A connection is called a Yang–Mills connection if it is a critical point of the Yang–Mills functional

86(2010) 163-188

GEOMETRY OF MINIMAL ENERGY YANG–MILLS CONNECTIONS

Mark Stern

Abstract

We prove that energy minimizing Yang–Mills connections on compact homogeneous 4-manifolds are either instantons or split into a sum of instantons on passage to the adjoint bundle. We prove related results for Calabi–Yau 3-folds and for 3−dimensional manifolds of nonnegative Ricci curvature.

1. Introduction

Let G be a compact Lie group and E a principal G−bundle on a complete oriented Riemannian manifold, M. LetAdenote a connection onE and ∇A the associated covariant derivative on the adjoint bundle, ad(E). The Yang–Mills energy of Ais

Y M(A) :=kFAk²,

whereF_Adenotes the curvature ofA, and k · kdenotes theL₂ norm. A connection is called a Yang–Mills connection if it is a critical point of the Yang–Mills functional. A Yang–Mills connection, A, is calledstable if the second variation of the Yang–Mills functional is nonnegative atA.

It is called a local minimum of the Yang–Mills functional if it minimizes Y M(A) among all nearby connections. We call a connection abelian if its curvature takes values in an abelian subalgebra of the adjoint bundle.

In four dimensions,F_Adecomposes into its self-dual and anti-self-dual components,

F_A=F_A⁺+F_A⁻,

where F_A^± denotes the projection onto the±1 eigenspace of the Hodge star operator. A connection is called self-dual (respectively anti-self- dual) if F_A = F_A⁺ (respectively F_A = F_A⁻). A connection is called an instanton if it is either self-dual or anti-self-dual. On compact ori- ented 4-manifolds, an instanton is always an absolute minimizer of the Yang–Mills energy. Not all Yang–Mills connections are instantons. See [SSU] and [SS] for (unstable) examples of SU(2) Yang–Mills connec- tions on S⁴ which are neither self-dual nor anti-self-dual. Moreover,

The author was partially supported by NSF grant DMS-0504890.

Received 03/27/2009.

163

not all stable abelian Yang–Mills connections are instantons; otherwise every harmonic 2-form representing an integral cohomology class would be self-dual or anti-self-dual, which is clearly false.

This leads to the

Converse question: In four dimensions are local minima for the Yang–

Mills energy necessarily direct sums of instantons and abelian connec- tions?

Partial positive results for low rankGwere obtained by Bourguignon, Lawson, and Simons in [BLS] and [BL], where they use a variational argument to show that if G = SU(2) or SU(3), and M is a compact oriented 4-dimensional homogeneous space, then the connection is either an instanton or abelian. In this note, we settle this question on the (possibly reduced) adjoint bundle ofE, for any compact group,G, when M is a complete nonnegatively curved homogeneous manifold. We also give related results for oriented 3−manifolds and Calabi–Yau geometries in three complex dimensions.

Our main result is the following theorem.

Theorem 1.1. LetE be a principalG−bundle on a complete oriented nonnegatively curved homogeneous Riemannian 4-manifold, M. Let A be a finite energy, locally minimizing Yang–Mills connection on E. The adjoint bundle, ad(E), contains two ∇A-stable subbundles, k⁺ and k⁻, satisfying F_A^± is a section of Λ²T^∗M ⊗k^±,

[k⁺, k⁻] = 0,

and the curvature of k⁺ is self-dual and that ofk⁻ is anti-self-dual.

Our proof of Theorem 1.1 extends the variational argument of Bour- guignon, Lawson, and Simons. LetA_tbe a smooth family of connections on E with A₀ =A. The assumption that A is a local minimum of the Yang–Mills energy implies the variational inequality

(1.2) d²

dt²Y M(At)_|t=0 ≥0.

The proof of the theorem relies on choosing useful families of test con- nections with the difference,At−A, constructed fromFA. In [BL], the test connection At = A+tiXF_A⁺ was used, where iX denotes interior multiplication by the vector field X, andX runs over a basis of Killing vector fields. Our results rely on recognizing this variation as only the first term in an infinite family of related variations.

The curvature is the only natural object from which to construct test variations, but we need a map from 2−forms to 1−forms in order to create test variations from the curvature 2−form. In homogeneous geometries, interior product with Killing vector fields provides such a map. More generally, this suggests we seek new information on the

geometry of locally minimizing Yang–Mills connections in special ge- ometries where there exist natural maps

Φ : Λ²T^∗M⊗ad(E)→Λ¹T^∗M⊗ad(E).

When such Φ exist, we can consider variations with ^dA_dt(0) = Φ(F_A) and seek additional results. Covariant constant 3-forms induce natural maps from 2-forms to 1-forms. Hence, one expects new results for G₂ manifolds, Calabi-Yau 3 folds, and oriented 3-dimensional manifolds.

We note that for manifolds of dimension greater than 4, interesting sta- ble Yang–Mills connections need not always exist. For example, Simons (see [BL]) proved the nonexistence of nonflat stable Yang–Mills con- nections on Sⁿ,n > 4. This nonexistence result has subsequently been generalized in many directions, for example [KOT], [OP], [P],[Sh], and [X]. Our results below include no newexistence results for minimizing connections in higher dimensions.

