)=(δμixj-δμjxi)h(t,|x|),
where h:M→R is a spatially radial function. Setting u=r2h and r=|x|, the bi-Yang-Mills system (1.3) becomes the following equation for u(t,r):
utttt-urrrr-3rurrr+2r2urr-2r3ur=k(t,r),
which is a linear nonhomogeneous biwave equation, where k(t,r) is a function of t and r.

Biwave maps are biharmonic maps on Minkowski spaces. It is interesting to study biwave maps since their equations are the fourth-order hyperbolic systems of partial differential equations, which generalize wave maps. This is the first attempt to study biwave maps and their relationship with wave maps. There are interesting and difficult problems involving local well posedness and global well posedness of biwave maps into Riemannian manifolds or Lie groups (or Riemannian symmetric spaces), that is, gauged biwave maps for future exploration.

In Section 2, we compute the first variation of the bi-energy functional of a biharmonic map using tensor technique, which is different but much easier than Jiang’s [4] original computation. In Section 3, we prove in Theorem 3.3 that if f:Rm,1→N1is a biwave map and f1:N1→N2 is a totally geodesic map, then f1∘f:Rm,1→N2 is a biwave map. Then we can apply this theorem to provide many biwave maps (see Example 3.4). We also can construct biwave nonwave maps as follow: Let h:Ω⊂Rm,1→Sn(1/2)be a wave map on a compact domain and let i:Sn(1/2)→Sn+1(1) be an inclusion map. The map f=i∘h:Ω→Sn+1(1) is a biwave nonwave map if and only if h has constant energy density, compare with Theorem 3.5. Afterwards, we show that if f:Ω→Nis a biwave map on a compact domain into a Riemannian manifold satisfying -|τ□f|t2+∑i=1m|τ□f|xi2-Rβγμ′α(-ftβftγ+∑i=1mfiβfiγ)τ□(f)μ≥0,
then f is a wave map (cf. Theorem 3.6). This theorem is different than the theorem obtained by Jiang [4]: if f is a biharmonic map from a compact manifold into a Riemannian manifold with nonpositive curvature, then f is a harmonic map. In Section 4, we verify that if f is a stable biwave map into a Riemannian manifold with positive constant sectional curvature satisfying the conservation law, then f is a wave map (cf. Theorem 4.5). We also prove that if h:Ω→Sn(1/2) is a wave map on a compact domain with constant energy density, then f=i∘h:Ω→Sn+1(1)is an unstable biwave map (cf. Theorem 4.7).

2. Biharmonic Maps

A biharmonic map f:(Mm,gij)→(Nn,hαβ) from an m-dimensional Riemannian manifold M into an n-dimensional Riemannian manifold N is the critical point of the bi-energy functional
E2(f)=12∫M∥(d+d*)2f∥2dv=12∫M∥(d*d)f∥2dv=12∫M∥τ(f)∥2dv,
where dv is the volume form on M.

Notations

d* is the adjoint of d and τ(f)=trace (Ddf)=(Ddf)(ei,ei)=(Deidf)(ei) is the tension field. Here D is the Riemannian connection on T*M⊗f-1TN induced by the Levi-Civita connections on M and N, and {ei} is the local frame at a point of M. The tension field has components
τ(f)α=gijfi|jα=gij(fijα-Γijkfkα+Γβγ′αfiβfjγ),
where Γijk and Γαβ′γ are the Christoffel symbols on M and N, respectively.

In order to compute the Euler-Lagrange equation of the bi-energy functional, we consider a one-parameter family of maps {ft}∈C∞(M×I,N) from a compact manifold M (without boundary) into a Riemannian manifold N. Here ft(x) is the endpoint of a segment starting at f(x)(=f0(x)), determined in length and direction by the vector field ḟ(x) along f(x). For a nonclosed manifold M, we assume that the compact support of ḟ(x) is contained in the interior of M (we need this assumption when we compute τ(f) by applying the divergence theorem). Then we have
ddtE2(ft)|t=0=Ė2(f)=∫M(Dtτf,τf)t=0dv.

