Complex flat connections on compact manifolds

arXiv:1606.03948v5 [math.DG] 11 May 2017

An energy gap for complex Yang-Mills equations Teng Huang

Abstract In this article, we study the complex Yang-Mills equations which introduced by Gagliardo-Uhlenbeck [9] over a Riemannian manifold X of dimension n ≥ 2. Then we using the energy gap result of pure Yang-Mills equation ([8] Theorem 1.1) to prove another energy gap result of complex-Yang-Mills equations if X satisfies certain conditions.

Keywords. complex Yang-Mills equations, energy gap, gauge theory

1 Introduction Let X be a oriented n-manifold endow with a smooth Riemannian metric, g. Let P be a principle bundle over X with structure group G. Supposing that A is a connection on P , then we denote by FA its curvature 2-form, which is a 2-form on X with values in the bundle associated to P with fiber the Lie algebra of G denoted by gP . We define by dA the exterior covariant derivative on section of Λ• T ∗ X ⊗ (P ×G gP ) with respect to √ the connection A. The curvature FA of the complex connection A := dA + −1φ is a two-form with values in P ×G (gCP ): FA = [(dA +

√

−1φ) ∧ (dA +

√

√ 1 −1φ)] = FA − [φ ∧ φ] + −1dA φ. 2

The complex Yang-Mills functional is defined in any dimension as the norm squared of the complex curvature ([9] Section 3) Z Z 2 Y MC (A, φ) := |FA | = (|FA − φ ∧ φ|2 + |dA φ|2 ). X

X

T. Huang: Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, PR China; e-mail: [email protected] Mathematics Subject Classification (2010): 58E15;81T13

1

Teng Huang

2

This reduces to the pure Yang-Mills functional when the extra field φ vanishes. The EulerLagrange equations for this functional are d∗A (FA − φ ∧ φ) + (−1)n ∗ [φ, ∗dA φ] = 0,

d∗A dA φ − (−1)n ∗ [φ, ∗(FA − φ ∧ φ)] = 0.

(1.1)

These can also be succinctly written as, d∗A FA = 0. There equations are not elliptic, even after the real gauge-equivalence, so it is necessary to add the moment map condition, d∗A φ = 0. √ In this article, we called A + −1φ is the solution of the complex Yang-Mills equation, it not only satisfies Equations (1.1) but also satisfies the moment map condition. In particular, it is easy to see if φ = 0, the complex Yang-Mills equations is reduce to the pure Yang-Mills equation d∗A FA = 0. There are many articles study the energy gap result of Yang-Mills connection. In [1, 2, 3, 10, 13], they are all require some positive hypothesis on the curvature tensors of a Riemannian metric. In [8], Feehan apply the Lojasiewicz-Simon gradient inequality ([8] Theorem 3.2) to remove the positive hypothesis on the Riemannian curvature tensors. In [11], the author given another prove of energy gap theorem of pure Yang-Mills equation without using Lojasiewicz-Simon gradient inequality. It is an interesting question to consider, whether the complex-Yang-Mills equations have the energy gap phenomenon? In this article, we give a positive answer for this question if X satisfies certain conditions. Theorem 1.1. Let X be a compact Riemannian manifold of dimension n ≥ 2 with smooth Riemannan metric g, P be a G-bundle over X, let p ∈ (1, ∞) be such that 2p > n and in the case n = 4 assume p ≥ 2. Then there exist a positive constant ε = ε(X, n, p) with following significance. Suppose that there exist flat connection on P and all flat connections are non-degenerate in the sense of Definition 3.6. If the pair (A, φ) is a C ∞ solutions of complex-Yang-Mills equations over X and the curvature FA of connection A obeying kFA kLp (X) ≤ ε then A is a flat connection and φ is vanish. √ If FA = 0 i.e. FA − φ ∧ φ = 0 and dA φ = 0, we called A = A + −1φ is a complexflat connection. One can see the complex-flat connection satisfies the complex-Yang-Mills equations. It’s easy to see there has a priori estimate of FA by φ. Using the gap Theorem 1.1 of complex-Yang-Mills equations, we have a gap result for the extra fields as follow:

An energy gap for complex Yang-Mills equations

3

Corollary 1.2. Let X be a compact Riemannian manifold of dimension n ≥ 2 with smooth Riemannan metric g, P be a G-bundle over X. Then there exist a positive constant ε = ε(X, n) with following significance. Suppose that there exist flat connection on P and all flat connections are non-degenerate in the sense of Definition 3.6. If the pair (A, φ) is a C ∞ -solutions of complex-flat connection over X and the L2 -norm of extra field φ obeying kφkL2 (X) ≤ ε then φ is vanish and A is a flat connection. In particular the moduli space of solutions for complex flat connections is non-connected. The organization of this paper is as follows. In section 2, at first we establish our notation and recall some basic definitions in differential geometry. Next, we recall some identities and some estimates for the solutions of complex Yang-Mills equations which proved by Gagliardo-Uhlenbeck [9]. At last, we recall energy gap result of pure YangMills equation due to Feehan [8]. Since the Theorem 2.6 plays an essential role in our proof of our main result, we include more detail concerning its proof by another way. In section, we define the least eigenvalue λ(A) of dA d∗A + d∗A dA |Ω1 (X,gP ) with respect to connection A. We extend the idea of Feehan’s [6] to prove that λ(A) has a lower bounded that is uniform with respect to [A] obeying kFA kLp (X) ≤ ε for a small enough ε = ε(X, n, p) under some conditions for g, G, P , and X. We conclude in Section 4 with the proofs of Theorem 1.1 and Corollary 1.2.

