Biwave Maps Into Manifolds

Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2009, Article ID 104274, 14 pages doi:10.1155/2009/104274

Research Article Biwave Maps into Manifolds

Yuan-Jen Chiang

Department of Mathematics, University of Mary Washington, Fredericksburg, VA22401, USA

Correspondence should be addressed to Yuan-Jen Chiang, [email protected]

Received 8 January 2009; Accepted 30 March 2009

Recommended by Jie Xiao

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.

Copyright q 2009 Yuan-Jen Chiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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,Montaldoand Oniciuc 9, Chiang and Wolak 10, Chiang and Sun 11, 12, Chang et al. 13,Wang14, 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,Tao20, 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. 2 International Journal of Mathematics and Mathematical Sciences

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,andletG be the Lie algebra of G. A connection A can be considered G μ as a -valued 1-form A Aμ x dx locally. The curvature of the connection A is given by the μ ν 2-form F Fμνdx dx with − Fμν ∂μAν ∂νAμ Aμ,Aν . 1.1

The bi-Yang-Mills Lagrangian is deﬁned 1 2 L2A δF dvM, 1.2 2 M where δ is the adjoint operator of the exterior diﬀerentiation d on the space of E-valued smooth forms on M E EndP, the endormorphisms of P. Then the Euler-Lagrange equation describing the critical point of 1.2 has the form

δd FδF 0, 1.3 which is the bi-Yang-Mills system. In particular, letting M R × R2 and G SO2,the 2 × group of orthogonal transformations on R , we have that Aμ x is a 2 2 skew symmetric ij matrix Aμ . The appropriate equivariant ansatz has the form ij i j − j i | | Aμ x δμx δμx h t, x , 1.4 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 ut, r:

3 2 2 u − u − u u − u kt, r, 1.5 tttt rrrr r rrr r2 rr r3 r which is a linear nonhomogeneous biwave equation, where kt, 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 m,1 → computation. In Section 3, we prove in Theorem 3.3 that if f : R N1 is a biwave map → ◦ m,1 → and f1 : N1 N2 is a totally geodesic map, then f1 f : R N2 is a biwave map. Then we can International Journal of Mathematics and Mathematical Sciences 3 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 → Sn1/ 2 be a wave map on a compact domain and let i : Sn1/ 2 → Sn 11 be an inclusion map. The map f i ◦ h : Ω → Sn 11 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 → N is a biwave map on a compact domain into a Riemannian manifold satisfying m m 2 2 α β γ β γ μ −| | | | − − ≥ τ f t τ f xi Rβγμ ft ft fi fi τ f 0, 1.6 i1 i1 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 → Sn1/ 2 is a wave map on a compact domain with constant energy density, then f i ◦ h : Ω → Sn 11 is an unstable biwave map cf. Theorem 4.7.

2. Biharmonic Maps

m → n A biharmonic map f : M ,gij N ,hαβ from an m-dimensional Riemannian manifold M into an n-dimensional Riemannian manifold N is the critical point of the bi-energy functional 2 1 ∗2 1 ∗ 2 1 2 E2 f d d f dv d d f dv τ f dv, 2.1 2 M 2 M 2 M where dv is the volume form on M.

Notations ∗ d is the adjoint of d and τ f trace Ddf Ddf ei,ei Dei df 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 α ij α ij α − Γk α Γα β γ τ f g fi|j g fij ij fk βγ fi fj , 2.2a

Γk Γ γ ﬀ where ij and αβ are the Christo el 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.Hereft x is the endpoint of a segment starting at f x f0 x , determined in length and direction by the vector ﬁeld f˙ x along fx. For a nonclosed manifold M, we assume that the compact support of f˙x is contained 4 International Journal of Mathematics and Mathematical Sciences in the interior of M we need this assumption when we compute τf by applying the divergence theorem. Then we have d | ˙ E2 ft t 0 E2 f Dtτf,τf t0dv. 2.3 dt M

α α Γα μ γ Let ξ ∂ft/∂t. The components of Dtτf are fi|j|t ∂fi|j /∂t μγ fi|j ξ . We can use the curvature formula on M × I → N and get

β γ f α f α R α f f ξμ, 2.4 i|j|t i|t|j βγμ i j

α α α where R is the Riemannian curvature of N.Butfi|t ft|i ξ|i , therefore, Dtτf has compo- α α β γ μ nents ξ|i|j Rβγμfi fj ξ . We can rewrite 2.3 as

d | E2 ft t0 Jf τf ,τf dv, 2.5 dt M where

α ij α ij α β γ μ Δ α α Jf ξ g ξ|i|j g Rβγμfi fj ξ ξ R df , df ξ 2.6

∗ − − is a linear equation for ξ τf, and ΔξD Dξ 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,and2.6.