Let ξ=∂ft/∂t. The components of Dtτf are fi|j|tα=(∂fi|jα/∂t)+Γμγ′αfi|jμξγ. We can use the curvature formula on M×I→N and get
fi|j|tα=fi|t|jα+Rβγμ′αfiβfjγξμ,
where R′ is the Riemannian curvature of N. But fi|tα=ft|iα=ξ|iα, therefore, Dtτf has components ξ|i|jα+Rβγμ′αfiβfjγξμ. We can rewrite (2.3) as
ddtE2(ft)|t=0=∫M(Jf(τf),τf)dv,
where
Jfα(ξ)=gijξ|i|jα+gijRβγμ′αfiβfjγξμ=Δξα+R′α(df,df)ξ
is a linear equation for ξ(=τ(f)), and Δ(ξ)=D*D(ξ) is an operator from f-1TN to f-1TN. Solutions of Jf(ξ)=0 are called Jacobi fields. Hence, we obtain the following definition from (2.3), (2.5), and (2.6).

Definition 2.1.

f:M→N is a biharmonic map if and only if the bitension field
τ2(f)α=Jf(τf)α=Δτ(f)α+R′α(df,df)τ(f)=gij(fijα-Γijkfkα+Γβγ′αfiβfjγ)+gijRβγμ′αfiβfjγτ(f)μ=0,
that is, the tension field τ(f), is a Jacobi field.

If τ(f)=0, then τ2(f)=0. Thus, harmonic maps are obviously biharmonic. Biharmonic maps satisfy the fourth-order elliptic systems of PDEs, which generalize harmonic maps. Our computation for the first variation of the bi-energy functional presented here using tensor technique is different but much easier than Jiang’s [4] original computation (it took him four pages).

Caddeo et al. [7] showed that a biharmonic curve on a surface of nonpositive Gaussian curvature is a geodesic (i.e., is harmonic) and gave examples of biharmonic nonharmonic curves on spheres, ellipses, unduloids, and nodoids.

Theorem 2.2 (see [<xref ref-type="bibr" rid="B14">4</xref>]).

Let f:Mm→Sm+1(1) be an isometric embedding of an m-dimensional compact Riemannian manifold M into an (m+1)-dimensional unit sphere Sm+1(1) with nonzero constant mean curvature. The map f is biharmonic if and only if ∥B(f)∥2=m, where B(f) is the second fundamental form of f.

Example 2.3.

In Sm+1(1), the compact hypersurfaces, whose Gauss maps are isometric embeddings, are the Clifford surfaces [28]:
Mkm(1)=Sk(12)×Sm-k(12),0≤k≤m.
Let f:Mkm(1)→Sm+1(1) be a standard embedding such that k≠m/2. Because ∥B(f)∥2=k+m-k=m and τ(f)=k-(m-k)=2k-m≠0,f is a biharmonic nonharmonic map by Theorem 2.2.

3. Biwave Maps

Let Rm,1 be an m+1 dimensional Minkowski space R×Rm with the metric (gij)=(-1,1,…,1) and the coordinates x0=t,x1,x2,…,xm and let (N,hαβ) be an n-dimensional Riemannian manifold. A wave map is a harmonic map on the Minkowski space Rm,1 with the energy functional
E(f)=12∫Rm,1(-|ft|2+|∇xf|2)dtdx=12∫Rm,1hαβ(-ftαftβ+∑i=1mfiαfiβ)dtdxi.
The Euler-Lagrange equation describing the critical point of (3.1) is
τ□α(f)=□fα+Γβγ′α(-ftβftγ+∑i=1mfiβfiγ)=0,
where □=-(∂2/∂t2)+Δx is the wave operator on Rm,1 and Γβγ′α are the Christoffel symbols of N. f is a wave map iff the wave field τ□α(f) (i.e., the tension field on a Minkowski space) vanishes. The wave map equation is invariant with respect to the dimensionless scaling f(t,x)→f(ct,cx),c∈R. But, the energy is scale invariant in dimension m=2.

If f:Rm,1→N is a smooth map from a Minkowski space Rm,1 into a Riemannian manifold N, then the bi-energy functional is, from (2.1),
E2(f)=12∫Rm,1∥(d+d*)2f∥2dtdx=12∫Rm,1∥d*df∥2dtdx=12∫Rm,1∥τ□(f)∥2dtdx.
The Euler-Lagrange equation describing the critical point of (3.3), from (2.5), is
(τ2)□(f)=Jf(τ□f)=Δτ□(f)+R′(df,df)τ□(f)=0.

Definition 3.1.

f:Rm,1→N from a Minkowski space into a Riemannian manifold is a biwave map if and only if the biwave field (i.e., the bitension field on a Minkowski space),
(τ2)□(f)α=Jf(τ□f)α=Δτ□(f)α+R′α(df,df)τ□(f)=□τ□(f)α+Γμγ′α(-τ□(f)tμτ□(f)tγ+∑i=1mτ□(f)iμτ□(f)iγ)+Rβγμ′α(-ftβftγ+∑i=1mfiβfiγ)τ□(f)μ=0,
that is, the wave field τ□(f), is a Jacobi field on the Minkowski space.

