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Communications in Contemporary MathematicsVol. 13, No. 3 (2011) 463–486c© World Scientific Publishing CompanyDOI: 10.1142/S0219199711004403

CRITICAL POINTS OF YANG–MILLS–HIGGS FUNCTIONAL

CHONG SONG

LMAM, School of Mathematical Sciences, Peking UniversityBeijing 100871, P. R. China

songchong@amss.ac.cn

Received 9 November 2009Revised 7 July 2010

We use Sacks–Uhlenbeck’s perturbation method to find critical points of the Yang–Mills–Higgs functional on fiber bundles with 2-dimensional base manifolds. Such criticalpoints can be regarded as a generalization of harmonic maps from surfaces, and alsoa generalization of the so-called twisted holomorphic maps [15]. We prove an existenceresult analogous to the one for harmonic maps. In particular, we show that the so-calledenergy identity holds for the Yang–Mills–Higgs functional.

Keywords: Yang–Mills–Higgs functional; twisted holomorphic map; energy identity; har-

monic map.

Mathematics Subject Classification 2010: 58E15, 35J50, 53C80

1. Introduction

Suppose Σ is a compact Riemann surface with a Riemannian metric g, G is acompact Lie group with a metric, and P is a principal G-bundle on Σ. Let F bea compact symplectic manifold which supports a Hamiltonian action of G with amoment map µ : F → g∗, and π : F = P ×G F → Σ be the associated bundle withfiber F . Since µ is by definition G-invariant, it extends to a map on the bundleµ : F → g∗. Denote the W 1,2-completion of the space of metric connections on P

and the space of sections on F by A1,2 and S1,2 respectively. (See the Appendix forthe definitions.) Then the Yang–Mills–Higgs functional is defined on A1,2×S1,2 by

L(A, φ) := ‖dAφ‖2L2 + ‖FA‖2

L2 + ‖µ(φ) − c‖2L2, ∀(A, φ) ∈ A1,2 × S1,2

where FA is the curvature of A and c ∈ g∗ is fixed.The Yang–Mills–Higgs functional is composed of the energy functional, the

Yang–Mills functional and the Higgs potential. In particle physics, it describes asystem where the Higgs field interacts with the gauge field. It also corresponds tothe Ginzburg–Landau theory which is used to model superconductivity.

463

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464 C. Song

If there is a bi-invariant metric on g (that is, a metric invariant under the adjointaction of G), then it induces an equivariant isomorphism g g∗ which extends toan isomorphism of vector bundles P ×Ad g P ×Ad g∗ (note that we denote boththe adjoint representation on g and the coadjoint representation on g∗ by the samesymbol Ad). A well-known result about the Yang–Mills–Higgs functional is that itsminimum only depends on the topology of F and φ. Indeed, L can be rewritten as

L(A, φ) = ‖ΛFA + µ(φ) − c‖2L2 + 2‖∂Aφ‖2

L2 + 4‖F 0,2A ‖2

L2

+ 2∫

Σ

〈ΛFA, c〉 +∫

Σ

trFA ∧ FA ∧ ω[n−2]F ,

where ΛFA is the contraction of FA, ∂Aφ is the Ω0,1 part of dAφ, F 0,2A is the Ω0,2

part of FA and w[n−2]F = ωn−2

F /(n − 2)!. The minimizers of the Yang–Mills–Higgsfunctional are called twisted holomorphic maps. Since F 0,2

A disappears on a Riemannsurface Σ, a twisted holomorphic map (A, φ) satisfies

∂Aφ = 0

ΛFA + µ(φ) = c.(1.1)

Massive work has been done concerning Eq. (1.1). Recently Riera and Tian [16]gave a compactification of the moduli space of twisted holomorphic maps. Thiscompactness result serves as a crucial tool to construct general HamiltonianGromov–Witten invariants, which makes it possible to find a new quantum productin equivariant cohomology (see also [15]). On the other hand, many previous worksare concerned with the case where the fiber bundle is just a vector bundle. Forexample, Qing [13] studied the minimizers of the Yang–Mills–Higgs functional ona line bundle on a closed Riemann surface and defined a renormalized energy. Onemay refer to Jaffe and Taubes’ book [5] for more information.

Motivated by Riera and Tian’s work, we also consider Yang–Mills–Higgs func-tional on a general fiber bundle F where the fiber is a compact symplectic manifold.In particular, we are interested in finding minimax critical points of Yang–Mills–Higgs functional. These critical points only satisfy the following Euler–Lagrangeequation:

d∗AdAφ = (µ(φ) − c)∇µ(φ)

d∗AFA = −〈dAφ, φ〉,(1.2)

where d∗A is the adjoint operator of dA. This a second-order equation which isweaker than Eq. (1.1). Namely, twisted holomorphic maps also satisfy Eq. (1.2).In a certain sense, the relationship between twisted holomorphic maps and generalcritical points of the Yang–Mills–Higgs functional can be compared with the onebetween holomorphic maps and harmonic maps.

In order to find critical points of the Yang–Mills–Higgs functional, one maystart with some sequence (An, φn) of bounded energy and try to find a limit.

