Instantons on Cylindrical Manifolds 3

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Instantons on the higher dimension, proposed in [6] and studied in [5, 8, 9, 22, 23], are important both in mathematics [9, 22] and string theory [12]. In this paper, we consider the cylinder manifold Z = R M with metric × 2 gZ = dt + gM where M is a compact Riemannian manifold. We assume M admits a 3-form P and a 4-form Q satisfying

dP =4Q (1.2) d Q =(n 3) P. (1.3) ∗M − ∗M On Z,the 4-form [14, 15] can be deﬁned as

Q = dt P + Q. Z ∧ Then the instanton equation on the cylinder manifold Z is

F + Q F =0 (1.4) ∗ A ∗ Z ∧ A Remark 1.1. Manifolds with P and Q satisfying equations (1.2), (1.3) include nearly

Kähler 6-manifolds and nearly parallel G2-manifolds. (1)M is a nearly Kähler 6-manifold. It is defined as a manifold with a 2-form ω and a 3-form P such that dω =3 P and dP =2ω ω =: 4Q ∗M ∧ For a local orthonormal co-frame ea on M one can choose { } ω = e12 + e34 + e56 and P = e135 + e164 e236 e245, − − where a =1,..., 6, ea1...al = e1 ...el, and ∧ P = e145 + e235 + e136 e246, Q = e1234 + e1256 + e3456. ∗M − Here denotes the -operator on M. ∗M ∗ (2)M is a nearly parallel G2 manifold. It is defined as a manifold with a 3-form P (a G structure [4]) preserved by the G SO(7) such that 2 2 ⊂ dP = γ P ∗M for some constant γ R. For a local orthonormal co-frame ea, a = 1,..., 7, on M one ∈ can choose P = e123 + e145 e167 + e246 + e257 + e347 e356 − − and therefore

P =: Q = e4567 + e2367 e2345 + e1357 + e1346 + e1256 e1247. ∗M − − It is easy to check dP =4Q. Instantons on Cylindrical Manifolds 3

Constructions of solutions of the instanton equations on cylinders over nearly K¨ahler

6-manifolds and nearly parallel G2 manifold were considered in [1, 13, 15, 16]. In [16] section 4, the authors conﬁrm that the standard Yang-Mills functional is inﬁnite on their 2 solutions. In this paper, we assume the instanton A has L -bounded curvature FA. Then we have the following theorem.

Theorem 1.2. (Main theorem) Let Z = R M, here M is a closed Riemannian n- × manifold, n 4, which admits a smooth 3-form P and a smooth 4-form Q satisfying ≥ equations (1.2) and (1.3). Let A be a instanton over Z. Assume that the curvature F A ∈ L2(Z) i.e.

FA FA < + ZZ h ∧∗ i ∞ Then the instanton is a ﬂat connection.

2 Esitimationof Curvatureof Yang-Millsconnectionwith torsion

Let Q be a smooth 4-form on n-dimensional manifold X. Let A be an anti-self-dual instanton which satisﬁes the instanton equation (1.1). Taking the exterior derivative of (1.1) and using the Bianchi identity, we obtain

d F + F =0, (2.1) A ∗ A ∗H ∧ A where the 3-form is defined by H = d( Q). (2.2) ∗ H ∗ The second-order equation (2.1) differs from the standard Yang-Mills equation by the last term involving a 3-from . This torsion term naturally appears in string-theory compacti- H fications with fluxes [2, 10, 11]. For the case d( Q)=0, the torsion term vanishes and the ∗ instanton equation (1.1) imply the Yang-Mills equation. The latter also holds true when the instanton solution A satisfies d( Q) F =0 as well, like the cases, on nearly Kähler ∗ ∧ A 6-manifolds, nearly parallel G2-manifolds and Sasakian manifolds [14]. In section 4.2 of [1] or in section 2.1 of [13], they online that in the instanton does not extremize the standard Yang-Mills functional in the torsionful case. Instead, they add a add a Chern-Simons-type term to get the following functional:

S(A)= Tr FA FA + FA FA Q , (2.3) − Z ∧∗ ∧ ∧∗ X This is the right functional which produces the correct Yang-Mills equation with torsion. And the instanton equations (1.1) can be derived from this action using a Bogomolny 4 Teng Huang argument. In the case of a closed form Q, the second term in (2.3) is topological invariant ∗ and the torsion (2.2) disappears from (2.1). In this section, we will derive monotonicity formula for Yang-Mills connection with torsion (2.1). Its proof follows Tian’s arguments about pure Yang-Mills connection in [22] with some modifications. Let X be a compact Riemannian n-manifold with metric g and E is a vector bundle over X with compact structure group G. For any connection A of E, its curvature form F takes value in Lie(G). The norm of F at any p X is given by A A ∈ n F 2 = F (e , e ), F (e , e ) , | A| h A i j A i j i i,jX=1 where e is any orthonormal basis of T X, and , is the Killing form of Lie(G). { i} p h· ·i As in [22], we consider a one-parameter family of diffeomorphisms ψ of X { t}|t|<∞ with ψ0 = idX . We fix a connection A0, and denote by its derivative D. Then for any connection A, we can define a one-parameter family A in the following way. Let τ 0 { t} t be the parallel transport on E associated to A along the path ψ (x) , where x X. 0 s 0

∗ We deﬁne a family of connections At := ψt (A) by deﬁning its covariant derivative as

t 0 −1 0 Dνs =(τt ) Ddψt(ν)(τt (s)) for any ν TX, s Γ(X, E). Then the curvature of A is written as ∈ ∈ t

F (X ,X )=(τ 0)−1 F (dψ (X ), dψ (X )) τ 0. At 1 2 t · A t 1 t 2 · t

It follows that

n 2 2 FA = FA(dψt(ei), dψt(ej)) (ψt(x))dVg. Z | | Z | | X X i,jX=1 where dV denotes the volume form of g, and e is any local orthonormal basis of TX. g { i} By changing variables, we obtain

n 2 −1 −1 2 −1 FA = FA(dψt(ei(ψ (x))), dψt(ej(ψ (x))) Jac(ψ )dVg. Z | | Z | t t | t X X i,jX=1 Instantons on Cylindrical Manifolds 5

Let ν be the vector ﬁeld ∂ψt on X. Then we deduce from the above that ∂t |t=0

d ∗ Y M(At) t=0 = iνFA,dAFA dt | ZX h i

= Tr iνFA FA (d Q) ZX ∧ ∧ ∗ n 2 (2.4) = FA divν +4 FA([ν, ei], ej), FA(ei, ej) dVg Z | | h i X i,jX=1 n 2 = FA divν 4 FA( e ν, ej), FA(ei, ej) dVg Z | | − h ∇ i i X i,jX=1 Fix any p X, let r be a positive number with following properties: there are normal ∈ p coordinates x1,...,xn in the geodesic ball Brp (p) of (X,g), such that p = (0,..., 0) and for some constant c(p),

g δ c(p)( x 2 + ... + x 2), | ij − ij| ≤ | 1| | n|

2 2 dgij c(p) x1 + ... + xn , | | ≤ p| | | | where ∂ ∂ g = g , . ij ∂x ∂x i j 2 2 ∂ Let r(x) := x1 + ... + xn be the distance function from p. Deﬁne ν(x) = ξ(r)r ∂r , p where ξ is some smooth function with compact support in Brp (p). Let e1,...,en be ∂ { } any orthonormal basis near p such that e1 = ∂r . Since x1,...,xn are normal coordinates, ∂ we have ∂ =0. It follows that ∇ ∂r ∂r

′ ∂ ′ ∂ ∂ ν =(ξr) =(ξ r + ξ) . (2.5) ∇ ∂r ∂r ∂r Moreover, for i 2, ≥ n ∂ ν = ξr( )= ξ b e . (2.6) ∇ei ∇ei ∂r ij j Xj=1 where b δ = O(1)c(p)r2. Applying (2.5) and (2.6) to the variation formula (2.4), | ij − ij| we obtain

∗ 2 ′ 2 iνFA,d FA = FA (ξ r +(n 4)ξ + O(1)c(p)r ξ)dVg Z h A i Z | | − X X (2.7) ′ ∂ 2 4 (ξ r yFA )dVg. − ZX |∂r | where ∂ yF = F ( ∂ , ). ∂r A A ∂r · 6 Teng Huang

r We choose, for any τ small enough. ξ(r) = ξτ (r) = η( τ ), where η is smooth and satisﬁes: η(r)=1 for r [0, 1], η(r)=0 for r [1 + ǫ, ), ǫ< 0 and η′(r) 0. Then ∈ ∈ ∞ ≤ ∂ τ (ξ (r)) = rξ′ (r). (2.8) ∂τ τ − τ Plugging this into (2.7), we obtain