Deﬁnition 2.1. f : M → N is a biharmonic map if and only if the bitension ﬁeld

α α Δ α τ2 f Jf τf τ f R α df , df τ f 2.7 ij α − Γk α Γα β γ ij α β γ μ g fij ij fk βγ fi fj g Rβγμfi fj τ f 0, that is, the tension field τf, is a Jacobi field. If τ f 0, then τ2 f 0. Thus, harmonic maps are obviously biharmonic. Bihar- monic 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 4. Let f : Mm → Sm 11 be an isometric embedding of an m-dimensional compact Riemannian manifold M into an m 1-dimensional unit sphere Sm 11 with nonzero constant mean curvature. The map f is biharmonic if and only if Bf2 m, where Bf is the second fundamental form of f. International Journal of Mathematics and Mathematical Sciences 5

Example 2.3. In Sm 11, the compact hypersurfaces, whose Gauss maps are isometric embed- dings, are the Cliﬀord surfaces 28:

1 − 1 Mm1 Sk √ × Sm k √ , 0 ≤ k ≤ m. 2.8 k 2 2

m → m1 2 Let f : Mk 1 S 1 be a standard embedding such that k / m/2. Because B f k m − k m and τfk − m − k2k − m / 0,f is a biharmonic nonharmonic map by Theorem 2.2.

3. Biwave Maps

m,1 × m − Let R be an m 1 dimensional Minkowski space R R with the metric gij 1, 1,...,1 0 1 2 m and the coordinates x t, x ,x ,...,x 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

m 1 − 2 ∇ 2 1 − α β α β E f ft xf dt dx hαβ ft ft fi fi dt dxi. 3.1 2 Rm,1 2 Rm,1 i1

The Euler-Lagrange equation describing the critical point of 3.1 is

m α α Γα − β γ β γ τ f f βγ ft ft fi fi 0, 3.2 i1

− 2 2Δ m,1 Γα ff where ∂ /∂t x is the wave operator on R and βγ are the Christo el symbols of ff α N. f is a wave map i the wave field τ f i.e., the tension field on a Minkowski space vanishes. The wave map equation is invariant with respect to the dimensionless scaling ft, x → fct, 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,

2 1 ∗2 E2 f d d f dt dx 2 Rm,1 3.3

1 ∗ 2 1 2 d df dt dx τ f dt dx. 2 Rm,1 2 Rm,1

The Euler-Lagrange equation describing the critical point of 3.3,from2.5,is

Δ τ2 f Jf τf τ f R df , df τ f 0. 3.4 6 International Journal of Mathematics and Mathematical Sciences

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,

α α Δ α α τ2 f Jf τf τ f R df , df τ f m α α μ γ μ γ τf Γ −τf τf τf τf μγ t t i i i1 3.5 m α β γ β γ μ − Rβγμ ft ft fi fi τ f 0, i1 that is, the wave ﬁeld τf, 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:

2 − ∈ ∞ × m ut, x u utttt 2uttxx uxxxx 0, t, x 0, R , 3.6 ∂ ∂u u u ,u u ,u u , u u , t, x ∈ {t 0} × Rm, 0 t 1 0 ∂t ∂t 1 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 ut, x can be solved by 18, 29.

