In this paper, we define anF-Yang-Mills functional, and hence F-Yang-Mills fields

ARCHIVUM MATHEMATICUM (BRNO) Tomus 49 (2013), 125–139

STABILITIES OF F-YANG-MILLS FIELDS ON SUBMANIFOLDS

Gao-Yang Jia and Zhen-Rong Zhou

Abstract. In this paper, we define anF-Yang-Mills functional, and hence F-Yang-Mills fields. The first and the second variational formulas are calcu- lated, and the stabilities ofF-Yang-Mills fields on some submanifolds of the Euclidean spaces and the spheres are investigated, and hence the theories of Yang-Mills fields are generalized in this paper.

1. Introduction

Let P(M, G) be a principal bundle over a compact Riemannian manifold M with structure groupG (a Lie group), and letE =P×ρV be a vector bundle associated with P(M, G), whose standard fibre is some vector space V, where ρ:G→ GL(V) is a representation ofG. Denote the space of E-valuedp-forms by Ω^p(E) = Γ(∧^pT^∗M ⊗E), and the space of connections of E by C_E. Let g_E=P×_AdGgbe the adjoint vector bundle wheregis the Lie algebra ofG. It is known that, for any ∇,∇⁰∈ C_E, we have∇ − ∇⁰∈Ω¹(g_E). For each∇ ∈ C_E, the curvature 2-formR^∇∈Ω²(gE) is defined byR^∇_X,Y = [∇X,∇Y]− ∇_[X,Y]. IfGis a semisimple Lie group, there is a natural invariant metric ongE which is defined by the Killing form, and this metric induces a one on Ω²(gE). With respect to this induced metric, the Yang-Mills functional is defined as follows:

(1) S(∇) = 1

2 Z

M

kR^∇k².

If a connection∇ofE is a critical point of the Yang-Mills functional, we call it a Yang-Mills connection, the associated curvature tensor is called a Yang-Mills field.

For a connection∇, its variation is a family∇^tof connections with|t|< ε(a small positive number) and∇⁰=∇. If

(2) d²

dt² _t=0

S(∇^t)≥0

holds for any variations of a Yang-Mills connection ∇, then we call the Yang-Mills connection (and the corresponding Yang-Mills field) to be stable. Otherwise, we call it unstable.

2010Mathematics Subject Classification: primary 58E20.

Key words and phrases:F-Yang-Mills field, stability.

Research supported by National Science Fundation of China No.10871149.

Received August 12, 2011, revised June 2013. Editor P. W. Michor.

DOI: 10.5817/AM2013-2-125

In the paper [2, 1], J. P. Bourguignon and H. B. Lawson obtained a well known result on stabilities of Yang-Mills fields as follows:

Theorem 1([1]). Forn >4, any nonzero Yang-Mills fields on Sⁿ are unstable.

Whenn= 4, we haveS(∇)≥4π²|p1(E)|for any connection∇(where,p₁(E) is the pontryagin number ofE, a topological invariant), and the equality holds if and only if the connection ∇is self-dual or anti-self-dual (in this case, the connection is called an instanton). Hence any self-dual or anti-self-dual connection is stable.

Conversely, any stable Yang-Mills connection (or field) onS⁴withG= SU₂,SU₃,U₂ is either self-dual or anti-self-dual (see [1]). On the other hand, an infinite number of unstable Yang-Mills fields onS² withG=SU(2) are constructed by L. M. Sibner, R. J. Sibner and K. Uhlenbeck in [4].

Y. L. Xin in [5] discussed the stabilities of Yang-Mills fields on submanifolds of the Euclidean space, and obtained the following

Theorem 2 ([5]). Let Mⁿ be a compact submanifold of R^n+k, and satisfy the following condition:

(3) 2h^µ_tih^µ_tjδkl−h^µ_tth^µ_ijδkl+ 2h^µ_ijh^µ_kl ≤bδijδkl,

where h^µ_ij is the second fundamental tensor with respect to a local orthonormal frame of M,1≤i, j, k, l≤n, n+ 1≤µ≤n+k, andb <0. Then any nonzero Yang-Mills fields onM are unstable.

OnSⁿ⊆Rⁿ⁺¹, we can choose a local orthonormal field of frame ofRⁿ⁺¹, such thathⁿ⁺¹_ij =δij. Then the condition in Theorem 2 becomes asn >4. Therefore, Theorem 2 is a generalization of Theorem 1.