2 Tr(iνFA FA d( Q)) ν FA max d( Q) | Z ∧ ∧ ∗ | ≤ Z | |·| | · x∈X | ∗ | X X (2.11) 2 max d( Q) τ(1 + ǫ) FA dVg x∈X ≤ | ∗ | ZBτ(1+ǫ)(p) | | Then, by integrating on τ and letting ǫ tend to zero, we have already proved:

Theorem 2.1. Let rp, c(p) and a be as above. Then for any 0 <σ<ρ

4−n aρ 2 4−n aσ 2 ρ e FA dVg σ e FA dVg ZBρ(p) | | − ZBσ(p) | |

4−n ar ∂ 2 4 r e yFA dVg (2.12) ≥ ZBρ(p)\Bσ (p) |∂r | ρ aτ 4−n 2 + e τ (a max d( Q) O(1)cpτ) FA dVg dτ Z − x∈X | ∗ | − Z | | σ Bτ (p) Next, we prove a mean value inequality about the energy dense F 2. The Bochner- | A| Weitzenb¨ock formula ([3], Theorem 3.1) is

(d d∗ + d∗ d )F = ∗ F + F (Ric g +2R)+ A(F ). A A A A A ∇A∇A A A ◦ ∧ R A Instantons on Cylindrical Manifolds 7

Since A is a instanton and the Bianchi identity dAFA =0, we re-write the left hand

(d d∗ + d∗ d )F = d (d( Q) F ) A A A A A A ∗ ∗ ∧ A Hence d (d( Q) F ) n (d( Q)) F + n d( Q) F | A ∗ ∗ ∧ A | ≤ |∇ ∗ | A| | ∗ ||∇A A| due to for all X ,X ,...,X T X, where n = dimX, 1 2 n−2 ∈ x d∗ (d( Q) F ) (X ,X ,...,X ) A ∗ ∧ A 1 2 n−2 = ( ) (d( Q) F )(e ,X ,...,X ), − ∇A ei ∗ ∧ A i 1 n−2 Xi = (d( Q)) F (e ,X ,...,X ) − ∇ei ∗ ∧ A i 1 n−2 Xi (d( Q)) ( ) F (e ,X ,...,X ), − ∗ ∧ ∇A ei A i 1 n−2 Xi Then we have

(d d∗ + d∗ d )F , F n (d( Q)) F 2 + n d( Q) F F |h A A A A A Ai| ≤ |∇ ∗ |·| A| | ∗ |·|∇A A|·| A| 1 (2.13) C F 2 + C (ε F 2 + F 2) ≤ 1| A| 2 |∇A A| ε| A| where ε is a positive constant. The quadratic A(F ) Ω2(g) can be expressed with the help of a local orthonormal R A ∈ frame (e1, e2,...,en) of TX as

n A(F )(X ,X )=2 [F (e ,X ), F (e ,X )]. R A 1 2 A j 1 A j 2 Xj=1

The estimate of the Laplacian now follow from

∗ F 2 = 2 F 2 2 F , ∗ F −∇ ∇| A| − |∇A A| − h A ∇A∇A Ai 2 F , F (Ric g +2R) +2 F , A(F ) ≤ h A A ◦ ∧ i h A R A i 1 + C F 2 + C (ε F 2 + F 2) 2 F 2 1| A| 2 |∇A A| ε| A| − |∇A A|

We choose ε small enough such that C2ε< 2, then we have

∆ F 2 C F 2 + c F 3. | A| ≤ | A| | A| Thus, we get ∆ F C F + c F 2. (2.14) | A| ≤ | A| | A| 8 Teng Huang

Theorem 2.2. Let A be any Yang-Mills connection with torsion of a G-bundle E over X. Then there exist constants ε = ε(X, n, Q) > 0 and C = C(X, n), such that for any p X and p