2 Fundamental preliminaries We shall generally adhere to the now standard gauge-theory conventions and notation of Donaldson and Kronheimer [5] and Feehan [8]. Throughout our article, G denotes a compact Lie group and P a smooth principal G-bundle over a compact Riemannnian manifold X of dimension n ≥ 2 and endowed with Riemannian metric g, gP denote the adjoint bundle of P , endowed with a G-invariant inner product and Ωp (X, gP ) denote the smooth p-forms with values in gP . Given a connection on P , we denote by ∇A the corresponding covariant derivative on Ω∗ (X, gP ) induced by A and the Levi-Civita connection of X. Let dA denote the exterior derivative associated to ∇A . For u ∈ Lp (X, gP ), where 1 ≤ p < ∞ and k is an integer, we denote k Z X 1/p kukLpk,A(X) := |∇jA u|p dvolg , j=0

X

where ∇jA := ∇A ◦ . . . ◦ ∇A (repeated j times for j ≥ 0). For p = ∞, we denote kuk

L∞ k,A (X)

:=

k X j=0

ess sup |∇jA u|. X

Teng Huang

4

2.1 Identities for the solutions In this section, we recall some basic identities that are obeyed by solutions to complex Yang-Mills connections. A nice discussion of there identities can be found in [9]. Theorem 2.1. (Weitezenböck formula) d∗A dA + dA d∗A = ∇∗A ∇A + Ric(·) + ∗[∗FA , ·] on Ω1 (X, gP )

(2.1)

where Ric is the Ricci tensor.

√ Proposition 2.2. ([9] Theorem 4.3) If dA + −1φ is a solution of the complex Yang-Mills equations, then ∇∗A ∇A φ + Ric ◦ φ + ∗[∗(φ ∧ φ), φ] = 0. (2.2) By integrating 2.2 over X, we have a identity k∇A φk2L2 (X) + hRic ◦ φ, φiL2(X) + 2kφ ∧ φk2L2 (X) = 0.

(2.3)

Then Gagliardo-Uhlenbeck gives a following vanish result about the extra fields. Corollary 2.3. ([9] Corollary 4.5) If X is a compact manifold with positive Ricci curvature, solutions of the complex Yang-Mills equations reduce to solutions of the pure Yang-Mills equations with φ = 0. √ Proposition 2.4. (Energy Identity) If dA + −1φ is a solution of the complex Yang-Mills equations, then Y MC (A, φ) = kFA k2L2 (X) − kφ ∧ φk2L2 (X) Proof. Under the moment condition d∗A φ = 0, the complex Yang-Mills functional is written as: Z Y MC (A, φ) = (|FA − φ ∧ φ|2 + |dA φ|2 + |d∗A φ|) ZX = |FA |2 + |φ ∧ φ|2 − 2hFA , φ ∧ φi + |dA φ|2 + |d∗A φ| ZX = |FA |2 + |φ ∧ φ|2 + |∇A φ|2 + hRic ◦ φ, φi X

= kFA k2L2 (X) − kφ ∧ φk2L2 (X) .

The last identity, we using the Equation (2.3).

As a application of the maximum principle, Gagliardo-Uhlenbeck obtain a L∞ priori estimate for the extra fields. Theorem 2.5. ([9] Corollary 4.6) Let G be a compact Lie group, P be a G-bundle over a closed, smooth manifold X of dimension n ≥ 2 and endowed with a smooth Riemannian metric, g. Then there is a constant, C = C(X, g, n), with the following significance. If (A, φ) is a smooth solution of complex Yang-Mills connection, then kφkL∞ (X) ≤ CkφkL2 (X) .


5

2.2 Energy gap for Yang-Mills connections We recall the energy gap result of Yang-Mills connection as following Theorem 2.6. Let X be a compact Riemannian manifold of dimension n ≥ 2 with smooth Riemannan metric g, P be a G-bundle over X. Then any Yang-Mills connection A over X with compact Lie group G is either satisfies Z |FA |2 dvolg ≥ C0 X

for a constant C0 > 0 depending only on X, n, G or the connection A is flat. The Theorem 2.6 had been proved by Feehan ([8] Theorem 1.1). In [11], the author re-proof the energy gap theorem of Yang-Mills connection without using the LojasiewiczSimon gradient inequality. Here, we give a proof in detail for the readers convenience. We recall a priori estimate for the curvature of a Yang-Mills connection over a closed manifold. Theorem 2.7. ([16] Theorem 3.5) Let X be a compact manifold of dimension n ≥ 2 and endowed with a Riemannian metric g, let A be a smooth Yang-Mills connection with respect to the metric g on a smooth G-bundle P over X. Then there exist constants ε = ε(X, n, g) > 0 and C = C(X, n, g) with the following significance. If the curvature FA obeying kFA kL2 (X) ≤ ε, then kFA kL∞ (X) ≤ CkFA kL2 (X) . Proof. For small enough ε, from Theorem 2.7, we obtain kFA kLp (X) ≤ CkFA kL∞ (X) ≤ CkFA kL2 (X) , f or 2p > n. We choose ε sufficiently small such that kFA kLp (X) satisfies the hypothesis of Theorem 3.10. Hence there exist a flat connection Γ on P , and a gauge transformation g such that kg ∗ (A) − ΓkLp1 (X) ≤ C(p)kFA kLp (X) f or 2p ≥ n, and d∗Γ (g ∗(A) − Γ) = 0. We denote A˜ := g ∗ (A) and a := g ∗ (A) − Γ, the curvature of A˜ is FA˜ = dΓ a + a ∧ a.