On a Kahler m−fold with Kahler formω, the curvature decomposes as

F_A=F_A^2,0+F_A0^1,1+ 1

m(ΛF_A)ω+F_A^0,2,

where Λ denotes the adjoint of exterior multiplication byω, andF_A0^1,1= F_A^1,1−_m¹(ΛF_A)ω. Define the energyE^′(A) =kF_A^0,2k².

Theorem 1.3. Let Abe a G−connection on a bundle, E, on a com- plete Calabi Yau threefold, M. Assume that A is a stable critical point ofE^′. IfM is noncompact, assume also thatF_A^0,2∈L⁴. ThenF_A^0,2 is co- variant constant and takes values in an abelian subbundle ofad(E)⊗C. If Hol(M) =SU(3), then E is holomorphic.

Some of our results for 3-manifolds are presumably already known (see for example [JT, Chapter II, Corollary 2.3] for the case ofR³), but we include them here as they fall in the same family of techniques as the preceding results. We say an n−manifold haslocal flat factors if it is locally isometric to a product of an open subset in R^k,k≥1, and an (n−k)−manifold.

Theorem 1.4. Let E be a bundle on a complete 3-dimensional man- ifold, M, with nonnegative Ricci curvature. Let A be a stable finite energy Yang–Mills connection on E. If M is noncompact, assume also that FA∈L⁴. Then

∇AFA= 0.

Moreover, F_A takes values in a flat abelian subbundle of ad(E), and F_A= 0 unless M has local flat factors.

The additional assumption here that F_A ∈ L₄ if M is noncompact is easily established under additional geometric hypotheses of bounded geometry. (See Remark 2.18.) We remark that applying the preceding

theorem toS³ gives an analytic proof (see Section 4) of the triviality of π₂(G) for all compact Lie groupsG.

Acknowledgements. We wish to thank Savdeep Sethi for stimulating conversations and for his interest in this work. We thank Benoit Char- bonneau for helpful comments. We also thank the referee for his exten- sive comments, especially his suggested extension (Proposition 2.17) of our theorems to complete noncompact manifolds.

2. Preliminaries

Let M be a complete Riemannian manifold and E a principal G bundle over M, with G a compact Lie group. Let ad(E) denote the adjoint bundle of E, endowed with a G−invariant inner product. Let A^p(M, ad(E)) denote the smooth p−forms with values inad(E). Given a connection AonE, we denote by∇A the corresponding covariant de- rivative onA^∗(M, ad(E)) induced by Aand the Levi-Civita connection of M. Letd_A denote the exterior derivative associated to∇A.

We are interested in stable minima of the Yang–Mills energy Y M(A) =kF_Ak²,

where F_A denotes the curvature of A. Critical points of this energy satisfy the Yang–Mills equation

(2.1) d^∗_AF_A= 0,

where d^∗_A denotes the (L₂−)adjoint of d_A. In addition, all connections satisfy the Bianchi identity

(2.2) d_AF_A= 0.

If A_t is a smooth one parameter family of connections, then

(2.3) d

dtF_At =d_At(dA dt ).

More generally, ifψ∈A¹(M, ad(E)) then

(2.4) F_A+ψ =F_A+d_Aψ+ψ∧ψ.

Here we note that our convention on exterior products of ad(E) valued forms is normalized by

(dx^I ⊗v_I)∧(dx^J ⊗v_J) = 1

2(dx^I ∧dx^J)⊗[v_I, v_J].

As a notational convenience, we will often denote bye(w) exterior mul- tiplication on the left by a form w (possibly with ad(E) coefficients).

Its adjoint is denoted e^∗(w). Thus

e(w)h:=w∧h, and hf, e(w)hi=he^∗(w)f, hi.

If A minimizes the Yang–Mills energy, then of course it satisfies the inequality

(2.5) kFAk² ≤ kF_A+ψk²,

for all smooth compactly supported ψ. Replacing ψ by tψ in (2.5), using (2.4), and taking the limit as t→0 leads to the second variation inequality

(2.6) 0≤ kdAψk²+ 2hFA, ψ∧ψi.

When considering noncompact manifolds, we will consider variations withψnotcompactly supported. Let{yj}^∞_j=1be a sequence of functions, 0≤y_j ≤1, withlim_j→∞y_j = 1 pointwise and|dyj|uniformly bounded.

If we assume merely that ψ∈C¹∩L²∩L⁴, then replacingψby y_jψin (2.6) yields 0≤ kdAψk²+ 2hFA, ψ∧ψiupon passing to the limit. Hence we may apply this variational inequality to ψ∈C¹∩L²∩L⁴.

The ψ we will most often consider in (2.6) will be constructed from F_A and its derivatives. Hence we need to establish some standard es- timates for ∇^k_AF_A to ensure these test variations are in L² ∩L⁴. Let M be a complete noncompact homogeneous 4−manifold. Let A be a finite energy Yang–Mills connection on a G−bundle E over M. Let B_r(p) denote the ball of radius r about p. The following lemma is a specialization of [N, Lemma 3.1] to our context.