Proof. By scalling, we may assume that ρ =1. Deﬁne a function

2 1 f(r)=(1 2r) sup FA (x), r [0, ]. − x∈Br(p) | | ∈ 2

1 1 Then f(r) is continuous in [0, 2 ] with f( 2 )=0, and f attains its maximum at a certain r0 1 in [0, 2 ]. First we claim that f(r0) 64 if ε is sufﬁciently small. Assume that f(r0) > 64. Put ≤ 1 b = supx∈B (p) FA (x)= FA (x0) by taking σ = (1 2r0), we get r0 | | | | 4 − sup F sup F (x) | A| ≤ | A| x∈Bσ(x0) x∈Br0+σ(p) (1 2r )2 (2.16) 0 F x b. − 2 sup A ( )=4 ≤ (1 2r0 2σ) x∈B (p) | | − − r0

2 Clearly, 16σ b 64; i.e., σ√b 2. Deﬁne a scaled metric g = bg. Then the norm F e ≥ ≥ | A|g of F is equal to b−1F with respect to g. Hence A A e e sup FA ge 4, (2.17) x∈B2(x0,ge) | | ≤ where B2(x0, g) denotes the geodesic ball of g with radius 2 and centered at x0. Using (2.17), we deduce from (2.14) that in B (x , g), e 2 0 e e ∆e F e (C +4c) F e. (2.18) g| A|g ≤ | A|g Then, by using the mean-value theorem, we obtain

1 2 2 1= FA ge(x0) c FA gedVge . (2.19) | | ≤ Z e | | B1(x,g) e where c is some uniform constant.

e Instantons on Cylindrical Manifolds 9

However, by the monotonicity (Theorem 2.1),

2 n−4 2 FA gedVge =(√b) FA dVg ZB1(x0,ge) | | ZB 1 (x0) | | √b

1 4−n a 2 ( ) e 2 FA dVg ≤ 2 ZB 1 (x0) | | 2 n−4 a ε2 e 2 ≤ Combining this with (2.19), we obtain

n−4 a 1 cε2 e 2 . ≤ It is impossible since we can choose ε e= ε(X, n, Q) sufﬁciently small. The claim is proved. Thus, we have

sup FA (x) 4f(r0) 256. x∈B 1 (p) | | ≤ ≤ 4 It follows from this and (2.14) with g replaced by g that for some uniform constant c′,

∆e F c′ F . (2.20) g| A| ≤ | A| Then (2.15) follows from (2.20) and a standard Moser iteration.

3 Asymptotic Behavior and Conformal Transformation

3.1 Chern-Simons Functional

The main aim of this section is to get the relationship between gauge theory on an n- dimensional manifold M and the gauge theory on the n +1-dimensional manifold Z = R M. The main idea is that a connection on R M can be regard as one-parameter × × families of connections on M by local trivialisation. Let t be the standard parameter on the factor R in the R M and let xj n be local coordinates of M. A connection A × { }j=1 over the cylinder Z is given by a local connection matrix

n i A = A0dt + Aidx . Xi=1

1 n where A0 and Ai dependence on all n +1 variable t, x ,...,x . We take A0 =0 (some- times called a temporal gauge). In this situation, the curvature in a mixed xi-plane is given by the simple formula ∂A F = i . 0i ∂t 10 Teng Huang

n i ˙ ∂A We denote A = i=1 Aidx and A = ∂t , then the curvature is given by P F = F + dt A.˙ A A ∧ M has a Riemannian metric and -operator . If φ is a 1-form on M then, for -operator ∗ ∗M ∗ defined on Z = R M with respect to the product metric, we have × (dt φ)= φ. ∗ ∧ ∗M Then the instanton equation is equivalent to A˙ = P F , (3.1) ∗M −∗M ∧ A F = A˙ P Q F . (3.2) ∗M A − ∧∗M −∗M ∧ A Let E M be a vector bundle, the space is an affine space modelled on Ω1(g ) so, → A E fixing a reference connection A , we have 0 ∈A = A + Ω1(g ). A 0 E We define the Chern-Simons functional by 2 CS(A) := Tr a dA a + a a a M P, − Z ∧ 0 3 ∧ ∧ ∧∗ M fixing CS(A0)=0. This functional is obtained by integrating of the Chern-Simons 1- form