Teng Huang

6 The connection A˜ also satisfies Yang-Mills equation 0 = d∗A˜ FA˜ .

(2.4)

Hence taking the L2 -inner product of (2.4) with a, we obtain 0 = (d∗A˜ FA˜, a)L2 (X) = (FA˜, dA˜ a)L2 (X) = (FA˜, dΓ a + 2a ∧ a)L2 (X) = (FA˜, FA˜ + a ∧ a)L2 (X) .

Then we get kFA kL2 (X) = kFA˜ k2L2 (X)

= −(FA˜, a ∧ a)L2 (X)

≤ kFA˜kL2 (X) ka ∧ akL2 (X) = kFA kL2 (X) ka ∧ akL2 (X)

where we using the fact |Fg∗ (A) | = |FA | since Fg∗ (A) = g ◦ FA ◦ g −1 . If n ≥ 4: ka ∧ akL2 (X) ≤ Ckak2L4 (X)

≤ Ckak2Ln (X)

≤ Ckak2 n2 ≤

L1 (X) CkFA k2Ln (X)

≤ CkFA k2L∞ (X)

≤ CkFA k2L2 (X) . n

where we have applied the Sobolev embedding L12 ֒→ Ln . If n = 2, 3, ka ∧ akL2 (X) ≤ Ckak2L4 (X)

≤ CkFA k2L4 (X)

≤ CkFA k2L∞ (X)

≤ CkFA k2L2 (X) . By combining the preceding inequalities we find that kFA k2L2 (X) ≤ CkFA k3L2 (X) , We can choose kFA kL2 (X) sufficiently small to such that CkFA kL2 (X) < 1, hence FA ≡ 0. Then we complete the proof.


7

3 Eigenvalue bounds for Laplacian ∆A In this section, at first, we recall some background material concerning flat connections on principal G-bundle that will be useful in the sequel. Next, we shows that the least eigenvalue λ(Γ) of d∗Γ dΓ + dΓ d∗Γ has a positive lower bound λ that is uniform with respect to [Γ] ∈ M(P ) under the given conditions on X and P . The way is similar to Feehan +,∗g has a positive lower bound µ0 that is [7] proved the least eigenvalue µg (A) of d+,g A dA +,g uniform with respect to [A] ∈ B(P, g) obeying kFA kL2 ≤ ε, for a small enough ε and under the given sets of conditions on g, G, P and X.

3.1 Continuity for the least eigenvalue of ∆A Returning to the setting of connections on a principal G-bundle, P , over a real manifold, X, we recall the equivalent characterizations of flat bundles ([12] Section 1.2), that is, bundles admitting a flat connection. Proposition 3.1. ([12] Proposition 1.2.6) For a smooth principal G-bundle P over a smooth manifold, X, the following conditions are equivalent: (1) P admits a flat structure, (2) P admits a flat connection, (3) P is defined by a representation π1 (X) → G. We note [5] Proposition 2.2.3 that the gauge-equivalence classes of flat G-connections over a connected manifold, X, are in one-to-one correspondence with the conjugacy classes of representations π1 (X) → G. We denote M(P ) := {Γ : FΓ = 0}/GP , is the moduli space of gauge-equivalence class [Γ] of flat connection Γ on P . From [17, 19], we know Proposition 3.2. Let G be a compact Lie group, P be a G-bundle over a closed, smooth manifold X of dimension n ≥ 2 and endowed with a smooth Riemannian metric, g. Then the moduli space M(P ) is compact. The definition of the least eigenvalue of ∆A on L2 (X, Ω1 (gP )) as follow is similar to the Definition 3.1 on [15]. Definition 3.3. (Least eigenvalue of ∆A ) Let G be a compact Lie group, P be a Gbundle over a closed, smooth manifold X of dimension n ≥ 2 and endowed with a smooth Riemannian metric, g. Let A be a connection of Sobolev class L21 on P . The least eigenvalue of ∆A on L2 (X, Ω1 (gP )) is λ(A) :=

inf

v∈Ω1 (gP )\{0}

h∆A v, viL2 . kvk2

(3.1)

Teng Huang

8

The method of prove the continuity of the least eigenvalue of ∆A with respect to the +,∗ connection is similar to Feehan prove the continuity of the least eigenvalue of d+ A dA with respect to the connection in [6, 7]. n We give a priori estimate for v ∈ Ω1 (X, gP ) when the curvature FA with L 2 -norm sufficiently small. Lemma 3.4. Let G be a compact Lie group, P be a G-bundle over a closed, smooth manifold X of dimension n ≥ 2 and endowed with a smooth Riemannian metric, g. Then there are positive constants, c = c(X, g) and ε = ε(X, g) ∈ (0, 1], with the following significance. If A is a connection on P over X such that kFA kLn/2 (X) ≤ ε,

(3.2)

kvk2L2 (X) ≤ c(kdA vk2L2 (X) + kd∗A vk2L2 (X) + kvk2L2 (X) ).

(3.3)

and v ∈ Ω1 (X, gP ), then 1

Proof. The Weitzenböck formula for v ∈ Ω1 (X, gP ), namely, (dA d∗A + d∗A dA )v = ∇∗A ∇A v + Ric ◦ v + ∗[∗FA , v]. Hence k∇A vk2L2 (X) ≤ kd∗A vk2L2 (X) + kdA vk2L2 (X) + ckvk2L2 (X) + |h∗[∗FA , v], viL2(X) |, where c = c(g). By Hölder inequality, we see that |h∗[∗FA , v], viL2(X) | ≤ kFA kLn/2 (X) kvk2L2n/(n−2) (X)

(3.4)

≤ ckFA kLn/2 (X) kvk2L2 (X) , 1

for some c = c(g). Combining the preceding inequalities and Kato inequality ∇|v| ≤ |∇A v| yields kvk2L2 (X) ≤ (k∇A vk2L2 (X) + kvk2L2 (X) ) 1

