Biwave Maps into Manifolds

International Journal of Mathematics and Mathematical Sciences Volume 2009, Article ID 104274,14pages

doi:10.1155/2009/104274

Research Article

Biwave Maps into Manifolds

Yuan-Jen Chiang

Department of Mathematics, University of Mary Washington, Fredericksburg, VA 22401, 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 iffis a biwave map into a Riemannian manifold under certain circumstance, thenfis a wave map. We verify that iffis a stable biwave map into a Riemannian manifold with positive constant sectional curvature satisfying the conservation law, thenf is a wave map. We finally obtain a theorem involving an unstable biwave map.

Copyrightq2009 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 1in 1964. Afterwards, there were two reports on harmonic maps by Eells and Lemaire 2,3in 1978 and 1988. Biharmonic maps, which generalized harmonic maps, were first studied by Jiang4,5in 1986. In this decade, there has been progress in biharmonic maps made by Caddeo et al. 6, 7, Loubeau and Oniciuc 8, Montaldo and Oniciuc9, Chiang and Wolak10, Chiang and Sun11,12, 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 Machedon16,17, Shatah and Struwe

18,19, Tao 20,21, Tataru 22,23, and so forth. Furthermore, Nahmod et al. 24also studied wave maps fromR×Rm_{intocompactLie groups or Riemannian symmetric spaces,}

that is, gauged wave maps whenm ≥ 4, and established global existence and uniqueness, provided that the initial data are small. Moreover, Chiang and Yang25, Chiang and Wolak

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 projectionπ, and letGbe the Lie algebra ofG. A connectionAcan be considered as aG-valued 1-formAAμxdxμlocally. The curvature of the connectionAis given by the

2-formF Fμνdxμdxνwith

Fμν∂μAν−∂νAμ

Aμ, Aν

. 1.1

The bi-Yang-Mills Lagrangian is defined

L2A

1 2

M

δF2dvM, 1.2

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 of1.2has the form

δdFδF0, 1.3

which is the bi-Yang-Mills system. In particular, lettingM R×R2 _{andG SO2, the}

group of orthogonal transformations onR2_{,we have thatA}

μxis a 2×2 skew symmetric

matrixAijμ.The appropriate equivariant ansatz has the form

Aijμx

δi_μxj−δjμxi

ht,|x|, 1.4

whereh:M → Ris a spatially radial function. Settingur2_{handr |x|,the bi-Yang-Mills}

system1.3becomes the following equation forut, r:

utttt−urrrr−

3

rurrr

2

r2urr−

2

r3ur kt, r, 1.5

which is a linear nonhomogeneous biwave equation, wherekt, ris a function oftandr. 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 groupsor Riemannian symmetric spaces, that is, gauged biwave maps for future exploration.

In Section2, 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 Section3, we prove in Theorem3.3that iff : Rm,1 _{→ N}

1is a biwave map

apply this theorem to provide many biwave mapssee Example3.4. We also can construct biwave nonwave maps as follow: Leth: Ω ⊂ Rm,1 _{→ S}n_1/√_{2be a wave map on a compact}

domain and leti:Sn_1/√_{2 → S}n1_{1be an inclusion map. The mapf i◦h:Ω → S}n1_1is

a biwave nonwave map if and only ifhhas constant energy density, compare with Theorem3.5. Afterwards, we show that iff:Ω → Nis a biwave map on a compact domain into a Riemannian manifold satisfying

−|τf|2t m

i1

|τf|2xi−R_βγμα

−f_tβf_tγ

m

i1

f_iβf_iγ τfμ ≥0, 1.6

thenf is a wave mapcf. Theorem3.6. This theorem is different than the theorem obtained by Jiang4: iffis a biharmonic map from a compact manifold into a Riemannian manifold with nonpositive curvature, thenfis a harmonic map. In Section4, we verify that iffis a stable biwave map into a Riemannian manifold with positive constant sectional curvature satisfying the conservation law, thenfis a wave mapcf. Theorem4.5. We also prove that ifh:Ω → Sn1/√2is a wave map on a compact domain with constant energy density, thenf i◦h:Ω → Sn1_{1is an unstable}

biwave mapcf. Theorem4.7.