Γ(β)A =ΓA(βA)= 2 Tr(FA βA) M P. − ZM ∧ ∧∗ We ﬁnd CS explicitly by integrating Γ over paths A(t) = A0 + ta, from A0 to any

A = A0 + a: 1 CS(A) CS(A0)= ΓA(t) A˙(t) dt − Z0 1 2 = 2 Tr (FA + tdA a + t a a) a M P dt − Z Z 0 0 ∧ ∧ ∧∗ 0 M 2 = Tr dA a a + a a a M P + C, − Z 0 ∧ 3 ∧ ∧ ∧∗ M where C = C(A0, a) is a constant and vanishes if A0 is an instanton. The co-closed condition d P = 0 implies that the Chern-Simons 1-form is closed. So it does not ∗M depend on the path A(t) [7, 20, 21]. Since

2 2 2 dTr(dA a a + a a a)= Tr(F F ), 0 ∧ 3 ∧ ∧ A0+a − A0 we can re-write Chern-Simons functional as 2 CS(A) CS(A0)= Tr dA a a + a a a M P − − Z 0 ∧ 3 ∧ ∧ ∧∗ M (3.3) 1 2 2 = Tr(F F ) M Q, −n 3 Z A − A0 ∧∗ − M the second formula holds because of equation (1.3). Instantons on Cylindrical Manifolds 11

3.2 Asymptotic Behavior

Let Z = R M be an (n+1)-manifold and M be an n-manifold. Let A be an instanton on × Z with ﬁnite energy, i.e. F 2 < . We use the Chern-Simons functional to study the Z | A| ∞ decay of instantons over theR cylinder manifold. We will see that, an instanton with L2(Z)- bounded curvature can be represented by a connection form which decays exponentially on the tube. We consider a family of bands B = (T 1, T ) M which we identify with the T − × model B = (0, 1) M by translation. So the integrability of F 2 over the end implies × | A| that 2 FA 0 as T . Z(T,T +1)×M | | → →∞ Proposition 3.1. Let Z = R M, here M is a closed Riemannian n-manifold, n 4, × ≥ which admits a smooth 3-form P and a smooth 4-form Q those satisfy equations (1.2) and (1.3). Let A be a instanton over Z, then at the end of Z there is a ﬂat connection Γ over M such that A converges to Γ, i.e. the restriction A converges (modulo gauge |M×{T } equivalence) in C∞ over M as T . →∞ Proof. We choose 1 ρ = Inj (T, T + 1) M,g , 2 × Z where Inj (t, t + 1) M > 0 denotes the injectivity radius of the manifold (T, T + × 1) M,g . It’s easy to see ρ is not dependent on t. Since Q = P + dt Q, we × Z ∗ Z ∗M ∧∗M obtain 2 2 2 max d( QZ) = max d M P + d( M Q) < , (x,t)∈Z | ∗ | x∈M | ∗ | | ∗ | ∞ and

max (d QZ ) max (d M P ) + max (d M Q) < . (x,t)∈Z |∇ ∗ | ≤ x∈M |∇ ∗ | x∈M |∇ ∗ | ∞ We denote ε = ε(Z, n, Q) as the constant in Theorem 2.2. Then there exist T sufﬁciently large such that t T , we have ≥

2 n−3 FA ερ , Z(T,T +1)×M | | ≤

Then for any point (t, x) (T, T + 1) M, we have ∈ ×

3−n 2 ρ FA ε. ZBρ(x,t) | | ≤ From Theorem 2.2, we have

C 1 3−n 2 2 FA (t, x) ρ FA | | ≤ ρ2 Z | | Bρ(x,t) 12 Teng Huang

It implies that for any sequence T there exist a ﬂat connection Γ over M such that, i →∞ after suitable gauge transformations,

A Γ, Ti → in C∞ over M.