Remark 3. The condition (3) means that for any tensorAij, we have 2h^µ_tih^µ_tjδ_kl−h^µ_tth^µ_ijδ_kl+ 2h^µ_ijh^µ_kl

A_ikA_jl≤bδ_ijδ_klA_ikA_jl.

If the integrand of the Yang-Mills functional is replaced by kR^∇k^p, then we can obtain a p-Yang-Mills functional, whose critical points are calledp-Yang-Mills connections, and the associated curvature tensors are called p-Yang-Mills fields.

The paper [3] investigated the stabilities ofp-Yang-Mills fields of Euclidean and sphere submanifolds, and generalized the related results of [1] and [5].

Let Mⁿ be a submanifold ofR^n+k orS^n+k, andh(·,·) the second fundamental form. Let 1 ≤ i, j ≤ n; n+ 1 ≤ µ ≤n+k. Choose a local orthonormal frame {e_i|i= 1,· · ·, n+k} ofR^n+k orS^n+k, such that, restrict toMⁿ,{e₁,· · ·, e_n}are tangent toM and{e_µ|µ=n+ 1,· · ·, n+k}are normal toM. Seth(e_i, e_j) =h^µ_ije_µ andH^µ=P

h^µ_ii. Define

C_ijklsr≡ −H^µh^µ_jl+ 2h^µ_jmh^µ_ml

δ_kiδ_sr+ 2h^µ_ikh^µ_jlδ_sr+ 2(p−2)h^µ_ikh^µ_srδ_jl. For example, if Mⁿ = Sⁿ, as a hypersurface of Rⁿ⁺¹, then we can choose an adapted local normal frame such that hij =hⁿ⁺¹_ij =δij. In this case, Cijklsr = (2p−n)δ_jlδ_kiδ_sr.

The paper [3] proved the following theorems:

Theorem 4([3]). LetMⁿbe a submanifold ofR^n+k, satisfyingC_ijklsr ≤bδ_ikδ_jlδ_sr, whereb <0. Then any nonzero p-Yang-Mills fields on M are unstable.

Theorem 5 ([3]). Let Mⁿ be a submanifold of S^n+k, satisfying C_ijklsr <(n− 2p)δ_ikδ_jlδ_sr. Then any nonzerop-Yang-Mills fields onM are unstable.

When p = 2, the condition in Theorem 4 is the same as that in Theorem 2.

So, Theorem 4 is a generalization of Theorem 2. If we consider Sⁿ as a totally geodesic submanifold ofS^n+p, then the condition in Theorem 5 isn >2p. Therefore Theorem 5 is another generalization of Theorem 1.

Remark 6. InequalityCijklsr≤(or <)aδikδjlδsr means that XC_ijklsrA_ijA_klB_stB_rt≤(or <)aX

δ_ikδ_jlδ_srA_ijA_klB_stB_rt for any tensorAij andBij.

Replacing the integrand of the Yang-Mills functional by F ^kR∇₂^k2

, where F is a non-negative function, we define an F-Yang-Mills functional, and hence F-Yang-Mills fields. These generalize theories ofp-Yang-Mills fields. In this paper, we investigate the stabilities ofF-Yang-Mills fields on submanifolds of the Euclidean space and the spheres, and our main results are in the following:

Theorem 7. LetMⁿ be a submanifold of R^n+k, which satisfies

(4) Cijklsr ≤bδikδjl,

whereb <0. Suppose that for t >0, we have

(5) (p−2)F⁰(t)≥2tF⁰⁰(t), F⁰(t)>0, F(t)>0. Then any nonzero F-Yang-Mills fieldR^∇ onMⁿ is unstable.

Theorem 8. LetMⁿ be a submanifold of S^n+k, which satisfies

(6) Cijklsr <(n−2p)δikδjl.

Suppose that fort >0, we have

(7) (p−2)F⁰(t)≥2tF⁰⁰(t), F⁰(t)>0, F(t)>0. Then any nonzero F-Yang-Mills fieldR^∇ onMⁿ is unstable.

Theorem 7 generalizes Theorem 4 and Theorem 8 generalizes Theorem 5.

Remark 9.