Lemma 2.7. There exist constants ǫ= ǫ(M, G) and C =C(M, G) such that if

r⁴⁻ⁿ Z

Br(p)

|F_A|²dv < ǫ, then sup

B_r/4(p)

|F_A|² ≤Cr⁻ⁿ Z

Br(p)

|F_A|²dv,

for r < ¹₂min{1,injectivity radius of M}, p∈M.

CoverM with a collection of balls,{Bs(pi)}^∞_i=1, of radiuss= ¹₈min{1, injectivity radius ofM}, and with a uniform bound on the number of balls containing any point ofM.

Corollary 2.8. For every δ > 0, sup_x∈B2s(pi)|F_A|(x)² < δ, for i sufficiently large.

Proof. AsAis finite energy, for eachδ >0, there existsN_δ so that we may remove a finite number of balls so that Cs⁻ⁿR

B_8s(pi)|FA|²dv < δ, fori > N_δ. Now Lemma 2.7 yields the desired bound. q.e.d.

We specialize the existence and properties of the Uhlenbeck gauge to our situation.

Lemma 2.9. (Uhlenbeck Gauge)[U1]. (See also[W, p.91 and p.145]).

LetAbe a connection on aG−bundle. Fixq > ⁿ₂. There exist constants

β(q) =β(M, G, q), b(q) =b(M, G, q), and Ck,q such that if (2.10)

Z

B2s(pi)

|FA|^qdv≤β(q),

then there exists a trivialization of E over B_2s(pi) in which the connec- tion form Aˆ satisfies

d^∗Aˆ= 0, and

kAkˆ _W1,q(Bs(pi)) ≤b(q)kF_Ak_L^q.

If we assume also thatAis a critical point of the Yang–Mills functional, then ∀k

(2.11) kAkˆ _Wk+1,q(Bs(pi)) ≤C_k,q(1 +kAkˆ _Wk,q(Bs(pi))+kAkˆ ³_Wk,q(Bs(pi))).

Corollary 2.12. |∇^k_AF_A| is pointwise bounded on M for all k.

Proof. By Corollary 2.8,|FA|is pointwise bounded, and the hypoth- esis (2.10) holds fori sufficiently large. The result now follows from an induction argument using (2.11) and the Sobolev embedding theorem.

q.e.d.

Lemma 2.13. For A a critical point of the Yang–Mills functional with finite energy on a homogeneous manifold, ∇^k_AF_A∈L₂ for all k.

Proof. The Yang–Mills equation and Bianchi identity imply thatFA

satisfies the equation

(2.14) 0 =∇^∗_A∇AF_A+ ( ˆF_A+ ˆR)(FA), where, in a local orthonormal frame,

Fˆ_A(FA) =−[Fij, F_jk]dxⁱ∧dx^k, and

R(Fˆ A) =−RijmkF_jmdxⁱ∧dx^k.

Here R denotes the Riemann curvature tensor. Assume now that the sequence of cutoff functions, {y_m}^∞_m=1, is C^k bounded for allk. Then

0 =h∇^∗_A∇AF_A+ ( ˆF_A+ ˆR)(FA), y²_mF_AiL2

(2.15)

⇒0 =k∇Ay_mF_Ak²_L2 − k[∇A, y_m]FAk²_L2 +h( ˆF_A+ ˆR)(FA), y_m²F_AiL2. From Corollary 2.12, we see that |FA|is bounded in sup norm. Taking m→ ∞, equation (2.15) now implies that

∇AFA∈L₂. Differentiating equation (2.14), we obtain

(2.16) 0 =∇^∗_A∇A∇AF_A+( ˆF_A+ ˆR)(∇AF_A)+[∇A,∇^∗_A∇A+( ˆF_A+ ˆR)]FA. The quantity [∇A,∇^∗_A∇A+ ( ˆF_A+ ˆR)]FA is a sum of terms involving at most one derivative of F and bounded coefficients. Hence taking

theL₂-inner product of (2.16) with ∇AFAand integrating by parts, we obtain

k∇²_AFAk² ≤c₁k∇AFAk²+c₂kFAk².

We now induct. Suppose that ∇^b_AF_A ∈ L₂, for b ≤ k. Differentiating (2.14) ktimes, gives∇^k_A(∇^∗_A∇_A+ ( ˆF_A+ ˆR))F_A= 0,and

0 =h∇^k_A(∇^∗_A∇A+ ( ˆFA+ ˆR))FA, y_m²∇^k_AFAi

≥ k∇Aym∇^k_AFAk²− kdym⊗ ∇^k_AFAk²− X

0≤j≤k

Cjk∇^j_AFAk²,

for some constants C_j determined by the sup norms of |∇^b_AF_A|, b≤ k and the other coefficients of [∇^k_A,∇^∗_A]. Taking the limit asm→ ∞gives

∇^k+1_A F_A∈L₂. q.e.d.

These bounds now allow us to construct test variations from covariant derivatives ofF_A.