≤ kd∗A vk2L2 (X) + kdA vk2L2 (X) + (c + 1)kvkL2 (X) + ckFA kLn/2 (X) kvk2L2 (X) . 1

for some c = c(g). Provided ckFA kLn/2 (X) ≤ 1/2, rearrangements gives (3.3). Following the idea of [6] Lemma 35.12, we also have a useful lemma as follow. n

Lemma 3.5. (L12 continuity of least eigenvalue of ∆A with respect to the connection) Let G be a compact Lie group, P be a G-bundle over a closed, smooth manifold X of dimension n ≥ 2 and endowed with a smooth Riemannian metric, g. Then there are n/2 positive constants, c = c(g) and ε = ε(g), with the following significance. If A0 is an L1


9 n/2

connection on P that obeys the curvature bounded (3.2) and A is an L1 P such that kA − A0 kLn/2 (X) ≤ ε

connections on

1,A0

then, we denote a := A − A0 , (1 − ckak2Ln (X) )λ(A0 ) − ckak2Ln (X) ≤ λ(A) ≤ (1 − ckak2Ln (X) )−1 (λ(A0 ) + ckak2Ln (X) ). Proof. For convenience, write a := A − A0 ∈ Ln (X, Ω1 ⊗ gP ). For v ∈ L21 (X, Ω1 ⊗ gP ), we have dA v = dA0 v + [a, v] and kdA vk2L2 (X) = kdA0 v + [a, v]k2L2 (X)

≥ kdA0 vk2L2 (X) − k[a, v]k2L2 (X)

≥ kdA0 vk2L2 (X) − 2kakLn (X) kvk2L2n/(n−2) (X)

≥ kdA0 vk2L2 (X) − 2c1 kakLn (X) kvk2L2

1,A0 (X)

,

where c1 = c1 (g) is the Sobolev embedding constant for L21 ֒→ L2n/(n−2) . Similarly, d∗A v = d∗A0 v ± ∗[a, ∗v] and kd∗A vk2L2 (X) = kd∗A0 v ± ∗[a, ∗v]k2L2(X)

≥ kd∗A0 vk2L2 (X) − k[a, ∗v]k2L2 (X)

≥ kd∗A0 vk2L2 (X) − 2kak2Ln (X) kvk2L2n/(n−2) (X)

≥ kd∗A0 vk2L2 (X) − 2c1 kak2Ln (X) kvk2L2

1,A0 (X)

,

Applying the a priori estimate (3.3) for kvkL21 (X) from Lemma 3.4, with c = c(g) and smooth enough ε = ε(g), yields kvk2L2 (X) ≤ c(kdA0 vk2L2 (X) + kd∗A0 vk2L2 (X) + kvk2L2 (X) ). 1

Combining the preceding inequalities gives kdA vk2L2 (X) + kd∗A vk2L2 (X) ≥ (kdA0 vk2L2 (X) + kd∗A0 vk2L2 (X) ) − 4cc1 kak2Ln (X) kvk2L2 (X) − 4c1 ckak2Ln (X) (kdA0 vk2L2 (X) + kd∗A0 vk2L2 (X) ).

Now take v to be an eigenvalue of ∆A with eigenvalue λ(A) and kvkL2 (X) = 1 and also suppose that kA − A0 kLn (X) is small enough that 4c1 ckak2Ln (X) ≤ 1/2. The preceding inequality then gives λ(A) ≥ (1 − 4c1 kak2Ln (X) )(kdA0 vk2L2 (X) + kd∗A0 vk2L2 (X) ) − 4c1 ckak2Ln (X) . Since kvkL2 (X) = 1, we have (kdA0 vk2L2 (X) + kd∗A0 vk2L2 (X) ) ≥ λ(A0 ), hence λ(A) ≥ (1 − 4c1 kak2Ln (X) )λ(A0 ) − 4c1 ckak2Ln (X) .

Teng Huang

10

To obtain the upper bounded for λ(A), observe that FA = FA0 + dA0 a + a ∧ a and thus kFA kLn/2 (X) ≤ kFA0 kLn/2 (X) + kdA0 akLn/2 (X) + kak2Ln (X) ≤ kFA0 kLn/2 (X) + c′ (1 + kakLn/2

1,A0 (X)

′

)kakLn/2

1,A0 (X)

′ 2

≤ (1 + c )ε + c ε . where c′ = c′ (g). Hence A obeys the condition (3.2), for a constant ε′ := (1 + c′ )ε + c′ ε2 (for small enough ε). Therefore, exchange the roles of A and A0 yields the inequality, λ(A0 ) ≥ (1 − 4c1 kak2Ln (X) )λ(A) − 4c1 ckak2Ln (X) .

3.2 Uniform positive lower bound for the least eigenvalue of ∆A Our results in Subsection 3.1 assure the continuity of λ(·) with respect to the Uhlenbeck topology and they will be applied here. Before doing this, we recall the Definition 3.6. ([4] Definition 2.4) Let G be a compact Lie group, P be a G-bundle over a closed, smooth manifold X of dimension n ≥ 2 and endowed with a smooth Riemannian metric, g. The flat connection, Γ, is non − degenerate if ker ∆Γ |Ω1 (X,gP ) = {0}. If the Ricci tensor of Riemannian metric g on X is positive in the sense of Ric > 0 on X n