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Critical Points of Yang–Mills–Higgs Functional 465

The existence of a weak limit is obvious. But L does not satisfy the Palais–Smalecondition (C), so the convergence is not necessarily strong in general. The energycould probably accumulate at some point and the so-called blow-up phenomenonmight occur. Thus it is not easy to find critical points directly. However, as wewill see, the only part of the functional L which causes the problem is the energyof the section E(φ) = ‖dAφ‖2

L2 , which is invariant under conformal transforma-tions. In fact, if the fiber bundle is trivial and the connection A is trivial, thesection φ is just a map from Σ to F , and the energy E becomes the normal energyfunctional E(φ) = ‖dφ‖L2 , φ ∈ W 1,2(Σ, F ). In this case, the critical points of Yang–Mills–Higgs functional are just harmonic maps (with potential) from surfaces. Theblow-up phenomenon of harmonic maps is well known and has been extensivelyinvestigated. In Sacks and Uhlenbeck’s celebrated work [17], they considered theperturbed energy

Eα(φ) =∫

Σ

(1 + |dφ|2)αdV, for φ ∈W 1,2α(Σ, F ),

where α > 1. This functional satisfies the P.S. condition and attains its minimum inevery connected component. By letting α→ 1, they showed that the critical pointsof Eα subconverge to a harmonic map together with some harmonic spheres.

In this paper we will apply the perturbation method to the Yang–Mills–Higgsfunctional. Namely, we define for α > 1 the perturbed Yang–Mills–Higgs functional:

Lα(A, φ) :=∫

Σ

(1 + |dAφ|2)αdV + ‖FA‖2L2 + ‖µ(φ) − c‖2

L2 . (1.3)

We will show that Lα also satisfies the P.S. condition. So the minimax principleapplies to Lα for any α > 1. Then we investigate the convergence of the criticalpoints of Lα as α goes to 1.

Our main result is the following theorem, which is a generalization of corre-sponding result for harmonic maps from surfaces [4, 6, 7]:

Theorem 1.1. Suppose Σ is a compact Riemann surface, G is a compact Liegroup and P is a principal G-bundle on Σ. F is a compact symplectic manifoldwhich supports a Hamiltonian action of G with moment map µ : F → g∗, andπ : F = P ×G F → Σ is the associated bundle with fiber F . Let I be a compactparameter space without boundary, h0 : I → C0 ∩W 1,2(A × S ) be a continuousmap and H be the set of maps homotopic to h0. Define

β := infh∈H

supt∈I

L(h(t)).

Then there exists a sequence α→ 1, (Aα, φα) ∈ A ×S of critical points of Lα andfinitely many points x1, . . . , xd, such that

Lα(Aα, φα) = infh∈H

supt∈I

Lα(h(t)), (1.4)

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466 C. Song

Aα → A0 in C∞(Σ), (1.5)

φα → φ0 in C∞(Σ\x1, . . . , xd), (1.6)

where (A0, φ0) ∈ A × S is a critical point of L.Moreover, there exist at most finitely many harmonic spheres vi : S2 → F, 1 ≤

i ≤ k such that

β = limα→1

Lα(Aα, φα) = L(A0, φ0) + vol(Σ) +k∑

i=1

E(vi), (1.7)

where E(vi) =∫Σ|dvi|2dV is the normal energy.

The last identity (1.7) is the so-called energy identity, which asserts that there isno energy loss during the blow-up process. With this result, it becomes possible towork on compactification of the moduli space of critical points of the Yang–Mills–Higgs functional, as Riera and Tian did for the moduli space of twisted holomorphicmaps. This will be done in a forthcoming paper.

The energy identity for harmonic maps from surfaces proved to be a very inter-esting problem which has attracted many mathematicians’ attention in the lastdecades. Let’s recall some important results. Jost [4] first proved a version of energyidentity for minimax harmonic maps in his book. Not long ago Colding and Mini-cozzi [2] gave another proof in the case where the domain is a 2-sphere and used theidentity to prove the finite time extinction of Ricci flow. For the blow-up of heatflow of harmonic maps from surfaces, Qing [14] proved the energy identity whenthe target manifold is a sphere. Next, Ding and Tian [3] showed the identity holdsfor maps with L2-bounded tension fields and hence generalized Qing’s result togeneral target manifolds. (See [1, 9, 12, 22] for more related results.) However, theenergy identity for Sacks–Uhlenbeck type approximations for harmonic maps hasonly been proved very recently. Li and Wang [7] proved a weak energy identity fora Sacks–Uhlenbeck-type approximate sequence. Lamm [6] showed that combiningStruwe’s work [19], one can obtain the desired identity for a minimax α-harmonicmap sequence.

Here we generalize the energy identity to the Yang–Mills–Higgs functional. Theessential difficulties for the present functional and difference from harmonic mapsarise from the structure of the fiber bundle and the non-trivial connection A. For-tunately, when the L2-norm of the curvature FA is bounded, it turns out that theimpact brought by A is controllable. On the other hand, when taking limit of theapproximate sequence, there are only finitely many singularities where the energymay lost. Since near the singularities the fiber bundle can always be locally trivi-alized, the global structure of the bundle does not matter much, either. Moreover,after blowing up, the connection A vanishes and hence we get exactly the sameblow-up profile as in the harmonic map case.

We will prove Theorem 1.1 in the rest of this paper step by step. Now we givean outline. We first derive the Euler–Lagrange equation for the Yang–Mills–Higgs

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Critical Points of Yang–Mills–Higgs Functional 467

functional in Sec. 2. In Sec. 3, we introduce the perturbed functional and show thatit satisfies nice properties for α > 1, especially the P.S. condition which guaranteesthe minimax value can be attained as in (1.4). In Sec. 4, we obtain local estimateson both the section and the connection. Then we use these estimates to show theconvergence (1.5), (1.6) and demonstrate what happens at singularities in Sec. 5.At last, we prove the energy identity (1.7). For completeness, we also include somebackground knowledge in the appendix.