Proposition 2.17. Let M be a complete noncompact homogeneous 4−manifold. LetA be a smooth finite energy Yang–Mills connection on a G−bundle E over M. Then ∇^k_AF_A∈L₂∩L₄∩C^∞,for all k.

Proof. We are left to show ∇^k_AF_A ∈ L₄. By Kato’s inequality,

∇^k+1_A F_A ∈ L₂ ⇒ d|∇^k_AF_A| ∈ L₂. Hence the scalar function |∇^k_AF_A| ∈ W^1,2. The Sobolev embedding theorem holds (see [He]) on manifolds of bounded geometry. Hence |∇^k_AFA| ∈L₄. q.e.d.

Remark 2.18. The assumption of homogeneous geometry is stronger than necessary in this section and can be replaced by weaker assump- tions of bounded geometry. The constants in Lemmas 2.7 and 2.9 are then replaced by constants depending on bounds on the injectivity ra- dius, the Riemannian curvature, and its derivatives. These geometric bounds would also then be used to bound the coefficients, C_j, in the proof of Lemma 2.13.

3. Conservative Decompositions

Suppose that Λ²T_M^∗ ⊗ad(E) decomposes into two orthogonal subbun- dles

(3.1) Λ²T_M^∗ ⊗ad(E) = Λ⁺(E)⊕Λ⁻(E),

such that ∇A preserves this decomposition. LetP^± denote the projec- tion onto these summands. We call such a decomposition conservative if there exists a, b∈R, not both zero, so that

(3.2) akP⁺F_Ak²+bkP⁻F_Ak² is independent of A.

The following elementary lemma clarifies the importance of conser- vative decompositions.

Lemma 3.3. Given a conservative decomposition, a connection min- imizesY M(A) if and only if it minimizeskP⁻F_Ak² (equivalently, if and only if it minimizes kP⁺F_Ak²).

Proof. By (3.1) we have kFAk²=kP⁺FAk²+kP⁻FAk². The lemma

then follows readily from (3.2). q.e.d.

Consequently, for energy minimizing connections and conservative de- compositions we have the additional critical point equation :

(3.4) 0 =d^∗_AP⁻F_A

and the refined variational inequalities:

(3.5) kP⁻FAk² ≤ kP⁻F_A+ψk², and

(3.6) 0≤ kP⁻d_Aψk²+ 2hP⁻F_A, ψ∧ψi.

The following elementary lemma shows how to begin to extract infor- mation about the curvature from the variational inequalities (3.5) and (3.6).

Lemma 3.7. Let ψ∈A¹(M, ad(E)) satisfy ψ∈L₂∩L₄, 0 =P⁻dAψ, and 0 =hP⁻FA, ψ∧ψi.

Then

e^∗(ψ)P⁻F_A= 0.

Proof. Consider the variation A_t = A+tψ+t³²w, for w ∈ A¹(M, ad(E)) arbitrary. Then expanding (3.5) we have

kP⁻FAk² ≤ kP⁻FAk²+ 2hP⁻FA, dA(tψ+t³²w)

+t²ψ∧ψ+ 2t⁵²ψ∧wi+t²kP⁻dAψk²+O(t³).

Invoking (3.4) and our hypotheses on ψ, this reduces to 0≤2hP⁻F_A,2t⁵²ψ∧wi+O(t³).

Replacing wby −w, we see that

0 =he^∗(ψ)P_A⁻, wi

for all w, and the lemma follows. q.e.d.

In the following sections we will consider 1−formsψconstructed from F_A that satisfy the hypotheses of Lemma 3.7 and use them to uncover information about F_A and A.

4. Dimension 4: (Anti-)Self-Duality and Homogeneous Spaces

In this section we assume that M is a 4-dimensional oriented Rie- mannian homogeneous space with nonnegative sectional curvature. De- note the group of isometries ofM byKand its Lie algebra byk. Identify kwith the Lie algebra of Killing vector fields onM. Fixing a base point o∈M and a metric on k induces a decompositionk=p⊕u, whereu is the Lie algebra of the isotropy group ofoand therefore also the kernel of the evaluation map k →T_oM. Because K is the product of an abelian and a compact group, we may choose the metric on k to be invariant under the adjoint action ofKand so that for everyxthe evaluation map k→T_xM is an isometry when restricted to the orthogonal complement of its kernel.

Let {X_j}^D_j=1 be a basis of k. Letφ_j,t:M →M,j = 1, . . . , D, t∈R, be the associated one parameter families of isometries. We define a pullback map

φ^∗_j,t : (Λ²T_M^∗ ⊗ad(E))_φj,t(x) →(Λ²T_M^∗ ⊗ad(E))x

by defining the action ofφ^∗_j,t on thead(E) factor to be parallel transport along the curve t → φ_j,t(x). Fix j. Away from a fixed point of φ_j,t, we may choose a local frame that is parallel on the integral curves of X_j through all points in a neighborhood of x. In such a frame the connection form, which we also denote A, satisfies

(4.1) ijA= 0, and ijFA=ijdAA, where i_j =i_Xj denotes interior multiplication byX_j.