m,1 → m,1 Let f : R N1 be a smooth map from a Minkowski space R into a Riemannian → manifold N1 and let f1 : N1 N2 be a smooth map between two Riemannian manifolds N1 ◦ m,1 → m,1 and N2. Then the composition f1 f : R N2 is a smooth map. Since R 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.LetD, D , D, D , D , D, D , D” be the connections m,1 −1 −1 ◦ −1 ∗ m,1 ⊗ −1 ∗ ⊗ −1 ∗ m,1 ⊗ on R ,TN1,f N1,f1 TN2, f1 f TN2,TR f TN1,TN1 f1 TN2,TR −1 −1 − ◦ N2 f1 TN2 1 f1 f TN2, respectively, and let R , ,R , be the curvatures on TN2,f1 TN2, respectively. We ﬁrst have the following two formulas: ” ◦ ◦ DXd f1 f Y Ddf Xdf 1 df Y df 1 DXdf Y , 3.7a for X, Y ∈ Rm,1, and

−1 N2 f1 TN2 R df 1 X ,df1 Y df 1 Z R X ,Y df 1 Z , 3.7b

∈ Γ for X ,Y,Z TN1 .

Theorem 3.3. m,1 → → If f : R N1 is a biwave map and f1 : N1 N2 is totally geodesic between two ◦ m,1 → Riemannian manifolds N1 and N2, then the composition f1 f : R N2 is a biwave map. International Journal of Mathematics and Mathematical Sciences 7

0 1 m m,1 Proof. Let x t, x ,...,x be the coordinates of a point p in R and let e0 ∂/∂t, e1 1, 0,...,0 ,e2 0, 1, 0,...,0 , ..., em 0,...,0, 1 be the frame at p.Weknowfrom 4 ” ” ” ” ” ∗ − ◦ ◦ that D D Dek Dek DDe ek . Since f1 is totally geodesic, we have τ f1 f df 1 τ f k ◦ by applying the chain rule of the wave ﬁeld to f1 f as 1 . Then we get

” ” ” ” ∗ ◦ ∗ ◦ D Dτ f1 f D D df 1 τ f 3.8 ” ” ◦ − ” ◦ D D df 1 τ f D df 1 τ f . ek ek Dek ek

Recalling that τ f Dej df ej , we derive from 3.7a that ” ” ◦ ◦ Dek df 1 τ f Dek df 1 Dejdf ej ◦ ◦ D df 1 Dej df ej df 1 Dek Dej df ej df 1 Dek τ f , Dej df ek 3.9

since f1 is totally geodesic. Therefore, we have ” ” ” ◦ ◦ ◦ Dek Dek df 1 τ f Dek df 1 Dek τ f df 1 Dek Dek τ f , 3.10 ” ◦ ◦ D df 1 τ f df 1 DDe ek τ f . Dek ek k

Substituting 3.10 into 3.8, we arrive at

∗ ” ∗ ” ◦ ◦ D D τ f1 f df 1 D Dτ f , 3.11

∗ where D D D D − D . ek ek Dek ek On the other hand, we have by 3.7b

N2 ◦ ◦ ◦ R d f1 f ei,τ f1 f d f1 f ei 3.12 −1 f1 TN2 ◦ N1 R df ei,τ f df 1 df ei df 1 R df ei,τ f df ei.

We obtain from 3.11 and 3.12

” ” ∗ ◦ N2 ◦ ◦ ◦ D D f1 f R d f1 f ei,τ f1 f d f1 f ei ∗ 3.13 ◦ N1 df 1 D Dτ f R df ei,τ f df ei ,

◦ ◦ that is, τ2 f1 f df 1 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. 8 International Journal of Mathematics and Mathematical Sciences

Example 3.4. Let N1 be a submanifold of N. Are the biwave maps into N1 also biwave maps ffi ff into N? The answer is a rmative i N1 is a totally geodesic submanifold of N,thatis,N1 1 n → ⊂ n | | geodesics are N geodesics. N1 is a geodesic γ t γ ,...,γ : R N R with γ˙ t 1 ff ff ⊥ m,1 → ◦ i γ˙ is parallel, that is, D∂/∂tγ˙ 0i γ¨ Tγ N. For a map u : R R, letting f γ u f 1,...,fn : Rm,1 → N ⊂ Rn, we have by 3.13 the following:

◦ ◦ 2 τ2 f dγ τ2 u dγ u, 3.14 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

n 1 n 1 × 1 1 | 2 ··· 2 1 S √ S √ √ x1,x2,...,xn1, √ x x 3.15 2 2 2 2 1 n 1 2

√ be a hypersphere of Sn 11. Then Sn1/ 2 is a biharmonic nonminimal√ submanifold of n1 − S 1 by Theorem 2.2 and√ Example 2.3.Letζ x1,...,xn1, 1/ 2 be a unit section of the normal bundle√ of Sn1/ 2 in Sn 11. Then the second fundamental form of the inclusion i : Sn1/ 2 → Sn 11 is BX, Y DdiX, 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 →√ Sn1/ 2 be a nonconstant wave map on a compact space-time domain Ω ⊂ Rm,1 and let i : Sn1/ 2 → Sn 11 be an inclusion. The map f i ◦ h : Rm,1 → Sn 11 is a biwave nonwave map if and only if h has constant energy density eh1/2|dh|2.

0 1 m Ω ⊂ m,1 Proof. Let x t, x ,...,x be the coordinate of a point p in R 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 diτ h trace Ddidh, dh −2ehζ, 3.16 since h is a wave map. We can derive the following at the point p by straightforward calculation:

∗ f f f f − − − D Dτ f Dei Dei τ f Dei Dei 2e h ζ − 2eieiehζ 2ehdhei,dheiζ 4df eiehei 3.17 2ehDdhei,ei,

Sn 1 − R df ei,τ f df ei dhei,dheiτ f 2dhei,dheiehζ. International Journal of Mathematics and Mathematical Sciences 9

Therefore, we obtain

− Δ τ2 f 2 ehζ 4df grad eh . 3.18

√ √ Suppose that f i ◦ h : Ω → Sn1/ 2 ×{1/ 2}→Sn11 is a biwave nonwave map Δ τ f / 0 .Astheζ-part of τ2 f , e 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 ⊂ m,1 R 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 m m 2 2 α β γ β γ μ − − − ≥ τ f t τ f xi Rβγμ ft ft fi fi τ f 0, 3.19 i1 i1 then f is a wave map.

Proof. Since f is a biwave map, we have by 3.4

Δ τ2 f τ f R df , df τ f . 3.20

0 1 m Ω ⊂ m,1 Recall that x t, x ,...,x are the coordinates of a point in R and e0 ∂/∂t, e1 1, 0,...,0 ,e2 0, 1, 0,...,0 ,..., em 0,...,0, 1 . We compute

1 2 ∗ Δ τf D τ f ,D τ f D Dτ f ,τ f 2 ei ei m m α β γ β γ μ − − Dei τ f ,Dei τ f Rβγμ ft ft fi fi τ f ,τ f i0 i1 m m 2 2 α β γ β γ μ − − − τ f t τ f xi Rβγμ ft ft fi fi τ f ,τ f . i1 i1 3.21

By applying the Bochner’s technique from 3.19 and the assumption that the compact Ω 2 supports of ∂f/∂xi and Dei ∂f/∂xi are contained in the interior of , we know that τ f is constant, that is, dτf0. If we use the identity 2 div df , τ f dz τ f df , d τ f dz, z t, x 3.22 Ω Ω 10 International Journal of Mathematics and Mathematical Sciences and the fact dτf0, then we can conclude that τf0 by applying the divergence theorem. m 2 2 Corollary 3.7. If f : Ω → N is a biwave map on a compact domain such that |τf| ≥|τf| i 1 xi t β γ m β γ μ α − ≤ and Rβγμ ft ft i1 fi fi τ f 0,thenf 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 Ω ∈ Γ −1 | interior of . Let V f TN be a vector ﬁeld such that ∂f/∂t t0 V .Ifweapplythe 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

2 2 1 d | Δ N E2 f t0 V R df ei ,V df ei dz 2 dt2 Ω N

N N 2R df ei ,V Dei τ f 2R df ei ,τ f Dei V>dz,

where z t, x ∈ R × Rm,D is the Riemannian connection on TN, and V is the vector ﬁeld along f.

m,1 → 2 2 | ≥ Deﬁnition 4.2. Let f : R N be a biwave map. If d /dt E2 f t0 0, then f is a stable biwave map. If we consider a wave map, that is, τf0 as a biwave map, then by 4.1 we have 2 2 | ≥ d /dt E2 f t0 0andf 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 Sfefg − f h, where ef −10 1/2|df |2 is the energy function and g . The map f satisfies the conservation law if 0 I div Sf0.