2. The Yang–Mills–Higgs Functional

Suppose Σ is a compact Riemann surface, G is a compact Lie group, and P is aprincipalG-bundle on Σ. Let F be a compact symplectic manifold with a compatibleRiemannian metric h, which supports a Hamiltonian action of G with moment mapµ : F → g∗. Let π : F = P ×G F → Σ be the associated bundle with fiber F .Then for every connection A ∈ A1,2 and section φ ∈ S1,2, the Yang–Mills–Higgsfunctional is given by:

L(A, φ) := ‖dAφ‖2L2 + ‖FA‖2

L2 + ‖µ(φ) − c‖2L2 , (2.1)

where FA is the curvature of A and c ∈ g∗ is fixed.Since the base manifold Σ is of real dimension 2, the L2-norm of dAφ is confor-

mally invariant while the L2-norm of the curvature FA is not. But the norm ‖FA‖L2

behaves in a good way when we rescale the metric. Actually, one can easily provethe following lemma.

Lemma 2.1. Let 0 < r < 1 be a real number. Let A be a connection on the trivialprincipal bundle G ×Dr → Dr over the disk Dr ⊂ C of radius r, and φ : Dr → F

a section of the corresponding fiber bundle. Consider the rescaling λr : D → Dr

which sends z in the unit disk D to rz ∈ Dr. Then

‖dλ∗rA(λ∗rφ)‖L2(D) = ‖dAφ‖L2(Dr), (2.2)

‖Fλ∗rA‖L2(D) = r‖FA‖L2(Dr). (2.3)

Lemma 2.1 implies that

L(λ∗r(A, φ)) ≤ L(A, φ), 0 < r < 1.

This lemma can also be understood as a formula under a conformal change. Namely,if we denote the L2-norm of dAφ and FA over a domain U of Σ with metric g byE1(A, φ, g) and E2(A, g), respectively. Then we have

E1(A, φ, g) = E1(A, φ, r2g),

E2(A, g) = rE2(A, r2g).

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468 C. Song

The Euler–Lagrange equation for L is:d∗AdAφ = (µ(φ) − c)∇µ(φ)

d∗AFA = −〈dAφ, φ〉.(2.4)

From the first equation of (2.4), we can see that if the bundle and the connectionare trivial, i.e. F = Σ× F,A = 0, then φ is just a map from Σ to F which satisfies

d∗dφ = (µ(φ) − c)∇µ(φ).

In this case, φ is the so-called harmonic map with potential. The second equationof (2.4) should be understood as follows. For all B ∈ TAA = Ω1(P ×Ad g), we have

〈FA, dAB〉 = −〈d∗AFA, B〉 = 〈dAφ,Bφ〉.Next we rewrite Eq. (2.4) explicitly in local coordinates. Suppose U is a domain

in Σ and ψ : U × F → π−1(U) is a local trivialization of F . By writing φ(x) =(x, u(x)) ∈ U × F under this trivialization, we can identify φ with the map u :U → F . We may also identify dAφ = dAu and µ(φ) = µ(u) since their values aredecided by u. One important characteristic of the functional L is its invariance undergauge transformations. More precisely, suppose s ∈ G, then L(s∗A, sφ) = L(A, φ).Thus we can always choose a good gauge as a representative. This is a theorem ofUhlenbeck [20]:

Theorem 2.2. Let p ≥ 1 and consider a trivial bundle E = U×G on a disk U ∈ C.There exists a constant δ0 > 0 such that for any connection A ∈ Ω1(E ×Ad g)W 1,p ,

if its curvature satisfies ‖FA‖L1 ≤ δ0, then A is gauge equivalent by an elements ∈ W 2,p(U,G) to a connection A which satisfies

(1) d∗A = 0;(2) x ·A = 0 for any x ∈ ∂D;(3) ‖A‖W 1,p ≤ C‖FA‖Lp.

Such a gauge in the above theorem is called Coulomb gauge. When it comes tolocal analysis, we will always identify φ with u and use the Coulomb gauge in thefollowing context of this paper.

Now suppose A is in Coulomb gauge which satisfies d∗A = 0 in U , we have

d∗AdAφ = (d∗ +A∗)(d +A)u

= d∗du+ d∗(Au) +A∗du+A∗Au

= d∗du− 2Adu−A2u. (2.5)

Here we used the skew-symmetric property ofA (see (A.1) in the Appendix). EmbedF into a Euclidean space Rk and denote the Laplace–Beltrami operator on functionsfrom Σ to Rk by ∆. If u is smooth enough, we can write

d∗du = −∆u− Γ(u)(du, du),

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Critical Points of Yang–Mills–Higgs Functional 469

where Γ is the second fundamental form of F . Thus we get the equation for u:

∆u+ Γ(u)(du, du) + 2Adu+A2u+ (µ(u) − c)∇µ(u) = 0. (2.6)

Next, we can also rewrite the equation for A as:

∆A+ [A, dA] + [A, [A,A]] + 〈dAu, u〉 = 0, (2.7)

since FA = dA + [A,A]. Here the Laplace operator on 1-form A is ∆ = d∗d + dd∗

and should not cause any confusion.