Given a local frame{s_a}_aforad(E), we write anad(E) valuedp−form f asP

af^a⊗s_a.Then we have φ^∗_j,tdAf =X

a

φ^∗_j,t(df^a)⊗φ^∗_j,tsa+X

a,b

(−1)^pφ^∗_j,tf^a⊗φ^∗_j,t(A^b_as_b)

=d_Aφ^∗_j,tf + 2(φ^∗_j,tA−A)∧φ^∗_j,tf.

(4.2)

In four dimensions (oriented) we have the decomposition of Λ²T^∗M⊗ ad(E) given by the decomposition into self-dual and anti-self-dual sum- mands:

Λ²T^∗M⊗ad(E) = (Λ²₊T^∗M⊗ad(E))⊕(Λ²₋T^∗M⊗ad(E)).

Thus the projections onto the summands are given by P^±= 1

2(1± ∗),

where∗ denotes the Hodge star operator. Letp₁(A, E) denote the first pontrjagin form of E determined by the connection A. Recall that (4.3) kF_A⁺k²− kF_A⁻k² =c

Z

M

p₁(A, E),

where c > 0 is determined by normalization of the inner product on ad(E). Chern–Weil theory says that the right hand side is independent of A on compact manifolds. On noncompact manifolds it is constant under variations of A that decay suitably at ∞. Hence P^± define a conservative decomposition. Because d^∗_A = − ∗ d_A∗, we have for this decomposition

dAP^±FA= 0.

Having lifted the action of φ^∗_j,t to ad(E), we obtain an extension of the Lie derivativeL_j =L_Xj toad(E) valued forms. It satisfies the usual relation

Lj =dAij+ijdA, and of course

Ljf = d

du|u=0φ^∗_j,uf.

Because φj,t is an isometry we have

φ^∗_j,tP⁻=P⁻φ^∗_j,t, and hence, infinitesimally,

(4.4) [Lj, P⁻] = 0.

4.1. Variations.Set

F_A^±=P^±F_A.

The stability results in [BLS] and [BL] followed in large part from con- sideration of the variations A+ti_jF_A⁺. Our theorems in 4-dimensions rely on recognizing these variations as an approximation to the varia- tionsA+i_jF_j⁺(t), where

F_j⁺(t) :=

Z t

0

φ^∗_j,sF_A⁺(x)ds.

Heuristically, this variation may be thought of as an attempt to test whether the isometry invariance of the Yang–Mills energy extends to isometry invariance when only a self-dual component of the connection is shifted by the isometry.

Next we use (4.2) to expandF_A+i

jF_j⁺(t). F_A+i

jF_j⁺(t) =FA+dAijF_j⁺(t) +ijF_j⁺(t)∧ijF_j⁺(t)

=F_A+L_jF_j⁺(t)−2ij

Z t

0

(A−φ^∗_j,sA)∧φ^∗_j,sF_A⁺(x)ds +i_jF_j⁺(t)∧i_jF_j⁺(t)

=F_A+ Z t

0

d

dsφ^∗_j,sF_A⁺(x)ds + 2i_j

Z t

0

Z s

0

∂

∂uφ^∗_j,uA∧φ^∗_j,sF_A⁺(x)duds +i_j

Z t

0

Z t

0

φ^∗_j,uF_A⁺(x)∧i_jφ^∗_j,sF_A⁺(x)duds.

Using (4.1), we have

∂

∂uφ^∗_j,uA=φ^∗_j,ui_jd_AA=φ^∗_j,ui_jF_A. This and additional manipulations give

(4.5) F_A+i

jF_j⁺(t) =F_A⁻+φ^∗_j,tF_A⁺−2Φj(t), where

Φj(t) = Z t

0

Z s

0

φ^∗_j,ui_jF_A∧i_jφ^∗_j,sF_A⁺(x)duds

−1 2

Z t

0

Z t

0

i_jφ^∗_j,uF_A⁺(x)∧i_jφ^∗_j,sF_A⁺(x)duds

= Z t

0

Z s

0

φ^∗_j,ui_jF_A∧i_jφ^∗_j,sF_A⁺(x)duds

−1 2

Z t

0

Z s

0

φ^∗_j,uijF_A⁺∧ijφ^∗_j,sF_A⁺(x)duds

−1 2

Z t

0

Z t

s

ijφ^∗_j,uF_A⁺(x)∧ijφ^∗_j,sF_A⁺(x)duds.

Changing the order of integration in the last term and cancelling reduces the preceding to

(4.6) Φ_j(t) = Z t

0

Z s

0

i_jφ^∗_j,uF_A⁻(x)∧i_jφ^∗_j,sF_A⁺(x)duds.

For later application it is useful to Taylor expand Φ_j. We have for all integers B,

(4.7) Φ_j(t) =

a+b=B

X

a,b≥0

t^a+b+2ijL^a_jF_A⁻(x)∧ijL^b_jF_A⁺(x)

(a+ 1)!b!(a+b+ 2) +O(t^B+3).