Then Proposition 3.7 as follow shows that if the curvature FA with L 2 -norm sufficiently small, then the least eigenvalues function λ[·] : M(P ) → [0, ∞) admits a uniform positive lower bound, λ0 = λ(g), λ(A) ≥ λ0 , ∀[A] ∈ M(P ). Proposition 3.7. Let X be a compact manifold of dimension n ≥ 2 with Riemannian metric g, P be a principal G-bundle with G is a compact Lie group. Then there is a positive constant ε with the following significance. Suppose the Ricci tensor of metric g is positive. If A is a smooth connection on P and the curvature FA obeying kFA kL n2 (X) ≤ ε and λ(A) is defined as in (3.6), then λ(A) ≥ inf Ric(x) > 0. x∈X


11

Proof. The Bochner-Weitenb”ock formula gives Z 2 ∗ 2 2 kdA vkL2 (X) + kdA vkL2 (X) = k∇A vkL2 (X) + hRic ◦ v, vi + 2hFA , v ∧ vi ≥ inf

x∈X

X 2 Ric(x)kvkL2 (X)

− kFA kL n2 (X) kvkL2n/n−2 (X)

≥ inf Ric(x)kvk2L2 (X) − cε(kdA vk2L2 (X) + kd∗A vk2L2 (X) + kvkL2 (X) ) x∈X

≥ ( inf Ric(x) − cε)kvk2L2(X) − cε(kdA vk2L2 (X) + kd∗A vk2L2 (X) ) x∈X

Then we obtain kdA vk2L2 (X) + kd∗A vk2L2 (X) ≥

1 ( inf Ric(x) − cε)kvk2L2 (X) . 1 − cε x∈X

Let ε → 0+ and choose v such that kdA vk2L2 (X) + kd∗A vk2L2 (X) = λ(A)kvk2L2(X) . Then we complete the proof. Remark 3.8. In fact, the extra fields are vanish under the condition the positive Ricci tensor of a Riemannian metric of X (See Corollary 2.3). In general Riemannin metric, we do not know that ker ∆Γ |Ω1 (X,gP ) = {0}, where Γ is any flat connection on P , unless we assume a topological hypothesis for X, such as π1 (X) = {1}, so P ∼ = X × G if only if P is flat ([5] Theorem 2.2.1). In this case, Γ is gauge-equivalent to the product connection and ker ∆Γ |Ω1 (X,gP ) ∼ = H 1 (X, R), so the hypothesis for X ensure the kernel vanishing. We then have the Proposition 3.9. Let G be a compact Lie group, P be a G-bundle over a closed, smooth manifold X of dimension n ≥ 2 and endowed with a smooth Riemannian metric, g. Assume all flat connections are non-degenerate, Then there is constant λ > 0, with the following significance. If Γ is a flat connection, then λ(Γ) ≥ λ, where λ(Γ) is as in Definition 3.3. Proof. The conclusion is a consequence of the fact that M(P ) is compact, λ[·] : M(P ) ∋ [Γ] → λ(Γ) ∈ [0, ∞), to M(P ) is continuous by Lemma 3.5, the fact that λ(Γ) > 0 for [Γ] ∈ M(P ). We consider the open subset of the space B(P, g) defined by Bε (P, g) := {[A] ∈ B(P, g) : kFA kLp (X) ≤ ε}, where p is a constant such that 2p > n. At first, we review a key result due to Uhlenbeck for the connections with Lp -small curvature (2p > n)[18].

Teng Huang

12

Theorem 3.10. ([18] Corollary 4.3) Let X be a closed, smooth manifold of dimension n ≥ 2 and endowed with a Riemannian metric, g, and G be a compact Lie group, and 2p > n. Then there are constants, ε = ε(n, g, G, p) ∈ (0, 1] and C = C(n, g, G, p) ∈ [1, ∞), with the following significance. Let A be a Lp1 connection on a principal G-bundle P over X. If kFA kLp (X) ≤ ε, then there exist a flat connection, Γ, on P and a gauge transformation g ∈ Lp2 (X) such that (1) d∗Γ g ∗(A) − Γ = 0 on X, (2) kg ∗(A) − ΓkLp1,Γ ≤ CkFA kLp (X) and (3) kg ∗(A) − ΓkLn/2 ≤ CkFA kLn/2 (X) . 1,Γ

Then we have the Theorem 3.11. Let G be a compact Lie group, P be a G-bundle over a closed, smooth manifold X of dimension n ≥ 5 and endowed with a smooth Riemannian metric, g, and 2p > n. Assume all flat connections are non-degenerate, then there are positive constants, λ = λ(X, g, n) and ε = ε(X, g, n) with the following significance. If A is an Lp1 connection on P such that kFA kLp (X) ≤ ε, and λ(A) is as in Definition 3.3, then λ(A) ≥ λ/2, where λ is the constant in Proposition 3.9. Proof. For a connection A of class Lp1 on P with kFA kLp (X) ≤ ε, where ε is as in the hypotheses of Theorem 3.10. Then there exist a flat connection, Γ, on P and a gauge transformation g ∈ Lp2 (X) such that kg ∗(A) − ΓkLp1,Γ (X) ≤ CkFA kLp (X) . For kFA kLp (X) sufficiently small, we can apply Lemma 3.5 for A and Γ to obtain λ(A) ≥ (1 − ckg ∗ (A) − ΓkLn (X) )λ(Γ) − ckg ∗ (A) − ΓkLn (X)

≥ (1 − ckg ∗ (A) − ΓkLp1,Γ (X) )λ(Γ) − ckg ∗(A) − ΓkLp1,Γ (X) ≥ λ − CckFA kLp (X) (1 + λ),

We choose kFA kLp (X) sufficiently small such that |FA kLp (X) ≤ λ(A) ≥ λ/2.