3. The Perturbed Functional

In this section we follow the idea of Sacks and Uhlenbeck [17] for finding harmonicmaps and consider the perturbed functional on A1,2 × S1,2α for α > 1:

Lα(A, φ) :=∫

Σ

(1 + |dAφ|2)αdV + ‖FA‖2L2 + ‖µ(φ) − c‖2

L2 . (3.1)

A simple computation yields the Euler–Lagrange equation of this functional:d∗A[α(1 + |dAφ|2)α−1dAφ] − (µ(φ) − c)∇µ(φ) = 0

d∗AFA + α(1 + |dAφ|2)α−1〈dAφ, φ〉 = 0.(3.2)

A pair (A, φ) ∈ A1,2 × S1,2α is a critical point of Lα if it is a weak solutionof Eq. (3.2). As we did before for the Yang–Mills–Higgs functional L in the lastsection, locally we let U be a domain in Σ and ψ : U × F → π−1(U) a localtrivialization of F . Write φ(x) = (x, u(x)) and dA = d + A in U where A satisfies(1)–(3) of Theorem 2.2. Then φ being a weak solution to (3.2) is equivalent to sayu is a W 1,2α weak solution to the equation:

d∗[α(1 + |dAu|2)α−1dAu] − α(1 + |dAu|2)α−1AdAu− (µ(u) − c)∇µ(u) = 0, (3.3)

i.e. for any function v ∈ W 1,20 (U,Rk)(F is embedded in Rk), we have∫

Σ

α(1 + |dAu|2)α−1(〈dAu, dv〉 + 〈AdAu, v〉) +∫

Σ

(µ(φ) − c)〈∇µ(u), v〉 = 0.

If u is regular enough, Eq. (3.2) transforms to:

∆u+ 2(α− 1)〈d2u, dAu〉dAu

1 + |dAu|2 + Φ(A, u) = 0

∆A+ Ψ(A, u) = 0,

(3.4)

where

Φ(A, u) = Γ(u)(du, du) + 2Adu+A2u− 2(α− 1)〈Adu, dAu〉dAu

1 + |dAu|2

+(µ(u) − c)∇µ(u)α(1 + |dAu|2)α−1

and

Ψ(A, u) = [A, dA] + [A, [A,A]] + α(1 + |dAu|2)α−1〈dAu, u〉.

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470 C. Song

Note that in order to write the equation for u as above, we need to make sure thesecond-order derivative of u exists, which will be justified later (see Theorem 3.3).

In Urakawa’s book [21] (see also [11]), the author gave a detailed proof that theperturbed energy

Eα(u) =∫

Σ

(1 + |du|2)αdV

satisfies the Palais–Smale condition (C) on the Sobolev space of maps

W 1,2α(Σ, F ) = u ∈W 1,2α(Σ,Rk) : u(x) ∈ F ⊂ C0(Σ, F )

which is a C2 separable Banach manifold for α > 1. The key point in his proofis to take an embedding F ⊂ Rk and consider the functional Eα on W 1,2α(Σ,Rk)defined by

Eα(u) :=∫

Σ

(1 + |du|2)αdV, u ∈W 1,2α(Σ,RK).

If un is a P.S. sequence of Eα, there exists a subsequence, denoted by the sameletter, converging to a weak limit u0 such that

〈dEα(un), un − u0〉 → 0, as n→ ∞.

Combining this with the convexity of the function f(x) = (1 + |x|2)α, x ∈ R, onecan prove that the convergence of un → u0 is actually strong in W 1,2α.

In our case, the Sobolev space of sections S1,2α ⊂ C0(Σ,F) is also a C2 sepa-rable Banach manifold. If we consider the energy on S1,2α

Eα(φ) =∫

Σ

(1 + |dAφ|2)αdV

for a fixed connection A ∈ A1,2, we can prove that this functional also satisfies theP.S. condition almost in the same way.

To see this, we embed the total space F , which is a compact space, into a largeEuclidean space RK and consider S1,2α as a subspace of

W 1,2α(Σ,F) = φ | φ ∈ W 1,2α(Σ,RK), φ(x) ∈ F.Then we need to prove that every sequence φn ∈ S1,2α with Eα(φn) ≤ C and‖dEα(φn)‖ → 0 has a subsequence which converges strongly in S1,2α. The differenceis that now we have a non-trivial connection A and the norm of dEα(φ) is taken inthe dual space of φ∗TFv instead of φ∗TF . But note that A ∈ A 1,2 ⊂ Lq(A ), ∀ q <∞, and the co-invariant differential dA = πA d only differs from the normal one bya projection πA which has bounded Lq-norm, so the terms involving A can alwaysbe controlled and the proof still works.

A better way to understand this is perhaps that when α > 1, the Sobolev spaceW 1,2α is compactly embedded in C0 and the energy cannot accumulate at a point,i.e. there are no singularities and hence the convergence must be strong. Moreover,the connection A and the potential term ‖µ(φ) − c‖2

L2 do not cause much trouble.

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Critical Points of Yang–Mills–Higgs Functional 471

But here we need to consider the couple (A, φ), that is, we have to prove Lα

satisfies the P.S. condition on the product space A1,2 × S1,2α. Let’s first recall atheorem of Uhlenbeck [20]:

Theorem 3.1. Let 2p > dim Σ and An ∈ A1,p be a sequence of connections with∫Σ|FAn |p ≤ B. Assume Σ and G are compact. Then there exists a subsequence Am

and gauge transformations sm ∈ G2,p such that s∗mAm is weakly convergent in A1,p.The weak limit A0 satisfies

∫Σ |FA0 |p ≤ B.

Now we can prove:

Lemma 3.2. Lα satisfies the P.S. condition on the product space A1,2 ×S1,2α forα > 1.

Proof. Take a P.S. sequence (An, φn)n≥1 ∈ A1,2 × S1,2α which satisfies

(i) Lα(An, φn) ≤ B, where B > 0 is an upper bound.(ii) ‖DLα(An, φn)‖ → 0, where the norm is taken in T ∗

(An,φn)A1,2 × S1,2α.