2. Biharmonic Maps

A biharmonic mapf :Mm_{, g}

ij → Nn, hαβfrom anm-dimensional Riemannian manifold

M into an n-dimensional Riemannian manifold N is the critical point of the bi-energy functional

E2

f 1

2

M

dd∗2f2dv 1

2

M

d∗df2dv 1

2

M

τf2dv, 2.1

wheredvis the volume form onM.

Notations

d∗ is the adjoint of d and τf traceDdf Ddfei, ei Deidfei is the tension field. Here D is the Riemannian connection onT∗M⊗f−1TN induced by the Levi-Civita connections onMandN, and{ei}is the local frame at a point ofM. The tension field has

components

τfαgijf_iα_|jgijf_ijα−Γ_ijkf_kα Γ_βγαf_iβf_jγ, 2.2a

whereΓk ijandΓ

γ

αβare the Christoffel symbols onMandN, 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, Nfrom a compact manifoldM

without boundaryinto a Riemannian manifoldN. Hereftxis the endpoint of a segment

starting atfxf0x, determined in length and direction by the vector field ˙fxalong

in the interior of M we need this assumption when we compute τf by applying the divergence theorem. Then we have

d

dtE2

ft

|t0E˙2

f

M

Dtτf, τft0dv. 2.3

Letξ∂ft/∂t.The components ofDtτfaref_iα_|j|t ∂f_iα_|j/∂t Γμγαf μ

i|jξγ.We can use the

curvature formula onM×I → Nand get

f_iα_|j|tf_iα_|t|jR_βγμα f_iβf_jγξμ, 2.4

whereRis the Riemannian curvature ofN. Butfα

i|t ftα|i ξ|αi,therefore,Dtτf has

compo-nentsξα_|i|jR_βγμα f_iβf_jγξμ_{.We can rewrite2.3as}

d dtE2

ft

|t0

M

Jf

τf, τfdv, 2.5

where

J_fαξ gijξα_|i|jgijR_βγμα f_iβf_jγξμ ΔξαRαdf, dfξ 2.6

is a linear equation forξτf,andΔξ D∗Dξis an operator fromf−1_TNtof−1_TN.

Solutions ofJfξ 0 are called Jacobi fields. Hence, we obtain the following definition from

2.3,2.5, and2.6.

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

τ2fαJfτfα ΔτfαRα

df, dfτf

gijf_ijα−Γk_ijf_kα Γ_βγαf_iβf_jγgijR_βγμα f_iβf_jγτfμ0,

2.7

that is, the tension fieldτf, is a Jacobi field.

Ifτf 0, thenτ2f 0. Thus, harmonic maps are obviously biharmonic.

Bihar-monic maps satisfy the fourth-order elliptic systems of PDEs, which generalize harBihar-monic maps. Our computation for the first variation of the bi-energy functional presented here using tensor technique is different but much easier than Jiang’s4original computationit took him four pages.

Caddeo et al.7showed that a biharmonic curve on a surface of nonpositive Gaussian curvature is a geodesici.e., is harmonicand gave examples of biharmonic nonharmonic curves on spheres, ellipses, unduloids, and nodoids.

Theorem 2.2see4. Letf : Mm _{→ S}m1_{1be an isometric embedding of anm-dimensional}

compact Riemannian manifold Minto an m1-dimensional unit sphereSm1_{1with nonzero}

Example 2.3. InSm1_{1,the compact hypersurfaces, whose Gauss maps are isometric}

embed-dings, are the Clifford surfaces28:

Mm_k1 Sk

1

√

2

×Sm−k

1

√

2

, 0≤k≤m. 2.8

Letf : Mm

k1 → Sm11be a standard embedding such thatk /m/2.BecauseBf 2

km−k mandτf k−m−k 2k−m /0, f is a biharmonic nonharmonic map by Theorem2.2.

3. Biwave Maps

LetRm,1_{be anm1 dimensional Minkowski spaceR×R}m_{with the metricg}

ij −1,1, . . . ,1

and the coordinatesx0 _{t, x}1_{, x}2_{, . . . , x}m _{and letN, h}

αβbe ann-dimensional Riemannian

manifold. A wave map is a harmonic map on the Minkowski spaceRm,1 _{with the energy}

functional

Ef 1

2

Rm,1

−ft2∇xf2

dt dx 1

2

Rm,1

hαβ

−f_tαf_tβ

m

i1

f_iαf_iβ dt dxi. 3.1

The Euler-Lagrange equation describing the critical point of3.1is

τα

ffα _Γα

βγ

−f_tβf_tγ

m

i1

f_iβf_iγ 0, 3.2

where−∂2_/∂t2 _Δ

xis the wave operator onRm,1andΓ_βγα are the Christoffel symbols of

N.f is a wave map iffthe wave fieldταf 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 dimensionm2.