Proposition 4.4. Let f : Rm,1 → N, h be a smooth map from a Minkowski space into a Riemannian manifold N, h. Then

m,1 div S f X − τ f ,dfX ,X∈ R . 4.2 International Journal of Mathematics and Mathematical Sciences 11

0 1 m m,1 Proof. Let x t, x ,...,x be the coordinates of a point in R ,e0 ∂/∂t, e1 1, 0,...,0 , −10 × ..., em 0, 0,...,1 and g , where I is an m m matrix. We calculate 0 I

−10 1 2 − ∗ div S f X Dei S f ei,X Dei df f h ei,X 2 0 I −10 1| |2 − ∗ Dei df ei,X Dei f h ei,X 2 0 I

− ∂f ∂f − ∂f ∂f − D , 1 e0,X D , I ei,X Dei f∗ei,f∗X ∂t ∂t ∂xi ∂xi

∂f ∂f ∂f ∂f − − D , e0,X D , ei,X Dei f∗ei,f∗X f∗ei,Dei f∗X ∂t ∂t ∂xi ∂xi

− − DXdf ei,f∗ei τ f ,f∗X f∗ei,Dei f∗X , 4.3

where the ﬁrst term and the third term are canceled out and Dei f∗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

2 2 1 d Δ N E2 ft V R df ei ,V df ei dz 2 dt2 Ω t0 N N 4.4 V, R τ f ,V τ f 2R df ei ,V Dei τ f Ω N 2R df ei ,τ f Dei V dz.

In particular, let V τf. Recalling that f is a biwave map and N has constant sectional curvature K>0, 4.4 can be reduced to

2 1 d N E2 f 4 R df ei ,τ f Dei τ f ,τ f dz 2 dt2 Ω t0 4.5 2 4K df ei ,Dei τ f τ f Ω − df ei ,τ f τ f ,Dei τ f dz. 12 International Journal of Mathematics and Mathematical Sciences

Since f satisﬁes the conservation law, by Deﬁnition 4.3,Proposition4.4,and4.2 we have df ei,τ f 0, 4.6 − − 2 df ei ,Dei τ f Dei df ei ,τ f τ f .

Substituting 4.6 into 4.5 and applying the stability of f,weget 2 1 d − 4 ≥ E2 ft 4K τf dz 0, 4.7 2 dt2 Ω t0 which implies that τf0, 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 11, then we have the following.

Lemma 4.6. Ω → n1 Let f : S 1 be a biwave map. The Hessian of the bi-energy functional E2 of f is

H E2 f X, Y If X ,Y dz, 4.8 Ω where Δf Δf Δf · ·− 2 If X X trace X, df df df X 2 dτ f ,df X 2 τ f X − 2 trace X, dτ f · df − 2 trace τ f ,dX· df · 4.9 f − τ f ,X τ f trace df ·, Δ X df · trace df , trace X, df· df · df · 2 2 f 4 − 2 df trace df ·,X df · 2 dX, df τ f − df Δ X df X,

− for X, Y ∈ Γf 1TSn 11. √ Theorem 4.7. Let√ h : Ω → Sn1/ 2 be a wave map on a compact domain with constant energy and let i : Sn1/ 2 → Sn11 be an inclusion map. Then f i ◦ h : Ω → Sn11 is an unstable biwave map.

Proof. We have the following identities from Theorem 3.5: 2 2 df 2eh, trace ζ, df· df · 0, dτ f ,df ζ −4eh ζ, 2 2 τf 4eh , trace ζ, dτ f df · 0, trace τ f ,dζ· df 0, 4.10 T 2 f f τ f ,ζ τ f 4eh ζ, trace df , Δ ζ df · Δ ζ ,

2 dζ, df τ f −4eh ζ. International Journal of Mathematics and Mathematical Sciences 13

Then we obtain the following formula from Lemma 4.6 and the previous identities: 2 Δf − 2 − Δf If ζ,ζ ζ 12e h 4eh ζ, ζ dz, 4.11 Ω which is strictly negative, where Δf ζ 2ehζ. Hence, f is an unstable biwave map.

Acknowledgment

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

References

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