We need to show there is a subsequence which converges strongly.By (i) and Theorem 3.1, there exists a subsequence Am and gauge transfor-

mations sm ∈ G2,2 such that A′m = s∗mAm converges weakly to A0 in A1,2 which

satisfies∫Σ |FA0 |2 ≤ B. For convenience, we still denote the new sequence after

gauge transformation by (An, φn). Then we have

‖An −A0‖W 1,2 ≤ C and An A0 in A1,2

Since the space of connections A1,2 is an affine space, An −A0 can be viewed as antangent vector in TAnA1,2. Moreover, the norm ‖An − A0‖W 1,2 is bounded. So bycondition (ii) of the P.S. sequence, we have

〈DLα(An, φn), An −A0〉 → 0. (3.5)

Similar to the computation of Euler–Lagrange equation (3.2) for A, we have

〈DLα(An, φn), An −A0〉

=∫〈d∗An

FAn + α(1 + |dAnφn|2)α−1〈dAnφn, φn〉, An −A0〉

=∫〈FAn , dAn(An −A0)〉 + α

∫(1 + |dAnφn|2)α−1〈dAnφn, (An −A0)φn〉

= I + II .

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472 C. Song

Since An−A0 0 in W 1,2, ‖An−A0‖Lq → 0 for any q <∞ by Sobolev embeddingtheorems. It follows from Holder’s inequality that

I =∫〈dAn + [An, An], d(An −A0) + [An, An −A0]〉

=∫

|d(An −A0)|2 +∫〈dA0, dAn − dA0〉 −

∫〈d[An, An], An −A0〉

+∫〈dAn + [An, An], [An, An −A0]〉

→ ‖d(An −A0)‖2L2 .

For the second term, we have

|II | ≤ α

(∫D

(1 + |dAnφn|2)α

)α−1α

(∫D

|〈dAnφn, (An −A0)φn〉|α) 1

α

≤ C‖dAnφn‖L2α‖φn‖L∞‖An −A0‖L2α

→ 0.

Putting these two terms back in (3.5), we get

‖d(An −A0)‖2L2 → 0,

i.e. An actually converges strongly to A0 in A1,2. On the other hand, one can provethat for a fixed connection A0 ∈ A1,2, L satisfies the P.S. condition in S1,2α byUrakawa’s method. So there exists a subsequence of φnn≥1 which converges to alimit φ0 strongly. Thus we obtain a subsequence of (An, φn)n≥1 which convergesto (A0, φ0) strongly in A1,2 × S1,2α. This proves the lemma.

Next we show the regularity of the critical points of Lα. Though the theoremholds for all α > 1, we only prove for α − 1 small. The bootstrapping and gluingtechniques applied here are pretty much the same as in [16].

Theorem 3.3. The solutions of the Euler–Lagrange equation (3.4) of Lα in A1,2×S1,2α are smooth under some gauge transformation s ∈ G2,2 if α > 1.

Proof. First we prove the regularity on a local chart. Let U be a domain in Σ andψ : U×F → π−1(U) a local trivialization of F . Let φ(x) = (x, u(x)) and dA = d+Ain U where A satisfies (1)–(3) of Theorem 2.2 which satisfies the Euler–Lagrangeequation (3.2). Since φ ∈ S1,2α(U) and dim Σ = 2, by Sobolev embedding wehave φ ∈ C1−1/α(U,F) ⊂ C0(U,F). By Morrey [10, Chap. 1, Theorem 1.11.1’], theequation for u in (3.3) implies du ∈ W 1,2(U,F ). So dφ belongs to W 1,2(Σ,F), henceLq(Σ,F), ∀ q <∞ by Sobolev embedding. We may now differentiate and write theequation for u as in (3.4), i.e.

∆(A,u)u+ Φ(A, u) = 0,

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Critical Points of Yang–Mills–Higgs Functional 473

where the operator ∆(A,u) : W k+2,p(U,F ) →W k,p(U,F ) is given by

∆(A,u)v = ∆v + 2(α− 1)〈d2v, dAu〉dAu

1 + |dAu|2 .

This is a perturbation of the Laplace operator and when α− 1 is small enough, it’selliptic and has a nice inverse.

We now prove the smoothness of (A, u) by a two-step bootstrapping. For k ≥1, 2 ≥ p > 1, we have in real dimension 2 Sobolev multiplications W k,p ⊗W k,p →W k,p′

, where p′ > 1 is a real number slightly smaller than p. On the other hand,since µ is smooth, the composition µ(u) lies in the same Sobolev space as u does.Hence

‖Φ(A, u)‖W k,p′ ≤ C(‖du‖W k,p , ‖A‖W k,p),

‖Ψ(A, u)‖W k−1,p′ ≤ C(‖du‖W k,p , ‖A‖W k,p).

Now since du,A ∈ W 1,2, we have Φ(A, u) ∈ W 1,p(U,F ) for some 1 < p < 2. Itfollows from the property of ∆(A,u) that u ∈W 3,p(U ′, F ) for some smaller U ′ ⊂ U .On the other hand, we have Ψ(A, u) ∈ L2(U, g) and it follows from the ellipticequation for A in (3.4) that A ∈ A2,2(U ′). This in turn gives Φ(A, u) ∈ W 2,p′

(U ′, F )for a smaller number p′′ > 1 and the equation for u now implies u ∈ W 4,p′

(U ′′, F )for some smaller domain U ′′ ⊂ U ′. Then we go back to the equation for A againand get A ∈ A3,p′(U ′′) since Ψ(A, u) is now of class W 1,p′

(U ′). Iterate like thisagain and again. Note that although the number p and the domain U get smallereach time, we can always ensure that p > 1 and U does not shrink to a point. Atthe end we will get the smoothness of (A, u) at any domain smaller than U .