Iff : Rm,1 → N is a smooth map from a Minkowski spaceRm,1 into a Riemannian manifoldN, then the bi-energy functional is, from2.1,

E2

f 1

2

Rm,1

dd∗2f2dt dx

1

2

Rm,1

d∗df2dt dx 1

2

Rm,1

τf2dt dx.

3.3

The Euler-Lagrange equation describing the critical point of3.3, from2.5, is

τ2fJf

Definition 3.1. f:Rm,1 _{→ Nfrom a Minkowski space into a Riemannian manifold is a biwave}

map if and only if the biwave fieldi.e., the bitension field on a Minkowski space,

τ2fαJfτfα ΔτfαRαdf, dfτf

τfα Γα μγ

−τfμ_tτfγ_t

m

i1

τfμ_iτfγ_i

R_βγμα

−f_tβf_tγ

m

i1

f_iβf_iγ τfμ0,

3.5

that is, the wave fieldτf, is a Jacobi field on the Minkowski space.

Biwave maps satisfy the fourth-order hyperbolic systems of PDEs, which generalize wave maps. Ifτf 0, thenτ2f 0. Waves maps are obviously biwave maps, but

biwave maps are not necessarily wave maps.

Example 3.2. Letu : Rm,1 _{→ R be a function defined on a Minkowski space satisfying the}

following conditions:

2_{ut, x u u}

tttt−2uttxxuxxxx0, t, x∈0,∞×Rm,

uu0, utu1, uu0, ∂

∂tu

∂u

∂t u1, t, x∈ {t0} ×R

m_, 3.6

where the initial datau0 andu1 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.

Letf :Rm,1 _{→ N}

1be a smooth map from a Minkowski spaceRm,1into a Riemannian

manifoldN1and letf1 :N1 → N2be a smooth map between two Riemannian manifoldsN1

andN2. Then the compositionf1◦f : Rm,1 → N2 is a smooth map. SinceRm,1 is a

semi-Riemannian manifold i.e., a pseudo-Riemannian manifold, we can define a Levi-Civita connection onRm,1 _{by O’Neill 30. LetD,D, D, D, D}”_{, D, D, D}” _{be the connections}

onRm,1_{, TN}

1, f−1N1, f₁−1TN2, f1◦f−1TN2, T∗Rm,1⊗f−1TN1, T∗N1⊗f₁−1TN2, T∗Rm,1⊗

f1◦f−1TN2, respectively, and letRN2,, Rf

−1

1 TN2_{, be the curvatures onTN}

2, f1−1TN2,

respectively. We first have the following two formulas:

D”_Xdf1◦f

Y D_dfXdf1

dfY df1◦DXdfY, 3.7a

forX, Y∈Rm,1_,and

RN2_df

1

X, df1

Ydf1

ZRf1−1TN2_{X, Ydf}

1

Z, 3.7b

forX, Y, Z∈ΓTN1.

Theorem 3.3. Iff:Rm,1 _{→ N}

1is a biwave map andf1:N1 → N2is totally geodesic between two

Proof. Letx0 _{t, x}1_{, . . . , x}m _{be the coordinates of a pointpinR}m,1 _{and lete}

0 ∂/∂t, e1

1,0, . . . ,0, e2 0,1,0, . . . ,0, . . . , em 0, . . . ,0,1be the frame atp. We know from4

thatD”∗D”D_e”_kD”_ek −D”_D

ekek.Sincef1is totally geodesic, we haveτf1◦f df1◦τf by applying the chain rule of the wave field tof1◦fas1. Then we get

D”∗D”_τf1◦f

D”∗D”df1◦τf

D”_ekD”_ekdf1◦τf−D”D_ekek

df1◦τ f.