Under above, from (3.3) we can write Chern-Simons function as 1 CS(A(T )) CS(A( )) = Tr(FA FA) M Q. − ∞ −n 3 Z ∧ ∧∗ − M Lemma 3.2. Let A be an instanton with temporal gauge, then

′ CS(A(T )) CS(A(T )) = Tr(FA FA) − Z[T,T ]×M ∧∗ ′ (3.4) T ′ (n 3) CS(A(t)) CS(A∞) dt − − Z − T Proof. Using the method of previous section, we have d CS A(t) =Γ (A(˙t)) dt A(t) Then

T ′ T ′ ′ CS(A(T )) CS(A(T )) = dCS A(t) = ΓA(t) A˙(t) dt − Z Z T T

= 2 Tr(FA(t) dt A˙(t)) M P − Z[T,T ′]×M ∧ ∧ ∧∗ T ′ = Tr(F F ) Q + Tr(F F ) Q dt − A ∧ A ∧∗ Z A ∧ A ∧∗M Z[T,T ′]×M ZT ZM T ′ = Tr(FA FA) (n 3) CS(A(t)) CS(A( )) dt Z ∧∗ − − Z − ∞ [T,T ′]×M T

We set ∞ 2 J(T )= FA L2 = Tr(FA FA). ZT k k − Z[T,∞)×M ∧∗ On the one hand, we can express J(T ) as the integration of Tr(F F ) Q , since A A ∧ A ∧∗ Z is an instanton.

J(T )= Tr(FA FA) QZ Z[T,∞)×M ∧ ∧∗ From (3.4), taking the limit over ﬁnite tubes (T, T ′) M with T ′ + we see that × → ∞ ∞ J(T )= CS(A(T )) CS(A∞) (n 3) CS(A(t)) CS(A( )) dt (3.5) − − − Z − ∞ T Instantons on Cylindrical Manifolds 13 where A(T ) is the connection over M obtain by restriction to M T . From (3.5), we ×{ } can obtain the T derivative of J as d d J(T )= CS(A(T )) CS(A( )) +(n 3) CS(A(T )) CS(A ) (3.6) dT dT − ∞ − − ∞ On the other hand, the T derivative of J(T ) can be expressed as minus the integration over M T of the curvature density F 2, and this is exactly the n-dimensional curvature ×{ } | A| density F 2 plusing the density A˙ 2. By the relation (1.2) and (1.3) between the two | A(T )| | | components of the curvature for an instanton, we have

2 FA(T ) L2(M) = Tr(FA(T ) M FA(T )) k k − ZM ∧∗

= Tr FA(T ) ( A˙(T ) M P ) − Z ∧ − ∧∗ M

+ Tr(FA(T ) FA(T )) M Q ZM ∧ ∧∗ 2 = A˙ 2 (n 3) CS(A(T )) CS(A ) k kL (M) − − − ∞ Thus

d 2 J(T ) = 2 F 2 (n 3) CS(A(T )) CS(A ) (3.7) dT − k A(T )kL (M) − − − ∞ ˙ 2 = 2 A 2 +(n 3) CS(A(t)) CS(A ) (3.8) − k kL (M) − − ∞ From (3.6) and (3.7), we have d CS(A(T )) CS(A ) + 2(n 3) CS(A(T )) CS(A ) 0 dT − ∞ − − ∞ ≤ From (3.6) and (3.8), we have d CS(A(T )) CS(A ) 0 dT − ∞ ≤ It’s easy to see these imply that CS(A(t)) CS(A ) is non-negative and decays ex- − ∞ ponentially, 0 CS(A(T )) CS(A ) Ce−(2n−6)T (3.9) ≤ − ∞ ≤ We introduce a parameter δ and set ∞ δt 2 Lδ(T ) := e FA L2(M)dt ZT k k Theorem 3.3. Let A be an instanton with L2-bounded curvature on Z = R M, here × M is a closed Riemannian n-manifold, n 4, which admits a smooth 3-form P and a ≥ smooth 4-form Q those satisfy equations (1.2) and (1.3). Then there is a constant C, such that L (t) Ce(δ−2n+6)t δ ≤ where 0 <δ< 2n 6. − 14 Teng Huang

Proof. From (3.6), we get

2 d F 2 = CS(A(t)) CS(A ) (n 3) CS(A(t)) CS(A ) k AkL (M) −dt − ∞ − − − ∞ Then ∞ δt d Lδ(T )= e CS(A(t)) CS(A∞) − ZT dt − ∞ δt (n 3) e CS(A(t)) CS(A∞) − − ZT − ∞ δt ∞ δt = e CS(A(t)) CS(A∞) T + δe CS(A(t)) CS(A∞) − − | ZT − ∞ δt (n 3) e CS(A(t)) CS(A∞) − − ZT − ∞ δT δt e CS(A(T )) CS(A∞) + δe CS(A(t)) CS(A∞) ≤ − ZT − ∞ Ce(δ−2n+6)T + Cδe(δ−2n+6)tdt ≤ ZT Cδ =(C + )e(δ−2n+6)T 2n 6 δ − −

3.3 Conformal Transformation

We consider Z¯ = C(M), where C(M) is a cone over M with metric

2 2 2t 2 gZ¯ = dr + r gM = e (dt + gM ), where r := et.