Observe that if (s∗A, s∗φ) is smooth and s′ is any smooth gauge transformation,then ((s′s)∗A, (s′s)∗φ) is still smooth. In particular, we can take s′ near s−1 in theC0 norm (since s is in W 2,2 → C0). This means that the gauge transformationsending (A, φ) to a smooth pair may be taken arbitrarily close to identity in the C0

norm.Next we piece together and show there exists a global gauge transformation

s ∈ G2,2 such that (s∗A, s∗φ) is smooth. We have already shown that for anyx ∈ Σ there is a domain U and a local gauge transformation s ∈ G2,2(U) such that(s∗A, s∗φ) is smooth on U . At the intersection of two overlapping domains U1 andU2 the transformations s1 and s2 will differ by a smooth map s1s−1

2 : U1 ∩U2 → G.A standard gluing argument then gives the desired global gauge transformation.

We show for example how to glue s1 and s2 to a gauge s on U1 ∪ U2. Takea smooth function η defined in a neighborhood of U1 ∪ U2 such that η|U1\U2 = 1and η|U2\U1 = 0 (we may have to restrict to the complementary of a small tubularneighborhood of the boundary of U1 ∪U2 for this η to exist). By a previous commentwe may assume, without lose of generality, that s1 and s2 are close enough to the

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474 C. Song

identity e ∈ G so that on U1∩U2 we can write s1s−12 = expe h, where h : U1∩U2 →

TeG and exp is the exponential map of e. We then define

s =

s1, at U1\U2;expe(ηh)s2, at U1 ∩ U2;s2, at U2\U1.

4. Local Estimates

In this section we give some local estimates. Again let U be a domain in Σ and fixa trivialization ψ : U × F → π−1(U). Let φ(x) = (x, u(x)) and dA = d + A in U

where A satisfies (1)–(3) of Theorem 2.2. In particular, we assume ‖FA‖L2(U) < δ

for a small δ > 0.The main estimate in Sack and Uhlenbeck’s method is the so-called ε-regularity

theorem. Here we prove an analogous result for φ with small energy ‖dAφ‖L2 .

Lemma 4.1. Suppose (A, φ) ∈ A (D) × S (D) is a smooth pair which satisfiesEq. (3.4). Then there exist ε0 > 0 and α0 > 1 such that if ‖dAφ‖L2(D) < ε0 and1 ≤ α < α0, there is an estimate for any 1 < p ≤ 2, and uniform in 1 ≤ α < α0

‖u− u‖W 2,p(D′) < C(p,D′)(‖dAφ‖L2(D) + ‖FA‖L2(D)), (4.1)

where D′ ⊂ D is a smaller disk and u is the mean value of u over D.

Proof. Since F is embedded into an Euclidean space RK , it will be convenient toshift the origin and assume that u = 0. Then we have the inequality

‖u‖Lp ≤ Cp‖du‖Lp.

Because dA = d+A and u maps into a compact manifold F , we have

‖du‖Lp ≤ ‖dAu‖Lp + ‖Au‖Lp ≤ ‖dAu‖Lp + C‖A‖Lp .

So we get

‖u‖W 1,p ≤ C(‖dAu‖Lp + ‖A‖W 1,p). (4.2)

Let η be a cut-off function with support in D which equals to 1 in D′. From theequation for u in (3.4), one can compute

|∆(ηu)| ≤ C(η|∆u| + |du| + |u|)≤ C[η(α − 1)|d2u| + η|Φ(A, u)| + |du| + |u|]≤ C[(α − 1)|d2(ηu)| + |d(ηu)du| + |Adu| + |A2u| + |∇µ(φ)| + |du| + |u|]≤ C[(α − 1)|d2(ηu)| + |d(ηu)dAu| + |AdAu| + |A2u| + |du| + |u|].

By the standard Lp-estimate, we have

‖ηu‖W 2,p ≤ C[(α − 1)‖d2(ηu)‖Lp + ‖d(ηu)dAu‖Lp

+ ‖AdAu‖Lp + ‖A2u‖Lp + ‖u‖W 1,p ]. (4.3)

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Critical Points of Yang–Mills–Higgs Functional 475

Assume first 1 < p < 2. By Sobolev embedding W 1,p ⊂ Lq and Holder inequal-ity, we have

‖d(ηu)dAu‖Lp ≤ ‖d(ηu)‖Lq‖dAu‖L2 ≤ C‖d(ηu)‖W 1,p‖dAu‖L2 (4.4)

and

‖AdAu‖Lp ≤ ‖A‖Lq‖dAu‖L2 ≤ C‖A‖W 1,p‖dAu‖L2, (4.5)

where q = 2p/(2 − p). If (α− 1) is sufficiently small, we get by (4.2)–(4.5) that

‖ηu‖W 2,p ≤ C(‖d(ηu)‖W 1,p‖dAu‖L2 + ‖A‖W 1,2‖dAu‖L2

+ ‖dAu‖Lp + ‖A‖W 1,p). (4.6)

Thus if ‖dAu‖L2(D) is small, we arrive at

‖ηu‖W 2,p ≤ C(‖A‖W 1,2‖dAu‖L2 + ‖dAu‖Lp + ‖A‖W 1,p).

Now by (3) of Theorem 2.2, estimate (4.1) follows.Next for p = 2, we first derive the above estimate for p = 4/3. Such an estimate

gives a L4-bound for du by Sobolev embedding W 1, 43 ⊂ L4. Then one can apply

the interior L2-estimate to get (4.1) with p = 2.

As for the connection part A which behaves quite well in real dimension 2, wecan prove

Lemma 4.2. Suppose (A, φ) ∈ A (D) × S (D) is a smooth pair which satisfiesEq. (3.4). Then there is an estimate for any 1 < p < 2, and α ≥ 1 sufficientlysmall

‖A‖W 2,p(D′) < C(B,D′)(‖dAφ‖L2(D) + ‖FA‖L2(D)), (4.7)

where D′ ⊂ D is a smaller disk.