3.8

Recalling thatτf Dejdfej,we derive from3.7athat

D”_ekdf1◦τ

fD”_ekdf1◦Dejdf

ej D

D_ejdfekdf1

Dejdf

ej

df1◦Dek

Dejdf

ej

df1◦Dekτ

f,

3.9

sincef1is totally geodesic. Therefore, we have

D”_ekD”_ekdf1◦τfD”ek

df1◦Dekτ

fdf1◦DekDekτ

f,

D”_D

ekek

df1◦τ

fdf1◦DD_ekekτ

f.

3.10

Substituting3.10into3.8, we arrive at

D”∗D”τf1◦f

df1◦D

∗

Dτf, 3.11

whereD∗DDekDek−DDekek.

On the other hand, we have by3.7b

RN2_df

1◦f

ei, τf1◦f

df1◦f

ei

Rf1−1TN2_dfe

i, τfdf1

dfei

df1◦RN1

dfei, τfdfei.

3.12

We obtain from3.11and3.12

D”∗D”f1◦f

RN2_df

1◦f

ei, τf1◦f

df1◦f

ei

df1◦

D∗DτfRN1_dfe

i, τ

fdfei

,

3.13

that is,τ2f1◦f df1◦τ2f.Hence, iff is a biwave map andf1is totally geodesic,

thenf1◦f is a biwave map. Note that the total geodesicity off1cannot be weakened into a

Example 3.4. LetN1be a submanifold ofN. Are the biwave maps intoN1also biwave maps

intoN? The answer is affirmative iffN1 is a totally geodesic submanifold ofN, that is,N1

geodesics areNgeodesics.N1is a geodesicγt γ1, . . . , γn:R → N⊂Rnwith|γ˙t|1

iffγ˙ is parallel, that is,D∂/∂tγ˙ 0 iffγ¨ ⊥ TγN.For a mapu :Rm,1 → R,lettingf γ◦u

f1, . . . , fn:Rm,1 → N⊂Rn,we have by3.13the following:

τ2

fdγ◦τ2u dγ◦2u, 3.14

sinceγis a geodesic. Hence,fγ◦uis a biwave map if and only ifusolves the fourth-order homogeneous linear biwave equation2_{u0 as in Example3.2. It follows from Theorem3.3}

that there are many biwave mapsf:Rm,1 _{→ Nprovided by geodesics ofN.}

We also can construct examples of biwave nonwave maps from some wave maps with constant energy using Theorem3.5. Let

Sn

1

√

2

Sn

1

√

2

×

1

√

2

x1, x2, . . . , xn1,

1

√

2

|x2

1· · ·x2n1

1 2

3.15

be a hypersphere of Sn1_{1. ThenS}n_1/√_{2 is a biharmonic nonminimal submanifold of}

Sn1_{1by Theorem2.2and Example2.3. Letζ x}

1, . . . , xn1,−1/

√

2be a unit section of the normal bundle ofSn_1/√_2inSn1_{1.Then the second fundamental form of the inclusion}

i:Sn_1/√_{2 → S}n1_{1isBX, Y DdiX, Y −X, Yζ.By computation, the tension field}

ofiisτi −nζ, and the bitension field isτ2i 0.

Theorem 3.5. Leth:Ω → Sn_1/√_{2be a nonconstant wave map on a compact space-time domain}

Ω⊂Rm,1_{and leti:S}n_1/√_{2 → S}n1_{1be an inclusion. The mapf i◦h:R}m,1 _{→ S}n1_{1is a}

biwave nonwave map if and only ifhhas constant energy densityeh 1/2|dh|2_.

Proof. Letx0 _{t, x}1_{, . . . , x}m_{be the coordinate of a pointpinΩ⊂R}m,1_{and lete}

0 ∂/∂t, e1

1,0, . . . ,0, e2 0,1,0, . . . ,0, . . . , em 0, . . . ,0,1 be the frame atp. Recall thatζis the

unit section of the normal bundle. By applying the chain rule of the wave field tof i◦h,

we have

τfdiτh trace Ddidh, dh −2ehζ, 3.16

since h is a wave map. We can derive the following at the point p by straightforward calculation:

D∗Dτf−DfeiD

f eiτ

f−DefiD

f

ei−2ehζ

2eieiehζ−2ehdhei, dheiζ4dfeiehei

2ehDdhei, ei,

RSn1dfei, τfdfei −dhei, dheiτ

f2dhei, dheiehζ.