λ , 2Cc(1+λ)

then we have


13

3.3 The case of dimensional four In this section, we will show a similar theorem to Theorem 3.11 in the case of dimension four, but we only need suppose FA with L2 -norm sufficiently small. At fist, we recall a priori Lp estimate for the connection Laplace operator which proved by Feehan. Lemma 3.12. ([6] Lemma 35.5) Let X be a smooth manifold X of dimension n ≥ 4 and endowed with a smooth Riemannian metric, g and q ∈ (n, ∞). Then there are positive constant c = c(g, p) with the following significance. Let r ∈ ( n3 , n2 ) be defined by 1/r = 2/n + 1/q. Let A is a C ∞ connection on a vector bundle E over X. If v ∈ C ∞ (X, E), then then kvkLq (X) ≤ c(k∇∗A ∇A vkLr (X) + kvkLr (X) ). (3.5) We now apply Lemma to E = Ω1 ⊗ gP , then we give a priori Lp -estimate for ∆A . Lemma 3.13. Let G be a compact Lie group, P be a G-bundle over a closed, smooth manifold X of dimension n ≥ 4 and endowed with a smooth Riemannian metric, g and q ∈ (n, ∞). Then there are positive constants, c = c(g) and ε = ε(g), with the following significance. Let r ∈ ( n3 , n2 ) defined by 1/r = 2/n + 1/q. Let A is a C ∞ connection on P that each obeys the curvature bounded (3.2). If v ∈ Ω1 (X, gP ), then then kvkLq (X) ≤ c(k∆A vkLr (X) + kvkLr (X) ).

(3.6)

Proof. For v ∈ Ω1 (X, gP ), from the Weitzenböck formula, we have ∆A v = ∇∗A ∇A v + Ric · v + {v, FA }. Hence k∆A vkLr (X) ≤ k∇∗A ∇A vkLr (X) + ckvkLr (X) + k{v, FA }kLr (X) , and some c = c(g). Since 1/r = 2/n + 1/q by hypothesis, we see that k{v, FA }kLr (X) ≤ ckFA kLn/2 (X) kvkLq (X) , for some c = c(g). Combining the preceding inequalities with the Equation (3.5) yields kvkLq (X) ≤ k∇∗A ∇A vkLr (X) + ckvkLr (X) + ckFA kLn/2 (X) kvkLq (X) , for some c = c(g, q). Provided ckFA kLn/2 (X) ≤ 1/2, rearrangement gives (3.5). Hence, following the idea of [8] Lemma 35.13, we also have a useful lemma as follow. Lemma 3.14. (Lp continuity of least eigenvalue of ∆A with respect to the connection for 2 ≤ p < 4) Let G be a compact Lie group, P be a G-bundle over a closed, smooth fourmanifold X and endowed with a smooth Riemannian metric, g. Then there are positive

Teng Huang

14

constants, c = c(g) and ε = ε(g), with the following significance. If A0 and A are C ∞ connections on P that each obeys the curvature bounded (3.2), we denote a := A − A0 , then λ(A) satisfies λ(A) ≥ λ(A0 ) − c0 (1 + λ2 (A))kak2Lp (X) , and λ(A) ≤ λ(A0 ) + c0 (1 + λ2 (A0 ))kak2Lp (X) . Proof. For convenience, write a := A − A0 ∈ Lp (X, Ω1 ⊗ gP ). Define q ∈ [2, ∞) by 1/2 = 1/p + 1/q and consider v ∈ L21 (X, Ω1 ⊗ gP ). We used dA v = dA0 v + [a, v] and the Höler inequalities to give kdA vk2L2 (X) = kdA0 v + [a, v]k2L2 (X)

≥ kdA0 vk2L2 (X) − k[a, v]k2L2 (X)

≥ kdA0 vk2L2 (X) − 2kak2Lp (X) kvk2Lq (X) . Similarly, d∗A v = d∗A0 v ± ∗[a, ∗v] and kd∗A vk2L2 (X) = kd∗A0 v ± ∗[a, ∗v]k2L2 (X)

≥ kdA0 vk2L2 (X) − k[a, v]k2L2 (X)

≥ kdA0 vk2L2 (X) − 2kak2Lp (X) kvk2Lq (X) . For p > 4, we have 2 ≤ q < n and kvkLq (X) ≤ (vol(X))1/q−1/4 kvkL4 (X) , while for 2 < p ≤ 4, we have 4 ≤ q < ∞. Therefore, it suffices to consider the case 4 ≤ q < ∞. Applying the priori estimate (3.5) and r ∈ (4/3, 2) defined by 1/r = 1/2 + 1/q, yields kvk2Lq (X) ≤ c(k∆A vk2L2 (X) + kvk2Lr (X) ). By combining the preceding inequalities we get that kdA vk2L2 (X) + kd∗A vk2L2 (X)

≥ kdA0 vk2L2 (X) + kd∗A0 vk2L2 (X) − 2c1 kak2Lp (X) (k∆A vk2Lr (X) + kvk2Lr (X) )

≥ kdA0 vk2L2 (X) + kd∗A0 vk2L2 (X) − 2c0 kak2Lp (X) (k∆A vkL2 (X) + kvk2L2 (X) ), for c0 = c0 (p, q) = 2c1 vol(X)2/q , using the fact that kvkLr (X) ≤ vol(X)1/q kvkL2 (X) for r ∈ (4/3, 2) and 1/r = 1/2 + 1/q. By taking v ∈ L21 (X, Ω1 ⊗ gP ) to be an eigenvector of ∆A with eigenvalue λ(A) such that kvkL2 (X) = 1 and noting that k∆A vkL2 (X) = λ(A). We obtain the bound λ(A) ≥ kdA0 vk2L2 (X) + kd∗A0 vk2L2 (X) − c0 (1 + λ2 (A))kak2Lp (X) . But kdA0 vk2L2 (X) + kd∗A0 vk2L2 (X) ≥ λ(A0 ) and thus we have the inequality λ(A) ≥ λ(A0 ) − c0 (1 + λ2 (A))kak2Lp (X) .