(1) The condition (p−2)F⁰(t)≥2tF⁰⁰(t) is equivalent to ^F0(t)

t

p−2 2

⁰

≤0, i.e. ^F0(t)

t

p−2 2

is differential and non-increasing.

(2) Forp≥2, the following functions satisfy the condition 7: ¹_p(2t)^p2, ln(1 +t^p2), ln(t^p2 +√

1 +t^p), ^t

p

√ 2

1+t^p, arctan(t^p2),Rt^p2

0 e^−x2dx, etc.

(3) In general, iff: [0,∞)→(0,∞) is differential and non-increasing,F(t) = Rt^p2

0 f(x)dx, then ^F0(t)

t^p−22

is differential and non-increasing for p ≥ 2, and hence condition (7) is satisfied by such an F.

2. Variational formulas of F-Yang-Mills fields

Definition 10. LetF: [0,+∞)→[0,+∞) be aC^∞function. DefineSF:CE→R as following: For any∇ ∈ C_E, set

(8) SF(∇) =

Z

M

FkR^∇k² 2

,

which is called an F-Yang-Mills functional. The critical points of S_F are cal- led F-Yang-Mills connections, and the associated curvature tensors are called F-Yang-Mills fields.

Let ∇^t =∇+A^t be a variation of∇ ∈ CE, whereA^t ∈Ω¹(g_E) withA⁰ = 0.

Then the curvature of∇^tis given by

(9) R^∇t =R^∇+d^∇A^t+1

2[A^t∧A^t],

where, the compound operation [· ∧ ·] is defined as follows: For ϕ, ψ ∈ Ω(gE), [ϕ∧ψ]_X,Y = [ϕX, ψY]−[ϕY, ψX]. Here,d^∇is the wedge covariant differentiation.

By a straightforward calculation, we have

(10) d

dtSF(∇^t) = Z

M

d

dtFkR^∇tk² 2

= Z

M

F⁰kR^∇tk² 2

Dd

dtR^∇t, R^∇tE

= Z

M

F⁰kR^∇tk² 2

D d^∇d

dtA^t+hd

dtA^t∧A^ti , R^∇tE

.

Let D= _dt^dA^t|t=0 and letδ^∇ be the adjoint operator of d^∇ with respect to the inner product. The above equality becomes as

(11)

d

dtSF(∇^t) _t=0

= Z

M

F⁰kR^∇k² 2

hd^∇D, R^∇i

= Z

M

D

D, δ^∇F⁰kR^∇k² 2

R^∇E

.

Hence the Euler-Lagrange equation of S_F(·) is

(12) δ^∇F⁰kR^∇k²

2

R^∇= 0.

In order to discuss the stabilities of F-Yang-Mills fields, we need the second variational formula. A direct calculation yields

d

dtR^∇t =d^∇dA^t dt +1

2 d dt

A^t∧A^t (13)

and

d²

dt²R^∇t =d^∇d² dt²A^t

+hd²

dt²A^t∧A^ti

+hdA^t dt ∧dA^t

dt i

. (14)

Hence we have

(15) d

dt _t=0

R^∇t =d^∇D , d² dt²

_t=0

R^∇t =d^∇C+ [D∧D]. whereC=_dt^d2₂ |t=0A^t. Taking derivatives of (10), we have

d²

dt²SF(∇^t) = Z

M

d dt

hF⁰kR^∇tk² 2

Dd

dtR^∇t, R^∇tEi

= Z

M

F⁰⁰kR^∇tk² 2

Dd

dtR^∇t, R^∇tE² +

Z

M

F⁰kR^∇tk² 2

D d^∇(d²

dt²A^t), R^∇tE +

Z

M

F⁰kR^∇tk² 2

Dd²

dt²R^∇t, R^∇tE +

Z

M

F⁰kR^∇tk² 2

Dd

dtR^∇t, d dtR^∇tE

. (16)

Lettingt= 0, the above formula becomes as:

d²

dt²SF(∇^t) _t=0=

Z

M

F⁰⁰kR^∇k² 2

d^∇D, R^∇2

+ Z

M

F⁰kR^∇k² 2

d^∇C, R^∇

+ Z

M

F⁰kR^∇k² 2

d^∇C+ [D∧D], R^∇ +

Z

M

F⁰kR^∇k² 2

kd^∇Dk². (17)