Therefore, we obtain

τ2f−2Δehζ4dfgradeh. 3.18

Suppose that f i◦h : Ω → Sn_1/√_{2× {1/}√_{2} → S}n1_{1is a biwave nonwave map}

τf/0. As theζ-part ofτ2f,Δehvanishes, which implies thatehis constant since Ωis compact. The converse is obvious.

Letx0 _{t, x}1_{, . . . , x}m _{be the coordinates of a point in a compact space-time domain}

Ω ⊂ Rm,1 _{and e}

0 ∂/∂t, e1 1,0, . . . ,0, e2 0,1,0, . . . ,0, . . . , em 0, . . . ,0,1be the

frame at the point. Suppose thatf : Ω → N is a biwave map from a compact domain Ω into a Riemannian manifoldNsuch that the compact supports of∂f/∂xiandDei∂f/∂xiare contained in the interior ofΩ.

Theorem 3.6. Iff :Ω → Nis a biwave map from a compact domain into a Riemannian manifold such that

−τf2_t

m

i1

τf2_xi−R_βγμα

−f_tβf_tγ

m

i1

f_iβf_iγ τfμ ≥0, 3.19

thenfis a wave map.

Proof. Sincefis a biwave map, we have by3.4

τ2f ΔτfRdf, dfτf. 3.20

Recall thatx0 _{t, x}1_{, . . . , x}m_{are the coordinates of a point inΩ ⊂ R}m,1 _ande

0 ∂/∂t, e1

1,0, . . . ,0, e2 0,1,0, . . . ,0, . . . , em 0, . . . ,0,1.We compute

1

2Δτf

2

Deiτ

f, Deiτ

fD∗Dτf, τf

m

i0

Deiτ

f, Deiτ

f−

R_βγμα

−f_tβf_tγ

m

i1

f_iβf_iγ τfμ, τf

−τf2t m

i1

_τf2 xi−

R_βγμα

−f_tβf_tγ

m

i1

f_iβf_iγ τfμ, τf .

3.21

By applying the Bochner’s technique from 3.19 and the assumption that the compact supports of∂f/∂xiandDei∂f/∂xiare contained in the interior ofΩ,we know thatτf

2

is constant, that is,dτf 0. If we use the identity

Ωdiv

df, τfdz

Ω

and the factdτf 0,then we can conclude that τf 0 by applying the divergence

theorem.

Corollary 3.7. Iff :Ω → Nis a biwave map on a compact domain such thatm_i1|τf|2_xi ≥ |τf|2_t

andR_βγμα −f_tβf_tγm_i1f_iβf_iγτfμ≤0, thenfis a wave map.

Proof. The result follows from3.19immediately.

4. Stability of Biwave Maps

Letx0 t, x1, . . . , xmbe the coordinates of a point in a compact space-time domainΩ⊂Rm,1

and lete0 ∂/∂t, e1 1,0, . . . ,0, . . . , em 0,0, . . . ,1be the frame at the point. Suppose

thatf : Ω → Nis a biwave map from a compact space-time domainΩinto a Riemannian manifoldN such that the compact supports of ∂f/∂xi and Dei∂f/∂xi are contained in the interior ofΩ.Let V ∈ Γf−1_{TNbe a vector field such that∂f/∂t|}

t0 V. If we apply the

second variation of a biharmonic map in4to a biwave map, we can have the following.

Lemma 4.1. Iff : Ω → N is a biwave map from a compact domain into a Riemannian manifold, then

1 2

d2

dt2E2

f|t0

Ω

ΔVRNdfei, Vdfei 2

dz

Ω< V,

D_{df e}

iR

N_fe

i, τfV

D_τ

fR

N_dfe i, V

dfei RN

τf, Vτf

2RNdfei, V

Deiτ

f2RNdfei, τfDeiV > dz,

4.1

wherez t, x∈R×Rm_{, Dis the Riemannian connection onTN,andV is the vector field alongf.}

Definition 4.2. Letf :Rm,1 _{→ Nbe a biwave map. Ifd}2_/dt2_E

2f|t0≥0,thenfis a stable

biwave map.

If we consider a wave map, that is,τf 0 as a biwave map, then by4.1we have d2_/dt2_E

2f|t0≥0 andfis automatically stable.