Proof. The connection A of a critical point satisfies the elliptic equation (3.4):

∆A+ Ψ(A, u) = 0,

where

Ψ(A, u) = [A, dA] + [A, [A,A]] + α(1 + |dAu|2)α−1〈dAu, u〉.Since W 1,2 ⊂ Lq, ∀ q <∞ by Sobolev, we have for any 1 < p < 2

‖[A, dA]‖Lp ≤ C‖A‖Lp∗‖dA‖L2 ≤ C‖A‖2W 1,2 ,

and

‖[A, [A,A]]‖Lp ≤ C‖A‖L3p ≤ C‖A‖W 1,2 .

On the other hand, for 1α1

= 12 + α−1

α , we have

‖(1 + |dAu|2)α−1〈dAu, u〉‖Lα1 ≤ C‖(1 + |dAu|2)α‖α−1

α

L1 ‖dAu‖L2

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476 C. Song

where

‖(1 + |dAu|2)α‖L1 ≤ Lα(A, φ) ≤ B.

Thus if α is small enough such that α1 ≥ p, we have

‖Ψ(A, u)‖Lp ≤ C(B)(‖dAu‖L2 + ‖A‖W 1,2).

Then by interior Lp-estimate and Uhlenbeck’s Theorem 2.2, the estimate (4.7)follows.

Next we state a theorem of removable singularity which we will need later. Itis an extension of the original theorem of Sacks and Uhlenbeck for harmonic maps.For the proof of this theorem, we refer to Li and Wang’s paper [8].

Theorem 4.3. Let D be the unit disc in R2. If u : D\0 → F is a W 2,2loc -map with

finite energy and satisfies the following equation

∆u+ Γ(u)(du, du) = adu+ b,

where a ∈ C0(D) and b ∈ Lp(D,TF ) for some p > 2, then u can be extended to amap u ∈ W 2,p(D,F ).

5. Convergence and Blow-Up

In the previous sections, we obtained the existence of critical points of the perturbedfunctionals Lα and some uniform estimates. Now we can prove as α→ 1, these pairssubconverge to a critical point of the original functional L except finitely manysingularities.

Lemma 5.1. Let (Aα, φα) ∈ A (D) × S (D) be a sequence of critical points of Lα

for a sequence α → 1 with Lα(Aα, φα) ≤ B. Let ‖FAα‖L2(D) < δ0 and Aα be inCoulomb gauge which satisfies (1)–(3) of Theorem 2.2. Then if ‖dAφ‖L2(D) < ε0,

there exist a subsequence (Aα′ , φα′) and a limit (A0, φ0) ∈ A (D′) × S (D′) suchthat (Aα′ , φα′) → (A0, φ0) in C∞(D′), where D′ ⊂ D. Moreover, (A0, φ0) satisfiesthe Euler–Lagrange equation (2.4).

Proof. From the uniform estimate of Lemma 4.1, we have for 1 < p < 2 andD′′ ⊂ D

‖uα − uα‖W 2,p(D′′) < C(p,D′′)(ε0 + δ0)

for α < α0. From Lemma 4.2, we have

‖Aα‖W 2,p(D′′) < C(B,D′′)(ε0 + δ0).

Then we can use the same trick as we did in Theorem 3.3 to raise the order ofderivatives in the those estimates. Namely, once we have the W k,p bounds for Aand u, we get W k−1,p bounds for Φ(Aα, φα) and Ψ(Aα, φα). Then from the ellipticequations, we can raise the order and obtain W k+1,p estimates. At the end, we get

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Critical Points of Yang–Mills–Higgs Functional 477

the C∞ bounds on (Aα, φα) by the Sobolev embedding. It follows by Ascoli–Arzelatheorem that there exists a subsequence (Aα′ , φα′) which converges in C∞(D′) toa limit (A0, φ0), where D′ ⊂ D′′.

Now let α′ go to 1 in the Euler–Lagrange equations (3.2) for (Aα′ , φα′), it isobvious that (A0, φ0) satisfies (2.4).

To investigate the convergence over the whole surface Σ, we can cover the com-pact surface Σ by finitely many disks, with the disks of half the radius also coveringΣ, and the metric on these disks uniformly close to the flat metric. These disks canbe as small as we want and we can assume that each point of Σ is covered at mostl times, where l is uniform as the radius of the disks go to 0. We have the localconvergence in every small disk by Lemma 5.1. But to get a global limit, we needto piece them together using a gluing technique of Uhlenbeck [20].

Lemma 5.2. Let (Aα, φα) ∈ A × S be a sequence of critical points of Lα for asequence α → 1 with Lα(Aα, φα) ≤ B. Then there exist a subsequence β ⊂ αand a finite number of points x1, . . . , xd ⊂ Σ such that

Aβ → A0 in C∞(Σ)

φβ → φ0 in C∞(Σ\x1, . . . , xd).Moreover (A0, φ0) extends to a smooth critical point of L on Σ.

Proof. From the bound on the curvature and the scaling property of ‖FAα‖L2 byLemma 2.1, we deduce that for any δ > 0, there exist a small r(δ) and a rescalingλr : D → Dr which sends z in the unit disk D to rz ∈ Dr, such that for any x ∈ Σ

‖Fλ∗rAα‖L2(D) = r(δ)‖FAα‖L2(Dr) ≤ r(δ)B < δ.