Biwave maps satisfy the fourth-order hyperbolic systems of PDEs, which generalize wave maps. If τ□(f)=0, then (τ2)□(f)=0. Waves maps are obviously biwave maps, but biwave maps are not necessarily wave maps.

Example 3.2.

Let u:Rm,1→R be a function defined on a Minkowski space satisfying the following conditions:
□2u(t,x)=□(□u)=utttt-2uttxx+uxxxx=0,(t,x)∈(0,∞)×Rm,u=u0,ut=u1,u=u0,∂∂tu=∂u∂t=u1,(t,x)∈{t=0}×Rm,
where the initial data u0 and u1 are given. Since this is a fourth-order homogeneous linear biwave equation with constant coefficients, it is well known that u(t,x) can be solved by [18, 29].

Let f:Rm,1→N1 be a smooth map from a Minkowski space Rm,1 into a Riemannian manifold N1 and let f1:N1→N2 be a smooth map between two Riemannian manifolds N1 and N2. Then the composition f1∘f:Rm,1→N2 is a smooth map. Since Rm,1 is a semi-Riemannian manifold (i.e., a pseudo-Riemannian manifold), we can define a Levi-Civita connection on Rm,1 by O’Neill [30]. Let D, D′,D̅,D̅′,D̅”,D̂,D̂′,D̂” be the connections on Rm,1,TN1,f-1N1,f1-1TN2,(f1∘f)-1TN2,T*Rm,1⊗f-1TN1,T*N1⊗f1-1TN2,T*Rm,1⊗(f1∘f)-1TN2, respectively, and let RN2(,),Rf1-1TN2(,) be the curvatures on TN2,f1-1TN2, respectively. We first have the following two formulas:
D̂X”d(f1∘f)(Y)=(D̂df(X)′df1)df(Y)+df1∘D̅Xdf(Y),
for X,Y∈Rm,1, and
RN2(df1(X′),df1(Y′))df1(Z′)=Rf1-1TN2(X′,Y′)df1(Z′),
for X′,Y′,Z′∈Γ(TN1).

Theorem 3.3.

If f:Rm,1→N1 is a biwave map and f1:N1→N2 is totally geodesic between two Riemannian manifolds N1 and N2, then the composition f1∘f:Rm,1→N2 is a biwave map.

Proof.

Let x0=t,x1,…,xm be the coordinates of a point p in Rm,1 and let e0=∂/∂t,e1=(1,0,…,0),e2=(0,1,0,…,0),…,em=(0,…,0,1) be the frame at p. We know from [4] that D̅”*D̅”=D̅ek”D̅ek”-D̅Dekek”. Since f1 is totally geodesic, we have τ□(f1∘f)=df1∘τ□(f) by applying the chain rule of the wave field to f1∘f as [1]. Then we get
D̅”*D̅τ□”(f1∘f)=D̅”*D̅”(df1∘τ□(f))=D̅ek”D̅ek”(df1∘τ□(f))-D̅Dekek”(df1∘τ□(f)).
Recalling that τ□(f)=D̂ejdf(ej), we derive from (3.7a) that
D̅ek”(df1∘τ□(f))=D̅ek”(df1∘D̂ejdf(ej))=(D̂D̂ejdf(ek)′df1)(D̂ejdf(ej))+df1∘D̅ek(D̂ejdf(ej))=df1∘D̅ekτ□(f),
since f1 is totally geodesic. Therefore, we have
D̅ek”D̅ek”(df1∘τ□(f))=D̅ek”(df1∘D̅ekτ(f))=df1∘D̅ekD̅ekτ□(f),D̅Dekek”(df1∘τ(f))=df1∘D̅Dekekτ□(f).
Substituting (3.10) into (3.8), we arrive at
D̅”*D̅”τ□(f1∘f)=df1∘D̅*D̅τ□(f),
where D̅*D̅=D̅ekD̅ek-D̅Dekek.