By (12), we have:

(18) Z

M

F⁰kR^∇k² 2

d^∇C, R^∇

= Z

M

D

C, δ^∇F⁰kR^∇k² 2

R^∇E

= 0. Therefore, we obtain

d²

dt²SF(∇^t) _t=0=

Z

M

F⁰⁰kR^∇k² 2

d^∇D, R^∇2

+F⁰kR^∇k² 2

[D∧D], R^∇

+F⁰kR^∇k² 2

kd^∇Dk². (19)

Definition 11. ForD∈Ω¹(gE), the index of anF-Yang-Mills fieldR^∇ is defined as

I(D) = Z

M

F⁰⁰kR^∇k² 2

d^∇D, R^∇2

+ Z

M

F⁰kR^∇k² 2

[D∧D], R^∇ +

Z

M

F⁰kR^∇k² 2

kd^∇Dk². (20)

If for anyD∈Ω¹(g_E), there holdsI(D)≥0, then we callR^∇ stable. Otherwise, it is unstable.

3. Lemmas

Forϕ∈Ω²(g_E),ω∈Ω²(M)⊗Hom (X(M),X(M)), let

(21) (ϕ◦ω)X,Y = 1

2

Xϕe_j,ω_X,Ye_j.

We use R to express the Riemannian curvature tensor of M, Ric for the Ricci operator. OnM, we take a local orthonormal frame field{ei}_i=1,···,n, and adopt the Einstein convention of summation. The range of the indices i, j, k, l, m is {1, . . . , n}. Let

(22) (Ric∧I)X,Y = Ric(X)∧Y +X∧Ric(Y) and

(23) R^∇(ϕ)_X,Y =X

[R^∇_e

j,X, ϕ_ej,Y]−[R^∇_e

j,Y, ϕ_ej,X] . Here, Ric∧I∈Ω²(M)⊗Hom (X(M),X(M)), andX∧Y is defined as:

(24) (X∧Y) (Z) =hX, ZiY − hY, ZiX . For anyϕ∈Ω²(gE), we have (see [1])

(25) ∆ϕ=∇^∗∇ϕ−ϕ◦(Ric∧I+ 2R) +<^∇(ϕ). Hence we have

(26) 1

2∆kϕk²=

∆^∇ϕ, ϕ

− k∇ϕk²− hϕ◦(Ric∧I+ 2R), ϕi −

<^∇(ϕ), ϕ .

Lemma 12. For an F-Yang-Mills fieldR^∇, we have Z

M

F⁰⁰kR^∇k² 2

kR^∇k²k∇kR^∇kk²

+ Z

M

F⁰kR^∇k² 2

kR^∇k²+ Z

M

F⁰kR^∇k² 2

<^∇(R^∇), R^∇ +

Z

M

F⁰kR^∇k² 2

R^∇◦(Ric∧I+ 2R), R^∇

= 0. (27)

Proof. By a straightforward calculation, we get

∆FkR^∇k² 2

=−X

∇_ei∇_eiFkR^∇k² 2

=−X

∇_ei

F⁰kR^∇k² 2

∇_eikR^∇k² 2

=−F⁰⁰kR^∇k² 2

kR^∇k²k∇kR^∇k k²−1

2F⁰kR^∇k² 2

∆kR^∇k². (28)

In (26), takingϕ=R^∇, and then substituting the result into (28), we get

∆FkR^∇k² 2

=−F⁰⁰kR^∇k² 2

kR^∇k²k∇kR^∇k k²

−F⁰kR^∇k² 2

<^∇(R^∇), R^∇

+F⁰kR^∇k² 2

∆^∇R^∇, R^∇

−F⁰kR^∇k² 2

k∇R^∇k²−F⁰kR^∇k² 2

R^∇◦(Ric∧I+ 2R), R^∇ . (29)

Integrating (29) shows that it is sufficient to proveR

MF^{0 kR∇}₂^k2

h∆^∇R^∇, R^∇i= 0.