With this notation, (3.5) becomes

(4.8) kF_A⁻k² ≤ kF_A⁻−2P⁻Φ_j(t)k², or in a more useful form:

(4.9) hF_A⁻,Φj(t)i ≤ kP⁻Φj(t)k², Equivalently

(4.10) kF_A⁺k² =kφ^∗_j,tF_A⁺k²≤ kφ^∗_j,tF_A⁺−2P⁺Φj(t)k²,

The right hand side of (4.9) is evidently O(t⁴), implying the nonpos- itivity of the O(t²) terms in the left hand side. The Taylor expansion (4.7) gives for each j

(4.11) 0≥ hF_A⁻, i_jF_A⁻∧i_jF_A⁺i (noj sum).

Switching the roles ofP⁻ and P⁺, we similarly deduce (4.12) 0≥ hF_A⁺, i_jF_A⁺∧i_jF_A⁻i (noj sum).

Lemma 4.13. Let f⁺ be a self-dual 2-form andf⁻ an anti-self-dual 2-form. Let {e₁, e₂, e₃, e₄} be a local orthonormal frame for T M. Then P

ai_eaf⁺∧i_eaf⁺ is self-dual, and P

ai_eaf⁻∧i_eaf⁻ is anti-self-dual. If φ₁, φ₂, andφ₃ are ad(E) valued 2-forms, then

X

a

hφ₁, ie_aφ₂∧ie_aφ₃i=X

a

hφ₃, ie_aφ₁∧ie_aφ₂i.

Proof. This is an elementary computation. q.e.d.

As proved in [BLS],[BL], we now obtain our first commutation result:

Proposition 4.14.

0 = [F_st⁺, F_ij⁻], for all indices s, t, i, j.

Proof. Summing (4.11), we obtain 0≥X

j

hF_A⁻, ijF_A⁻∧ijF_A⁺i.

Because the evaluation mapk→T_mM is an isometry on the orthogonal complement to its kernel, the pointwise inner product,

X

j

hF_A⁻, i_jF_A⁻∧i_jF_A⁺i(m) =X

a

hF_A⁻, i_eaF_A⁻∧i_eaF_A⁺i(m),

for a local orthonormal frame {ea}⁴_a=1. Applying Lemma 4.13, we see that

X

a

hF_A⁻, i_eaF_A⁻∧i_eaF_A⁺i(m) =X

a

hieaF_A⁻∧i_eaF_A⁻, F_A⁺i(m) = 0.

Hence the inequality (4.11) is actually an equality for each j:

(4.15) 0 =hF_A⁻, i_jF_A⁻∧i_jF_A⁺i=hF_A⁺, i_jF_A⁻∧i_jF_A⁻i (noj sum).

Symmetrically we obtain

(4.16) 0 =hF_A⁻, i_jF_A⁺∧i_jF_A⁺i (noj sum).

On the other hand, we have

P⁻dAijF_A⁺=−P⁻ijdAF_A⁺+LjP⁻F_A⁺ = 0.

Hence ψ=i_jF_A⁺ satisfies the hypotheses of Lemma 3.7, implying e^∗(ijF_A⁺)F_A⁻ = 0.

Expanding this equality in components, using the duality relations, and allowing X_j(m) to run over a basis of T_mM, we obtain the claimed commutation result:

0 = [F_st⁺, F_ij⁻],

for all indices s, t, i, j. q.e.d.

4.2. Inductive hypothesis.In order to move beyond the commuta- tion of the self-dual with the anti-self-dual components of the curvature to the construction of ∇A stable subbundles k⁺ and k⁻ of ad(E) with self-dual (respectively anti-self-dual) curvature, we wish to prove the following proposition.

Proposition 4.17. [∇ⁱ_AF_A⁺,∇^j_AF_A⁻] = 0 for alli and j.

In this and the next subsection, we prove this proposition by induction on i+j.

Denote by A_N the inductive hypothesis:

(4.18) A_N : [∇ⁱ_AF_A⁺,∇^j_AF_A⁻] = 0, fori+j < N.

We have established A₁ in Proposition 4.14. We will show A_N implies A_N+1. Assume A_N holds, for some N ≥ 1. In the inductive hypoth- esis, powers of covariant derivatives can be replaced by powers of Lie derivatives, since they differ by lower order terms.

Observe that

(4.19) A_N ⇒Φj(t) =O(t^N+2).

Hence A_N and equation (4.9) imply

(4.20) hF_A⁻,Φj(t)i ≤O(t^2N+4).

Set

S_j(t) =hF_A⁻,Φj(t)i.

Then S_j(0) = 0, and

(4.21) S_j^′(t) =hF_A⁻, Z t

0

i_jφ^∗_j,uF_A⁻∧i_jφ^∗_j,tF_A⁺dui

=hF_A⁻, φ^∗_j,t Z 0

−t

i_jφ^∗_j,uF_A⁻∧i_jF_A⁺dui.

Hence S_j^′(0) =S_j^′′(0) = 0.