15

Interchanging the roles A and A0 in the preceding derivation yields λ(A) ≤ λ(A0 ) + c0 (1 + λ2 (A0 ))kak2Lp (X) .

We consider a sequence of C ∞ connection on E such that sup kFAi kL2 (X) < ∞. We denote Σ = {x ∈ X : lim lim sup kFAi k2L2 (Br )(x) ≥ ε˜}. rց0

→∞

the constant ε˜ ∈ (0, 1] as in [14] Theorem 3.2. We can see Σ is a finite points {x1 , . . . , xL } in X. In our artilce, we consider the open subset of the space B(P, g) defined by Bε = {[A] ∈ B(P, g) : kFA kL2 (X) ≤ ε}. We can choose ε sufficiently small such that any sequence {Ai }i∈N has the empty set Σ. From Sedlacek’s [14] Theorem 3.1 and [6] Theorem 35.15, we have Theorem 3.15. Let G be a compact Lie group and P be a principal G-bundle over a close, smooth four-dimensional X with Riemannian metirc, g. If {Ai }i∈N is a sequence of C ∞ connection on P such that kFAi kL2 (X) ≤ ε, there are (a) geodesic ball {Bα }α∈N such that ∪α∈N Bα = X; (b) the section σα,i : Bα → P ; (c) local connection one-forms, ∗ aα,i := σα,i Ai ∈ L21,Γ (Bα , Ω1 ⊗ g);and (d) transition functions gαβ,i ∈ L41,Γ (Bα ∩ Bβ ; G), such that the following hold for each α, β ∈ N, (1) d∗Γ aα,m = 0, for all m sufficiently large; (2) d∗Γ aα = 0, (3) gαβ,m ⇀ gαβ weakly in L41,Γ (Bαβ ; G), (4) Fα,m ⇀ Fα weakly in L2 (Bα , Ω2 ⊗ g) and (5) aα,m ⇀ aα weakly in L21 (Bα , Ω1 ⊗ g). ∗ where Fα,i = daα,i + [aα,i , aα,i ] = σα,i FAi and Fα := daα + [aα , aα ], and d∗ is the formal adjoint of d with respect to the flat metric defined by a choice of geodesic normal coordinates on Bα . From Sedlacek’s [14] Theorem 4.3 and [6] Theorem 35.17, we have Theorem 3.16. Let G be a compact Lie group and P be a principal G-bundle over a close, smooth four-dimensional X with Riemannian metirc, g. If {Ai }i∈N is a sequence of C ∞ connection on P , in the sense that Y M(Ai ) ց 0, as i → ∞, then the following hold. For each α, β ∈ N, (1) aα ∈ C ∞ (Bα , Ω1 (gP )) and a solution to the flat connection,

Teng Huang

16

T (2) gαβ ∈ C ∞ (Bα Bβ ; G), (3) The sequence, {aα }α∈N and {gαβ }α,β∈N define a C ∞ connection A∞ on a principal G-bundle P∞ over X. Then, We have Corollary 3.17. Assume the hypotheses of Theorem 3.16, then lim λ(Ai ) = λ(A∞ ),

i→∞

where λ(Γ) is as in Definition 3.3. ∗ Proof. From Theorem 3.16, σα,i Ai ⇀ σA∞ weakly in L21 (Bα , Ω1 ⊗ gP ). For L21 ⋐ Lp , (2 ≤ p < 4), hence ∗ kσα,i Ai − σ ∗ A∞ kLp (Bα ) → 0 strongly in Lp (Bα , Ω1 ⊗ gP ) as i → ∞,

for sequences of local sections {σα,m }m∈N of P ↾ Bα and a local section σα of P∞ ↾ Bα and X ∗ kAi − A∞ kLp (X) ≤ kσα,m Am − σ ∗ A∞ kLp (Bα ) . a

Hence from Lemma 3.14, we have lim λ(Ai ) ≥ λ(A∞ ) − c0 (1 + lim λ2 (Ai )) lim kai kLp (X) ,

i→∞

i→∞

i→∞

and lim λ(Ai ) ≤ λ(A∞ ) + c0 (1 + λ2 (A∞ )) lim kai kLp (X) ,

i→∞

i→∞

Then we obtain lim λ(Ai ) = λ(A∞ ).

i→∞

We then have Theorem 3.18. Let G be a compact Lie group, P be a G-bundle over a closed, smooth 4manifold and endowed with a smooth Riemannian metric, g. Assume all flat connections are non-degenerate, then there exist a positive constant ε = ε(P, g) ∈ (0, 1] with the following siginificance. If A is a C ∞ connection on P such that kFA kL2 (X) ≤ ε, and λ(A) is as in Definition 3.3, then λ(A) ≥

λ , 2

where λ is the positive constant in Propostion 3.9.


17

Proof. Suppose that the the constant λ ∈ (0, 1] does not exist. We may then choose a sequence {Ai }i∈N of connection on P such that kFAi kL2 (X) → 0 and λ(Ai ) → 0 as i → ∞. Since limi→∞ λ(Ai ) = λ(A∞ ) and λ(A∞ ) > 0 by A∞ is a flat connection, then it is contradict to our initial assumption regarding the sequence {Ai }i∈N .