By (12) and the Bianchi equalityd^∇R^∇= 0, we have Z

M

F⁰kR^∇k² 2

h∆^∇R^∇, R^∇i

= Z

M

D

d^∇◦δ^∇R^∇, F⁰kR^∇k² 2

R^∇E

+ Z

M

D

δ^∇◦d^∇R^∇, F⁰kR^∇k² 2

R^∇E

= Z

M

D

δ^∇R^∇, δ^∇F⁰kR^∇k² 2

R^∇E

+ Z

M

D

δ^∇◦d^∇R^∇, F⁰kR^∇k² 2

R^∇E

= 0. (30)

Let{Xa} be an orthonormal frame ofgE, and{ei} onM. Let

(31) R^∇_ei,ej =f_ij^aXa, ∇e_kR^∇

e_i,e_j =f_ijk^a Xa.

Then we havef_ij^a =−f_ji^a, f_ijk^a =−f_jik^a ,kR^∇k²= ¹₂f_ij^af_ij^a,k∇R^∇k²=¹₂f_ijk^a f_ijk^a . Lemma 13 ([3]). We have

(i) If Mⁿ is a submanifold ofR^n+k, then

(32)

R^∇◦(Ric∧I+ 2R), R^∇

=

(H^µh^µ_jl−h^µ_jmh^µ_ml)δki−h^µ_ikh^µ_jl f_ij^af_kl^a; (ii) If Mⁿ is a submanifold ofS^n+k, then

(33)

R^∇◦(Ric∧I+ 2R), R^∇

=

(H^µh^µ_jl−h^µ_jmh^µ_ml)δki−h^µ_ikh^µ_jl

f_ij^af_kl^a + 2(n−2)kR^∇k². 4. Stabilities of F-Yang-Mills fields

Theorem 14. Let Mⁿ be a submanifold ofR^n+k, which satisfies the following condition:

(34)

(−H^µh^µ_jl+ 2h^µ_jmh^µ_ml)δki+ 2h^µ_ikh^µ_jl

δsr+ 2(p−2)h^µ_ikh^µ_srδjl≤bδikδjlδsr, where b <0. If R^∇ is a nonzero F-Yang-Mills field on Mⁿ, then it is unstable, where fort >0 we have

(35) (p−2)F⁰(t)≥2tF⁰⁰(t), F⁰(t)>0, F(t)>0.

Proof. LetX andV be two tangent vectors toMⁿ, and letD=i_VR^∇, then we have D_X = (i_VR^∇)_X=R_V,X^∇ and

(d^∇D)_ei,ej = (∇eiD)_ej −(∇ejD)_ei

= ∇e_i(iVR^∇)

ej− ∇e_j(iVR^∇)

ei

=∇e_i(iVR^∇)e_j −(iVR^∇)_∇eie_j − ∇e_j(iVR^∇)e_i+ (iVR^∇)_∇eje_i

=∇e_i(iVR^∇)e_j − ∇e_j(iVR^∇)e_i−R^∇_V,∇

eiej +R_V,∇^∇

ejei. Because

∇e_j(iVR^∇)e_i =∇e_i R^∇_V,ej

= (∇e_jR^∇)V,e_i+R^∇_∇

ejV,ei+R^∇_V,∇

ejei, we have

(36) (d^∇D)e_i,e_j = (∇e_iR^∇)V,e_j −(∇e_jR^∇)V,e_i+R^∇_∇

eiV,ej−R^∇_∇

ejV,ei.

Let {EA |A= 1,2, . . . , n+k}be the canonical orthonormal base of R^n+k, and writeVA=v_Aⁱei as the tangent part ofEA. Let the indicesA,B, Crun from 1 to n+k, the indicesi,j from 1 ton, and the indiceµfromn+ 1 ton+k. Then we have

(37) v_A^Bv_A^C=δBC, ∇e_iVA=v_A^µh^µ_ijej. ForD_A=i_VAR^∇, A= 1,2, . . . , n+k, according to (20) we get

X

A

I_F(D_A) =X

A

Z

M

F⁰⁰kR^∇k² 2

d^∇D_A, R^∇2

+X

A

Z

M

F⁰kR^∇k² 2

DA∧DA, R^∇

+X

A

Z

M

F⁰kR^∇k² 2

d^∇D_A, d^∇D_A . (38)