It means that the cone C(M) is conformally equivalent to the cylinder

Z = R M × with the metric 2 gZ = dt + gM . Furthermore, we can show that the instanton equation on the cone Z¯ = C(M) is related with the instanton equation on the cylinder Z = R M , × (n−3)t ¯F + ¯Q ¯ F = e ( F + Q F )=0, ∗ A ∗ Z ∧ A ∗ A ∗ Z ∧ A where dimC(M)= dimZ = n +1, ¯ is the -operator in C(M).And ∗ ∗ 4t Q ¯ = e (dt P + Q). (3.10) Z ∧ Instantons on Cylindrical Manifolds 15

In the other word, equation on C(M) is equivalent to the equation on R M after rescal- × ing of the metric. So we can only consider the instanton equation

¯F + ¯Q ¯ F =0 ∗ A ∗ Z ∧ A on the cone C(M) over M. Since

(n−3) (n−3)t ¯ Q ¯ = e Q = e ( P + dt Q), ∗Z Z ∗ Z ∗M ∧∗M by direct calculate, ¯ Q ¯ is closed. This implies that the instantons also satisfy the pure ∗Z Z Yang-Mills equations with respect to the metric gZ¯. Proposition 3.4. Let A be a instanton on Z = R M, here M is a closed Riemannian × n-manifold, n 4, which admits a smooth 3-form P and a smooth 4-form Q those satisfy ≥ equations (1.2) and (1.3). Then the connection A is a Yang-Mills connection on C(M). After rescaling of the metric, F ¯F = e(n−3)tF F . A ∧ ∗ A A ∧∗ A 2 The curvature FA is L -bounded over Z. We shall prove that the curvature FA is still L2-bounded over C(M) by the following lemma. Lemma 3.5. Let A be a instanton on Z = R M with L2-bounded curvature F , i.e. × A

FA FA < + , ZR×M h ∧∗ i ∞ here M is a closed Riemannian n-manifold, n 4, which admits a 3-form P and a ≥ 4-form Q those satisfy equations (1.2) and (1.3). Then

FA ¯FA < + . ZR×M h ∧ ∗ i ∞ Proof. From theorem 3.3, we have ∞ (n−3)t 2 −(n−3)T Ln−3(T )= e FA L2(M)dt Ce ZT k k ≤ Then for any constant T [0, ), we have ∈ ∞ (n−3)t FA ¯FA = e FA FA ZR×M h ∧ ∗ i ZR×M h ∧∗ i

(n−3)t = + e FA FA Z(−∞,T ]×M Z[T,∞)×M h ∧∗ i

(n−3)T e FA FA + Ln−3(T ) < + ≤ ZR×M h ∧∗ i ∞

We only consider instantons with L2-bounded curvature on the cone of M. In the next section, we will give a vanishing theorem for Yang-Mills connection with ﬁnite energy on the cone of M. 16 Teng Huang

4 Vanishing Theorem for Yang-Mills

In this section, notations may be different from the previous sections. We use the conformal technique to give the vanishing theorem for Yang-Mills connection on the cone of M. Let M be a Riemannian n +1-manifold. Suppose X Γ(T M) is a conformal vector ∈ ﬁeld on (M,g), namely, g =2fg LX where f C∞(M). Here denotes the Lie derivative with respect to X. ∈ LX The vector ﬁeld X generates a family of local conformal diffeomorphism.

F = exp(tX): M M t →

This family of local conformal diffeomorphism can induce a bundle automorphism, F˜t, of the principal bundle P . Such a lift is readily obtained from a connection on P by setting

Ft = exp(tX) where X is the horizontal lift of X on P . If A is the connection form we have i e A since X is horizontal. Thus the Lie derivative of A is can be expressed in e X =e 0 e 1 terms of the curvature F : e A = i e dA + di e A = i e (F [A, A]) = i F . And e A X X X X A 2 X A ∗ L 2 − hence Ft A = A + tiX FA + o(t ). One can see the detailed process in [19]. Wee will consider the variation of the Yang-Mills functional under the family of diffeomorphism. 2 Y M(A, g)= FA dV olg ZM | | where dV olg = √detgdx is the volume form of M.