Definition 4.3. Let f : Rm,1 _{→ N, h be a smooth map from a Minkowski space into a}

Riemannian manifoldN, h.The stress energy is defined bySf efg−f∗h,whereef

1/2|df|2 _{is the energy function andg} −1 0 0 I

.The mapfsatisfies the conservation law if divSf 0.

Proposition 4.4. Letf:Rm,1 _{→ N, hbe a smooth map from a Minkowski space into a Riemannian}

manifoldN, h.Then

Proof. Letx0 _{t, x}1_{, . . . , x}m_{be the coordinates of a point inR}m,1_{, e}

0 ∂/∂t, e1 1,0, . . . ,0,

. . . , em 0,0, . . . ,1andg

_{−1 0}

0 I

,whereIis anm×mmatrix. We calculate

divSfX DeiS

fei, X Dei

1 2df

2

−1 0 0 I −f

∗_{h e}

i, X

Dei

1 2|df|

2

−1 0 0 I

ei, X−

Deif

∗_he

i, X

− D∂f ∂t, ∂f ∂t −1

e0, X

D∂f ∂xi , ∂f ∂xi

Iei, 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

−τf, f∗X−f∗ei, Deif∗X

,

4.3

where the first term and the third term are canceled out andDeif∗eiτf.

Theorem 4.5. LetΩ ⊂ Rm,1 _{be a compact domain and letN, hbe a Riemannian manifold with}

constant sectional curvatureK >0.Iff :Ω → Nis a stable biwave map satisfying the conservation law, thenfis a wave map.

Proof. BecauseN has constant sectional curvature, the second term of4.1disappears and

4.1becomes

1 2

d2

dt2E2

ft

t0

Ω

ΔVRNdfei, Vdfei 2 dz Ω

V, RNτf, Vτf2RNdfei, V

Deiτ

f

2RNdfei, τfDeiV

dz.

4.4

In particular, letV τf.Recalling thatf is a biwave map andNhas constant sectional

curvatureK >0,4.4can be reduced to

1 2

d2

dt2E2

f

t0

4

Ω

RNdfei, τfDeiτ

f, τfdz

4K

Ω

dfei, Deiτ

fτf2

−dfei, τfτf, Deiτ

fdz.

Sincefsatisfies the conservation law, by Definition4.3, Proposition4.4, and4.2we have

dfei, τf0,

dfei, Deiτ

f−Deidfei, τ

f−τf2.

4.6

Substituting4.6into4.5and applying the stability off, we get

1 2

d2

dt2E2

ft

t0

−4K

Ω

_τf4

dz≥0, 4.7

which implies thatτf 0,that is,f:Ω → Nis a wave map.

If we apply the Hessian of the bi-energy of a biharmonic map4to a biwave map

f:Ω → Sn1_{1, then we have the following.}

Lemma 4.6. Letf:Ω → Sn1_{1be a biwave map. The Hessian of the bi-energy functionalE} 2off

is

HE2fX, Y

Ω

IfX, Y

dz, 4.8

where

IfX Δf

Δf_{X Δ}f_{traceX, df·df· −df}2

X2dτf, dfX

τf2X−2 traceX, dτf·df−2 traceτf, dX·df·

−τf, Xτftracedf·,ΔfXdf·tracedf,traceX, df·df·df·

−2df2 tracedf·, Xdf·2dX, dfτf−df2Δf_Xdf4_X,

4.9

forX, Y ∈Γf−1TSn11.

Theorem 4.7. Leth:Ω → Sn_1/√_{2be a wave map on a compact domain with constant energy}

and leti:Sn_1/√_{2 → S}n1_{1be an inclusion map. Thenf i◦h:Ω → S}n1_{1is an unstable}

biwave map.

Proof. We have the following identities from Theorem3.5:

df22eh, traceζ, df·df·0, dτf, dfζ−4eh2ζ,

τf24eh2, traceζ, dτfdf·0, traceτf, dζ·df0,

τf, ζτf4eh2ζ, tracedf,Δf_ζdf·Δf_ζT_,

dζ, dfτf−4eh2ζ.

Then we obtain the following formula from Lemma4.6and the previous identities:

Ifζ, ζ

Ω

Δf_ζ2_−12eh2_−4ehΔf_{ζ, ζdz, 4.11}

which is strictly negative, whereΔf_{ζ2ehζ. Hence,fis an unstable biwave map.}

Acknowledgment

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

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