On the other hand, we have by (3.7b)
RN2(d(f1∘f)(ei),τ□(f1∘f))d(f1∘f)(ei)=Rf1-1TN2(df(ei),τ□(f))df1(df(ei))=df1∘RN1(df(ei),τ□(f))df(ei).
We obtain from (3.11) and (3.12)
D̅”*D̅”(f1∘f)+RN2(d(f1∘f)(ei),τ□(f1∘f))d(f1∘f)(ei)=df1∘[D̅*D̅τ□(f)+RN1(df(ei),τ□(f))df(ei)],
that is, (τ2)□(f1∘f)=df1∘(τ2)□(f). Hence, if f is a biwave map and f1 is totally geodesic, then f1∘f is a biwave map. Note that the total geodesicity of f1 cannot be weakened into a harmonic or biharmonic map.

Example 3.4.

Let N1 be a submanifold of N. Are the biwave maps into N1 also biwave maps into N? The answer is affirmative iff N1 is a totally geodesic submanifold of N, that is, N1 geodesics are N geodesics. N1 is a geodesic γ(t)=(γ1,…,γn):R→N⊂Rn with |γ̇(t)|=1 iff γ̇ is parallel, that is, D∂/∂tγ̇=0 iff γ̈⊥TγN. For a map u:Rm,1→R, letting f=γ∘u=(f1,…,fn):Rm,1→N⊂Rn, we have by (3.13) the following:
(τ2)□(f)=dγ∘(τ2)□(u)=dγ∘□2u,
since γ is a geodesic. Hence, f=γ∘u is a biwave map if and only if u solves the fourth-order homogeneous linear biwave equation □2u=0 as in Example 3.2. It follows from Theorem 3.3 that there are many biwave maps f:Rm,1→N provided by geodesics of N.

We also can construct examples of biwave nonwave maps from some wave maps with constant energy using Theorem 3.5. Let
Sn(12)=Sn(12)×{12}={(x1,x2,…,xn+1,12)∣x12+⋯+xn+12=12}
be a hypersphere of Sn+1(1). Then Sn(1/2) is a biharmonic nonminimal submanifold of Sn+1(1) by Theorem 2.2 and Example 2.3. Let ζ=(x1,…,xn+1,-1/2) be a unit section of the normal bundle of Sn(1/2) in Sn+1(1). Then the second fundamental form of the inclusion i:Sn(1/2)→Sn+1(1) is B(X,Y)=Ddi(X,Y)=-(X,Y)ζ. By computation, the tension field of i is τ(i)=-nζ, and the bitension field is τ2(i)=0.

Theorem 3.5.

Let h:Ω→Sn(1/2) be a nonconstant wave map on a compact space-time domain Ω⊂Rm,1 and let i:Sn(1/2)→Sn+1(1) be an inclusion. The map f=i∘h:Rm,1→Sn+1(1) is a biwave nonwave map if and only if h has constant energy density e(h)=(1/2)|dh|2.

Proof.

Let x0=t,x1,…,xm be the coordinate of a point p in Ω⊂Rm,1 and let e0=∂/∂t,e1=(1,0,…,0),e2=(0,1,0,…,0),…,em=(0,…,0,1) be the frame at p. Recall that ζ is the unit section of the normal bundle. By applying the chain rule of the wave field to f=i∘h, we have
τ□(f)=di(τ□(h))+trace Ddi(dh,dh)=-2e(h)ζ,
since h is a wave map. We can derive the following at the point p by straightforward calculation:
D*Dτ□(f)=-DeifDeifτ□(f)=-DeifDeif(-2e(h)ζ)=2(eieie(h))ζ-2e(h)(dh(ei),dh(ei))ζ+4df[(eie(h))ei]+2e(h)Ddh(ei,ei),RSn+1(df(ei),τ□(f))df(ei)=-(dh(ei),dh(ei))τ(f)=2(dh(ei),dh(ei))e(h)ζ.
Therefore, we obtain
τ2□(f)=-2(Δe(h))ζ+4df(grade(h)).
Suppose that f=i∘h:Ω→Sn(1/2)×{1/2}→Sn+1(1) is a biwave nonwave map (τ□(f)≠0). As the ζ-part of τ2□(f), Δe(h) vanishes, which implies that e(h) is constant since Ω is compact. The converse is obvious.

Let x0=t,x1,…,xm be the coordinates of a point in a compact space-time domain Ω⊂Rm,1 and e0=∂/∂t,e1=(1,0,…,0),e2=(0,1,0,…,0),…,em=(0,…,0,1) be the frame at the point. Suppose that f:Ω→N is a biwave map from a compact domain Ω into a Riemannian manifold N such that the compact supports of ∂f/∂xi and Dei∂f/∂xi are contained in the interior of Ω.