λ := F 2dV ol | A| g is an n-form on M. For any η C∞(M) ∈ 0

0= d[(iX λ)η]= ηd(iX λ)+ dη iX λ ZM ZM ZM ∧

= η X λ + dη iX λ ZM L ZM ∧ that is

η Xλ = dη iX λ (4.1) ZM L − ZM ∧ where i stands for the inner product with the vector X. Now, let us compute λ. X LX Lemma 4.1. Let λ = Tr(F F ) and X be a smooth vector ﬁeld on M satisfying A ∧∗ A g =2fg, then LX λ =(n 3)fλ +2Tr d (i F ) F LX − A X A ∧∗ A Instantons on Cylindrical Manifolds 17

Proof. In local coordinates xi n , the n-form λ { }i=1 ij kl 1 n λ = Tr(FA FA)= g g trFikFjl detgdx ... dx ∧∗ X p ∧ ∧ is conformal of weight n 3, i.e. λ(A, e2f g) = e(n−3)f λ(A, g) for any f C∞. The − ∈ vector field X satisfies g =2fg, so LX t ∗ ∗ Ft g = exp(2 Fs f)g. Z0 Since λ is conformal with wight n 3, − t (F ∗λ)(A, g)= λ(F˜∗A, F ∗g)= exp (n 3) F ∗f λ(F˜∗A, g) t t t − Z t · t 0 = 1+(n 3)tf + o(t2) − Tr F F +2td (i F ) F + o(t2) × A ∧∗ A A X A ∧∗ A = Tr(F F + t(n 3)fF F ) A ∧∗ A − A ∧∗ A +2tT r(d (i F ) F )+ o(t2) A X A ∧∗ A ∗ ˜∗ 2 where we used the fact Ft f = f + o(t) and Ft A = A + tiX FA + o(t ). By the definition of Lie derivative d λ = (F ∗λ) =(n 3)fλ +2Tr d (i F ) F . (4.2) LX dt t |t=0 − A X A ∧∗ A

We consider M = R N with metric × 2t 2 gM = e (dt + gN ) where N is a compact Riemannian n-manifold, n 4, with metric gN . Then the vector ∂ ≥ ﬁeld X = ∂t satisﬁes

g = X e2t(dt2 + g )+ e2t( dt2)=2g , LX M · N LX M and in this case, f =1.

Theorem 4.2. Let (M,gM ) be a Riemannian manifold as above. Let A be a Yang-Mills 2 connection with L -bounded curvature FA, i.e.

2 FA < + ZM | | ∞ over M. Then A must be a ﬂat connection. 18 Teng Huang

Proof. From (4.1) and (4.2), we have

η(n 3)λ = dη iX λ 2 ηTr dA(iX FA) FA Z − − Z ∧ − Z ∧∗ M M M

= dη iX λ 2 Tr dA(ηiX FA) FA − Z ∧ − Z ∧∗ M M (4.3)

+2 Tr (dη iX FA) FA Z ∧ ∧∗ M 3 dη X λ ≤ ZM | |·| |· The second term in the second line vanishes since A is a Yang-Mills connection. We choose the cut-off function with η(t)=1 on the interval t T , η(t)=0 on the | | ≤ interval t 2T , and dη 2T −1. Then dη has support in T t 2T and X(t) =1, | | ≥ | | ≤ ≤| | ≤ | | 6 η(n 3)λ λ. ZM − ≤ T Z{T ≤|t|≤2T }×N Letting T we get →∞ (n 3)λ =0, ZM −

Then FA =0.

Acknowledgment

I would like to thank O. Lechtenfeld for kind comments regarding this and it companion article [1, 13] and S.K. Donaldson for helpful comments regarding his article [7], and H.N. S´aEarp for helpful comments regarding his article [20, 21]. This work is partially supported by Wu Wen-Tsun Key Laboratory of Mathematics of Chinese Academy of Sciences at USTC.

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