By (36) and (37), we have

(d^∇DA)e_i,e_j = (∇e_iR^∇)V_A,e_j −(∇e_jR^∇)V_A,e_i+R^∇_∇

eiV_A,e_j −R^∇_∇

ejV_A,e_i

=v_A^l(∇e_iR^∇)e_l,e_j−v_A^l(∇e_jR^∇)e_l,e_i+v^µ_Ah^µ_ilR^∇_el,ej −v_A^µh^µ_jlR^∇_el,ei, (39)

from which, we have hd^∇DA, R^∇i=1

2 R^∇_e

i,e_j,(d^∇DA)e_i,e_j

=1 2v^l_A

R^∇_e

i,e_j,(∇eiR^∇)_el,ej

−1 2v_A^l

R^∇_e

i,e_j,(∇ejR^∇)_el,ei +1

2v^µ_Ah^µ_il

R_e^∇_i,ej, R_e^∇_l,ej

−1

2v^µ_Ah^µ_jlhR^∇_ei,ej, R^∇_el,eii

=v^l_A

R^∇_ei,ej,(∇e_iR^∇)e_l,e_j

+v^µ_Ah^µ_ilhR^∇_ei,ej, R^∇_el,eji.

According to (37) one has X

A

hd^∇D_A, R^∇i²= v_A^lhR^∇_e

i,e_j,(∇_eiR^∇)_el,eji+v_A^µh^µ_ilhR^∇_e

i,e_j, R_e^∇

l,e_ji

× v^k_AhR^∇_es,et,(∇e_sR^∇)e_k,e_ti+v_A^λh^λ_skhR^∇_es,et, R^∇_ek,eti

=δ_lk R^∇_e

i,e_j,(∇eiR^∇)e_l,ej

R^∇_e

s,e_t,(∇esR^∇)e_k,et

+δµλh^µ_ilh^λ_skhR^∇_ei,ej, R_e^∇_l,ejihR^∇_es,et, R^∇_ek,eti

= R^∇_e

i,e_j,(∇eiR^∇)el,ej

R^∇_e

s,e_t,(∇esR^∇)el,et

+h^µ_ilh^µ_skhR^∇_ei,ej, R^∇_el,ejihR^∇_es,et, R^∇_ek,eti. (40)

Taking use of the Bianchi identity, we reach R^∇_ei,ej,(∇e_iR^∇)e_l,e_j

=−

R^∇_ei,ej,(∇e_lR^∇)e_j,e_i

−

R^∇_ei,ej,(∇e_jR^∇)e_i,e_l

=

R^∇_ei,ej,(∇e_lR^∇)e_i,e_j

−

R^∇_ej,ei,(∇e_jR^∇)e_l,e_i ,

from which we obtain (41) X

i,j

R_e^∇

i,e_j,(∇_eiR^∇)_el,ej

= 1 2

X

i,j

R^∇_e

i,e_j,(∇_elR^∇)_ei,ej

=hR^∇,∇_elR^∇i.

Substituting (41) into (40), we have X

A

hd^∇DA, R^∇i²=X

l

hR^∇,∇e_lR^∇i²+h^µ_ilh^µ_tmhR^∇_ei,ej, R^∇_el,ejihR^∇_et,es, R^∇_em,esi

=kR^∇k²

∇kR^∇k

2+h^µ_ilh^µ_tmhR^∇_ei,ej, R^∇_e

l,e_jihR_e^∇_t,es, R^∇_e

m,e_si. (42)

Therefore X

A

Z

M

F⁰⁰kR^∇k² 2

hd^∇DA, R^∇i²= Z

M

F⁰⁰kR^∇k² 2

kR^∇k²

∇kR^∇k

2

+ Z

M

F⁰⁰kR^∇k² 2

h^µ_ilh^µ_tmhR^∇_ei,ej, R^∇_el,ejihR^∇_et,es, R^∇_em,esi

= Z

M

F⁰⁰kR^∇k² 2

kR^∇k²

∇kR^∇k

2

+ Z

M

F⁰⁰kR^∇k² 2

h^µ_ilh^µ_tmf_ij^af_lj^af_ts^bf_ms^b , (43)

wheref_ij^a’s are the components ofR^∇_ei,ej. Because

h^µ_ilh^µ_tmf_ij^af_lj^af_ts^bf_ms^b =h^µ_ikh^µ_srf_ij^af_kj^af_st^bf_rt^b =h^µ_ikh^µ_srδ_jlf_ij^af_kl^af_st^bf_rt^b ,