Theorem 3.6.

If f:Ω→N is a biwave map from a compact domain into a Riemannian manifold such that
-|τ□f|t2+∑i=1m|τ□f|xi2-Rβγμ′α(-ftβftγ+∑i=1mfiβfiγ)τ□(f)μ≥0,
then f is a wave map.

Proof.

Since f is a biwave map, we have by (3.4)
(τ2)□(f)=Δτ□(f)+R′(df,df)τ□(f).
Recall that x0=t,x1,…,xm are the coordinates of a point in Ω⊂Rm,1 and e0=∂/∂t,e1=(1,0,…,0),e2=(0,1,0,…,0),…,em=(0,…,0,1). We compute
12Δ∥τ□(f)∥2=(Deiτ□(f),Deiτ□(f))+(D*Dτ□(f),τ□(f))=∑i=0m(Deiτ□(f),Deiτ□(f))-(Rβγμ′α(-ftβftγ+∑i=1mfiβfiγ)τ□(f)μ,τ□(f))=-|τ□f|t2+∑i=1m|τ□f|xi2-(Rβγμ′α(-ftβftγ+∑i=1mfiβfiγ)τ□(f)μ,τ□(f)).
By applying the Bochner’s technique from (3.19) and the assumption that the compact supports of ∂f/∂xi and Dei∂f/∂xi are contained in the interior of Ω, we know that ∥τ□(f)∥2 is constant, that is, dτ□(f)=0. If we use the identity
∫Ωdiv(df,τ(f))dz=∫Ω(|τ(f)|2+(df,dτ(f)))dz,z=(t,x)
and the fact dτ□(f)=0, then we can conclude that τ□(f)=0 by applying the divergence theorem.

Corollary 3.7.

If f:Ω→N is a biwave map on a compact domain such that ∑i=1m|τ□f|xi2≥|τ□f|t2 and Rβγμ′α(-ftβftγ+∑i=1mfiβfiγ)τ□(f)μ≤0, then f is a wave map.

Proof.

The result follows from (3.19) immediately.

4. Stability of Biwave Maps

Let x0=t,x1,…,xm be the coordinates of a point in a compact space-time domain Ω⊂Rm,1 and let e0=∂/∂t,e1=(1,0,…,0),…,em=(0,0,…,1) be the frame at the point. Suppose that f:Ω→N is a biwave map from a compact space-time domain Ω into a Riemannian manifold N such that the compact supports of ∂f/∂xi and Dei∂f/∂xi are contained in the interior of Ω. Let V∈Γ(f-1TN) be a vector field such that ∂f/∂t|t=0=V. If we apply the second variation of a biharmonic map in [4] to a biwave map, we can have the following.

Lemma 4.1.

If f:Ω→N is a biwave map from a compact domain into a Riemannian manifold, then
12d2dt2E2(f)|t=0=∫Ω∥ΔV+RN(df(ei),V)df(ei)∥2dz+∫Ω<V,(Ddf(ei)′RN)(f(ei),τ□(f))V+(Dτ□(f)′RN)(df(ei),V)df(ei)+RN(τ□(f),V)τ□(f)+2RN(df(ei),V)D̅eiτ□(f)+2RN(df(ei),τ□(f))D̅eiV>dz,
where z=(t,x)∈R×Rm,D′ is the Riemannian connection on TN, and V is the vector field along f.

Definition 4.2.

Let f:Rm,1→N be a biwave map. If (d2/dt2)E2(f)|t=0≥0, then f is a stable biwave map.

If we consider a wave map, that is, τ□(f)=0 as a biwave map, then by (4.1) we have (d2/dt2)E2(f)|t=0≥0 and f is automatically stable.

Definition 4.3.

Let f:Rm,1→(N,h) be a smooth map from a Minkowski space into a Riemannian manifold (N,h). The stress energy is defined by S(f)=e(f)g-f*h, where e(f)=(1/2)|df|2 is the energy function and g=(-100I). The map f satisfies the conservation law if divS(f)=0.

Proposition 4.4.

Let f:Rm,1→(N,h) be a smooth map from a Minkowski space into a Riemannian manifold (N,h). Then
divS(f)(X)=-〈τ□(f),df(X)〉,X∈Rm,1.

Proof.