In order to use the variational inequality (4.20) we need to estimate Sj(t). Taylor expandingS_j^′ gives

S_j^′(t) =hF_A⁻,

a+b=m

X

a,b≥0

t^a a!L^a_j

Z 0

−t

i_ju^b

b!L^b_jF_A⁻∧i_jF_A⁺dui+O(t^m+2) Hence

Sj(t) =hF_A⁻,

a+b=m

X

a,b≥0

(−1)^bt^a+b+2

a!(b+ 1)!(a+b+ 2)L^a_j[ijL^b_jF_A⁻∧ijF_A⁺]i+O(t^m+3)

=hF_A⁻,

a+b=m

X

a,b≥N

(−1)^bt^a+b+2

a!(b+ 1)!(a+b+ 2)L^a_j[ijL^b_jF_A⁻∧ijF_A⁺]i+O(t^m+3) Hence, usingAN to eliminate lower order terms, (4.20) implies (4.22) hF_A⁻,(−1)^NL^N_j [ijL^N_j F_A⁻∧i_jF_A⁺]i ≤0.

Lemma 4.23. If A_N holds then

hF_A⁻, L^N_j [ijL^N_j F_A⁻∧i_jF_A⁺]i= 0.

Proof. Set S(Xj, N) = hF_A⁻, L^N_j [ijL^N_j F_A⁻ ∧ ijF_A⁺]i. The inequality (−1)^NS(Xj, N) ≤ 0 holds when X_j is replaced by any Killing vector.

We will show that the average of S(Xj, N) over the unit sphere of k is zero. HenceS(X_j, N) is zero for eachj(and each choice of basis ofk). To see this we consider S(P

ky^kXk, N) and integrate the resulting degree 2N + 2 homogeneous polynomial iny over the unit sphere. Integration of homogeneous degree 2N+2 polynomials over the sphere projects onto the span of the radial function, (r²)^N+1. Expanding in a multi-index notation whereL_J =L_j1· · ·L_j|J|, we write

S(X

k

y^kX_k, N) = X

|I|=N,|J|=N dimk

X

m,p=1

y^Iy^Jy^my^phF_A⁻, L_I[imL_JF_A⁻∧i_pF_A⁺]i.

That integration over the unit sphere projects onto radial functions im- plies that upon integration of S, we are left with a linear combination of coefficients, hF_A⁻, L_I[imL_JF_A⁻ ∧i_pF_A⁺]i, of S where the indices are contracted pairwise. We will see that all such contractions vanish. The conditionA_Nallows us to replace the Lie derivatives by covariant deriva- tives, as the difference vanishes in the inner product. We can also drop commutators of derivatives by A_N, as all such commutations drop the degree of the differentiation and thus lead to terms which vanish byA_N. We can then use the Yang–Mills equation and A_N to equate to zero all

inner products containing P

kikLkF_A^± terms or P

kL²_kF_A^± terms. We also invoke Lemma 4.13 to remove terms with paired indices m=p on the interior products. Thusmandpmust both pair with elements ofI, for if p pairs with anL_jr, we can use the Leibniz formula,A_N and the Yang–Mills equation to eliminate the corresponding term. This leaves two Lj terms which must pair with each other, but since

0 =−(dAd^∗_A+^∗_AdA)F_A^±

=X

j

L²_jF_A^± (modulo terms vanishing in the inner product byA_N), we find the average vanishes. Hence

hF_A⁻, L^N_j [ijL^N_j F_A⁻∧i_jF_A⁺]i= 0,

as claimed. q.e.d.

Now we apply a variant of Lemma 3.7 to obtain a commutation result.

Lemma 4.24. If A_N holds then

e^∗(ijL^N_j F_A⁺)F_A⁻= 0.

Proof. Once again we letψbe a smooth compactly supportedad(E) valued 1−form. Then

kF_A⁻k²≤ kF⁻

A+ijF_j⁺(t)+t^pψk²

=kF_A⁻−2P⁻Φj(t) +t^pd_Aψ+t^2pψ∧ψ+ 2t^pψ∧i_jF_j⁺(t)k²

=kF_A⁻k²−4Sj(t) + 4t^phF_A⁻, ψ∧i_jF_j⁺(t)i+O(t^2N+4+t^2p).

Lemma 4.23 implies S_j(t) =O(t^2N+3). Hence Taylor expanding again, we get

0≤

N

X

b=0

4t^phF_A⁻, ψ∧ t^b+1

(b+ 1)!i_jL^b_jF_A⁺i+O(t^2N+3+t^2p)

= 4t^phF_A⁻, ψ∧ t^N+1

(N+ 1)!i_jL^N_j F_A⁺i+O(t^2N+3+t^2p+t^N+p+2).

Choosing p=N+³₂ gives

0≤ hF_A⁻, ψ∧i_jL^N_j F_A⁺i.

Replacingψwith−ψmakes the inequality an equality, and we conclude 0 =e^∗(ijL^N_j F_A⁺)F_A⁻

as desired. q.e.d.

4.3. An Algebraic Reduction.We now complete our induction ar- gument by proving the following proposition.

Proposition 4.25. The assumption A_N implies A_N+1.