4 Proof Main Theorem 1.1 Now, we begin to proof the energy gap result of complex Yang-Mills equations. At first, we prove the complex Yang-Mills equations will be reduce to pure Yang-Mills equation under the certain conditions for A, g, G, P , and X. Proposition 4.1. Let X be a compact Riemannian manifold of dimension n ≥ 2 with smooth Riemannan metric g, P be a G-bundle over X, let p ∈ (1, ∞) be such that 2p > n and in the case n = 4 assume p ≥ 2. Then there exist a positive constant ε = ε(X, n, p) with following significance. Suppose all flat connections on P are non-degenerate. If the pair (A, φ) is a C ∞ -solutions of complex-Yang-Mills equations over X and the curvature FA of connection A obeying kFA kLp (X) ≤ ε then φ is vanish. Proof. We can choose ε sufficient small and the curvature FA obeying kFA kLp (X) ≤ ε where p and ε are the constants in Theorem 3.11 and Theorem 3.18. Then there exist a positive constant λ such that kdA vk2L2 (X) + kd∗A vk2L2 (X) ≥ λ/2kvk2L2 (X) , ∀v ∈ Ω1 (X, gP ). We have a identity for the solution of complex-Yang-Mills equations: kdA φk2L2 (X) + kd∗A φk2L2 (X) + 2kφ ∧ φk2L2 (X) − 2hFA , φ ∧ φiL2 (X) = 0. Hence we have λ/2kφk2L2(X) ≤ kdA φk2L2 (X) + kd∗A φk2L2 (X) ≤ |hFA , φ ∧ φiL2 (X) | ≤ kFA kLn/2 (X) kφk2

≤

2n

L n−2 (X) CkFA kLp (X) kφk2L2 (X) .

We can choose kFA kLp (X) ≤ ε sufficiently small such that Cε ≤ λ/4, then the extra field φ is vanish.

Teng Huang

18 Proof Theorem 1.1

Proof. From Proposition 4.1 the complex-Yang-Mills equations reduce to the pure YangMills equation d∗A FA = 0 and the curvature obeying kFA kLp (X) ≤ ε, then by the energy gap of Yang-Mills connection, we obtain the connection A is flat. Proof Corollary 1.2 Proof. For a smooth solution (A, φ) of complex flat connection, from the identity FA = φ ∧ φ and we applying Theorem 2.5 to obtain kFA kLp (X) ≤ kφ ∧ φkLp (X) ≤ Ckφk2L2 (X) , where C = C(X, n, p). We can choose kφkL2 (X) sufficiently small such that kFA kLp (X) ≤ ε, where ε is the constant in Theorem 1.1. Then from Theorem 1.1, we can obtain φ is vanish and A is a flat connection. It is easy to see the map (A, φ) 7→ kφkL2 (X) is continuous, then the moduli space of complex flat connections is non-connected.

Acknowledgements I would like to thank Karen Uhlenbeck and Michael Gagliardo for helpful comments regarding their article [9] and Paul Feehan for helpful comments regarding his article [6, 8]. This work is partially supported by Wu Wen-Tsun Key Laboratory of Mathematics of Chinese Academy of Sciences at USTC.

References [1] Bourguignon J-P., Lawson H. B., Stability and isolation phenomena for Yang-Mills fields, Comm. Math. Phys. 79 (1981) 189–230. [2] Stability and gap phenomena for Yang-Mills fields, PANS 76 (1979) 1550–1553. [3] Dodziuk J., Min-Oo M., An L2 -isolation theorem for Yang-Mills fields over complete manifolds, Compositio Math. 47 (1982) 165–169. [4] Donaldson S. K., Floer homology groups in Yang-Mills theory, Vol. 147, Cambridge University Press, Cambridge, 2002. [5] Donaldson S. K., Kronheimer P. B., The geometry of four-manifolds, Oxford University Press, 1990. [6] Feehan P. M. N., Global existence and convergence of smooth solutions to Yang-Mills gradient flow over compact four-manifolds, arxiv:1409.1525. [7] Feehan P. M. N., Energy gap for Yang-Mills connections, I: Four-dimensional closed Riemannian manifolds, Adv. Math. 296 (2016), 55–84. [8] Feehan P. M. N., Energy gap for Yang-Mills connections, II: Arbitrary closed Riemannian manifolds, Adv. Math. Doi.org/10.1016/j.aim.2017.03.023


19

[9] Gagliardo M., Uhlenbeck K. K., Geometric aspects of the Kapustin-Witten equations, J. Fixed Point Theory Appl. 11 (2012), 185–198. [10] Gerhardt C., An energy gap for Yang-Mills connections, Comm. Math. Phys. 298 (2010), 515–522. [11] Huang T., A proof on energy gap for Yang-Mills connection, arXiv:1704.02772 (2017). [12] Kobayashi S., Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, vol. 15, Princeton University Press, Princeton, NJ, 1987. [13] Min-Oo M., An L2 -isolation theorem for Yang-Mills fields, Compositio Math. 47 (1982), 153–163. [14] Sedlacek S., A direct method for minimizing the Yang-Mills functional over 4-manifolds, Comm. Math. Phys. 82 (1982), 515–527. [15] Taubes C. H., Self-dual Yang-Mills connections on non-self-dual 4-manifolds, J. Diff. Geom. 17 (1982), 139–170 [16] Uhlenbeck K. K., Removable singularites in Yang-Mills fields, Comm. Math. Phys. 83 (1982), 11–29. [17] Uhlenbeck K. K., Connctions with Lp bounds on curvature, Comm. Math. Phys. 83 (1982), 31–42. [18] Uhlenbeck K. K., The Chern classes of Sobolev connections, Comm. Maht. Phys. 101 (1985), 445– 457. [19] Wehrheim K., Uhlenbeck compactness. European Mathematical Society, (2004)