Let x0=t,x1,…,xm be the coordinates of a point in Rm,1,e0=∂/∂t,e1=(1,0,…,0),…,em=(0,0,…,1) and g=(-100I), where I is an m×m matrix. We calculate
divS(f)(X)=DeiS(f)(ei,X)=Dei(12|df|2(-100I)-f*h)(ei,X)=Dei(12|df|2(-100I))(ei,X)-(Deif*h)(ei,X)=(-(D∂f∂t,∂f∂t)(-1))(e0,X)+(D∂f∂xi,∂f∂xi)I(ei,X)-Dei(f*ei,f*X)=(D∂f∂t,∂f∂t)(e0,X)+(D∂f∂xi,∂f∂xi)(ei,X)-(Deif*ei,f*X)-(f*ei,Deif*X)=((DXdf)ei,f*ei)-(τ□(f),f*X)-(f*ei,Deif*X),
where the first term and the third term are canceled out and Deif*ei=τ□(f).

Theorem 4.5.

Let Ω⊂Rm,1 be a compact domain and let (N,h) be a Riemannian manifold with constant sectional curvature K>0. If f:Ω→N is a stable biwave map satisfying the conservation law, then f is a wave map.

Proof.

Because N has constant sectional curvature, the second term of (4.1) disappears and (4.1) becomes
12d2dt2E2(ft)|t=0=∫Ω∥ΔV+RN(df(ei),V)df(ei)∥2dz+∫Ω〈V,RN(τ□(f),V)τ□(f)+2RN(df(ei),V)Deiτ□(f)+2RN(df(ei),τ□(f))DeiV〉dz.
In particular, let V=τ□(f). Recalling that f is a biwave map and N has constant sectional curvature K>0, (4.4) can be reduced to
12d2dt2E2(f)|t=0=4∫Ω〈RN(df(ei),τ□(f))Deiτ□(f),τ□(f)〉dz=4K∫Ω[〈df(ei),Deiτ□(f)〉∥τ□(f)∥2-〈df(ei),τ□(f)〉〈τ□(f),Deiτ□(f)〉]dz.
Since f satisfies the conservation law, by Definition 4.3, Proposition 4.4, and (4.2) we have
〈df(ei),τ□(f)〉=0,〈df(ei),Deiτ□(f)〉=-〈Deidf(ei),τ□(f)〉=-∥τ□(f)∥2.
Substituting (4.6) into (4.5) and applying the stability of f, we get
12d2dt2E2(ft)|t=0=-4K∫Ω∥τ□f∥4dz≥0,
which implies that τ□(f)=0, that is, f:Ω→N is a wave map.

If we apply the Hessian of the bi-energy of a biharmonic map [4] to a biwave map f:Ω→Sn+1(1), then we have the following.

Lemma 4.6.

Let f:Ω→Sn+1(1) be a biwave map. The Hessian of the bi-energy functional E2 of f is
H(E2)f(X,Y)=∫Ω(If(X),Y)dz,
where
If(X)=Δf(ΔfX)+Δf(trace (X,df·)df·-|df|2X)+2(dτ□(f),df)X+|τ□(f)|2X-2trace (X,dτ□(f)·)df-2trace (τ□(f),dX·)df·-(τ□(f),X)τ□(f)+trace(df·,ΔfX)df·+trace(df,trace(X,df·)df·)df·-2|df|2trace(df·,X)df·+2(dX,df)τ□(f)-|df|2ΔfX+|df|4X,
for X,Y∈Γ(f-1TSn+1(1)).

Theorem 4.7.

Let h:Ω→Sn(1/2) be a wave map on a compact domain with constant energy and let i:Sn(1/2)→Sn+1(1) be an inclusion map. Then f=i∘h:Ω→Sn+1(1) is an unstable biwave map.

Proof.

We have the following identities from Theorem 3.5:
|df|2=2e(h),trace(ζ,df·)df·=0,(dτ□(f),df)ζ=-4(e(h))2ζ,|τ□(f)|2=4(e(h))2,trace(ζ,dτ□(f))df·=0,trace(τ(f),dζ·)df=0,(τ□(f),ζ)τ□(f)=4(e(h))2ζ,trace(df,Δfζ)df·=(Δfζ)T,(dζ,df)τ□(f)=-4(e(h))2ζ.
Then we obtain the following formula from Lemma 4.6 and the previous identities:
(If(ζ),ζ)=∫Ω(|Δfζ|2-12e(h)2-4e(h)(Δfζ,ζ))dz,
which is strictly negative, where Δfζ=2e(h)ζ. Hence, f is an unstable biwave map.

Acknowledgment

The author would like to appreciate Professor Jie Xiao and the referees for their comments.

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