Proof. Assume A_N holds. By Lemma 4.24, e^∗(i_jL^N_j F_A⁺)F_A⁻ = 0 for all j. The Lie derivativeLj differs from the covariant derivative ∇j by zero order terms which commute with every element of ad(E). Hence we have

(4.26) 0 =e^∗(ij∇^N_j F_A⁺)F_A⁻.

Fix a pointxand a basis for the infinitesimal isometries so thatXj(x) = e_j,j= 1,2,3,4 is an orthonormal basis ofT_xM. Then expanding (4.26) in this frame, yields for allk and t,

(4.27) X

s

[∇^N_kF_ks⁺, F_st⁻] = 0 (no k sum).

In fact, replacinge_k byu¹e₁+· · ·u⁴e₄, for any (u₁, u₂, u₃, u₄)∈R⁴,we have for all t,

(4.28) X

a,s

[(X

j

u^j∇j)^Nu^aF_as⁺, F_st⁻] = 0.

Set

p_ik(u) = [(X

j

u^j∇_j)^NF_1i⁺, F_1k⁻],

and let p_ik,a := ^∂p_∂u^ika. In our local orthonormal frame, duality implies that the components of F_A⁺ and F_A⁻ (and their covariant derivatives) satisfy F₁₂^± = ±F₃₄^±, F₁₃^± = ±F₄₂^±, and F₁₄^± = ±F₂₃^±, and consequently each term P

ju^j∇jN

u^aF_as⁺, F_st⁻

in (4.28) is equal to p_ik(u) for some iand k. Thus we may expand (4.28) as

0 =u¹(p₂₂+p₃₃+p₄₄)−u²(p₃₄−p₄₃) (4.29)

+u³(p₂₄−p₄₂)−u⁴(p₂₃−p₃₂).

0 =u¹(p34−p₄₃) +u²(−p22+p₄₄+p₃₃) (4.30)

−u³(p₃₂+p₂₃)−u⁴(p₄₂+p₂₄).

0 =u¹(−p24+p₄₂)−u²(p23+p₃₂) (4.31)

+u³(−p₃₃+p₄₄+p₂₂)−u⁴(p₄₃+p₃₄).

0 =u¹(p₂₃−p₃₂)−u²(p₂₄+p₄₂)−u³(p₃₄+p₄₃) (4.32)

+u⁴(−p44+p₃₃+p₂₂).

In addition to these equations, we have the Yang–Mills equations and the Bianchi identities. These are best encoded as relations between the

derivatives of the pij as follows.

4

X

i=2

p_ik,i=N[(u^j∇j)^N−1(d^∗_AF⁺)₁, F_1k⁻] = 0, and

p_ik,a(u) =N[(u^j∇j)^N−1F_1i,a⁺ , F_1k⁻]

=N[(u^j∇j)^N−1F_1a,i⁺ , F_1k⁻] +N[(u^j∇j)^N−1F_ai,1⁺ , F_1k⁻]

=p_ak,i(u) +N[(u^j∇j)^N−1F_ai,1⁺ , F_1k⁻].

This reduces to the following 12 equations.

p_2k,2+p_3k,3+p_4k,4= 0.

(4.33)

p_2k,1−p_4k,3+p_3k,4= 0.

(4.34)

p_2k,4−p_4k,2−p_3k,1= 0.

(4.35)

p_2k,3+p_4k,1−p_3k,2= 0.

(4.36)

We may also useA_N to shift the derivative to the F_A⁻ term, yielding the following relations among the derivatives.

4

X

k=2

p_ik,k=−N[(u^j∇j)^N−1F_1i⁺,(d^∗_AF_A⁻)₁] = 0, and

p_ik,a=N[(u^j∇j)^N−1F_1i,a⁺ , F_1k⁻] =−N[(u^j∇j)^N−1F_1i⁺, F_1k,a⁻ ]

=−N[(u^j∇_j)^N−1F_1i⁺, F_ak,1⁻ ]−N[(u^j∇_j)^N−1F_1i⁺, F_1a,k⁻ ]

=N[(u^j∇j)^N−1F_1i,1⁺ , F_ak⁻] +p_ia,k. This yields 12 additional equations.

p_i2,2+p_i3,3+p_i4,4 = 0.

(4.37)

p_i2,1+p_i4,3−p_i3,4 = 0.

(4.38)

p_i2,4+p_i3,1−p_i4,2 = 0.

(4.39)

p_i2,3−p_i4,1−p_i3,2 = 0.

(4.40)

Homogeneity allows us to reconstruct the p_ik readily from their deriva- tives:

(4.41) p_ik= 1

Nu^ap_ik,a.

Hence, equations (4.33) - (4.40) allow in (4.29) - (4.32) the replacement of p₂₂, p₂₃, p₂₄, p₃₂, and p₄₂ by linear combinations of p₃₃, p₄₄, p₃₄, and p₄₃.It is more convenient, however, to differentiate (4.29) - (4.32) and make the replacement at the level of derivatives.

Denote the quantities on the righthand side of equations (4.29) - (4.32) byI₁,I₂,I₃, andI₄, respectively. DifferentiatingI_j and replacing