Note that r(δ) is uniform for α and x. Taking this into account, we define r0 = r(δ0),where δ0 is the constant in Uhlenbeck’s Theorem 2.2.

Cover Σ by finitely many geodesic balls Di with radius smaller than r0. ByUhlenbeck’s Theorem 2.2, we can find trivializations for any α and i

σi,α : P |Di → Di ×G,

such that the localized connections Ai,α on each ball Di are in Coulomb gaugewhich satisfies

‖Ai,α‖W 1,2(Di) ≤ C. (5.1)

And we have

φi,α(x) = σi,α φα(x) = (x, ui,α(x))

where ui,α : Di → F → Rk is a localized map.

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478 C. Song

On any nonempty intersection Di ∩ Dj we have transition functions gij,α :Di ∩ Dj → G such that σi,α = gij,ασj,α and they satisfy the cocycle conditiongij,α = gik,αgkj,α. And the connections satisfy Ai,α = g∗ij,αAj,α. This is equivalentto dgij,α = g−1

ij,αAj,α − Ai,αg−1ij,α. The transition functions gij,α take values in the

compact groupG. So using the bound (5.1) and the Sobolev multiplication theoremswe get uniform bounds of ‖gij,α‖W 2,2(Di∩Dj).

From the weak compactness, we can find possibly after choosing a subsequencethat as α→ 1

Ai,α Ai in W 1,2,

gij,α gij in W 2,2.

This implies strong convergence gij,α → gij in C0. In particular, the cocycle condi-tion is preserved, i.e. gij = gjkgki and hence gij defines a bundle isomorphic to P .On the other hand, the consistence conditions are also preserved, i.e. Ai = g∗ijAj ,thus Ai represents a connection A0 ∈ A1,2 on P . In fact, this construction isessentially the main idea in proving Theorem 3.1 which implies that, there exists asequence of gauge transformations sα ∈ G2,2 such that s∗αAα A0 in A1,2.

But in our situation, Lemma 4.2 gives a better bounds for every Di and 1 <p < 2 that

‖Aα‖W 2,p( 12 Di) ≤ C. (5.2)

Thus s∗αAα actually converges to A0 strongly in C0.Similarly, if φi,α → φi in W 2,p → C0, or equivalently if ui,α → ui in C0(Di, F ),

the consistence conditions are also preserved, i.e. φi = gijφj , and φi represents asection φ0 ∈ S1,2.

Now choose m0 large enough such that 2−m0 < r0. Cover Σ by finitely manydisks Dm

i = D(xmi , 2

−m) which are geodesic balls with center xmi and radius 2−m <

r0, such that each point x ∈ Σ is covered at most l times and the disks of half radius12D

mi also cover Σ. The above construction of trivializations applies to eachm ≥ m0.Since each point x ∈ Σ is contained in at most l balls, we have∑

i

∫Dm

i

|dAαφα|2 ≤ lB

and for each α there are at most d < lB/ε0+1 disks with∫

Dmi|dAαφα|2 > ε0, where

ε0 is the constant in Lemma 4.1. Suppose these are the first d disks Dmi i=1,...,d. In

the other “good” disks where the ε-regularity holds, we have the C∞ convergencein 1

2Dmi by Lemma 5.1. Since the consistence conditions are preserved, the limits in

different disks can be patched together. Thus there exists a subsequence α(m) ⊂α such that (Aα(m), φα(m)) → (A0, φ0) in C∞(Σ\(⋃d

i=1Dmi )). Since Σ is compact,

xmi m≥m0 sub-converges to a point xi for each 1 ≤ i ≤ d.Then we can choose a diagonal subsequence β of the sequences α(m). It is

obvious that as β → 1

(Aβ , φβ) → (A0, φ0) in C∞(Σ\x1, . . . , xd).

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Critical Points of Yang–Mills–Higgs Functional 479

Moreover (A0, φ0) satisfies the Euler–Lagrange equations (2.4) in Σ\x1, . . . , xd.Now by (5.2) and Sobolev embedding W 2,p ⊂ C0, it obvious that A0 ∈ C0(Σ) andone can check Eq. (2.6) of φ0 satisfies the requirements of removable singularityTheorem 4.3. So φ0 extends to a W 2,p map all over Σ. The proof of Theorem 3.3then gives the regularity. Thus our proof is finished.

In the convergence process to harmonic maps, Sack and Uhlenbeck discoveredthe so-called blow-up phenomenon which generates harmonic spheres. In our case,it is almost the same because after rescaling, the connection vanishes on the singu-larities. Moreover, since the blow-up only occurs at a single fiber, the structure ofthe fiber bundle does not make any influence, either. In particular, we have

Lemma 5.3. Let (Aα, φα) be a sequence of critical points of Lα on a trivialbundle D × G → D over the unit disk D ∈ C with Lα(Aα, φα) ≤ B. Suppose(Aα, φα) → (A, φ) in C∞(D\0) with the origin a singular point such that

limα→1

∫Dr

|dAαφα|2 > ε0

for any r > 0. Then there exist finitely many non-trivial harmonic spheresv1, . . . , vk : S2 → F such that

limr→0

limα→1

∫Dr

|dAαφα|2 ≥k∑

i=1

∫S2

|dvi|2.

Proof. Let ρα be a sequence of real numbers which converges to 0 as α → 1. Forany R > 0, define the dilation map λα : D(R) → D(Rρα) by λα(x) = ραx. Thenwe can define the rescaled map vα : D(Rρα) → F by

vα(x) = λ∗α u(x) = u(ραx)

and the pull-back connection by

Aα(x) = λ∗αAα(x) = ραAα(ραx).

The energy of u is scaling invariant by Lemma 2.1, i.e.∫

D(Rρα)

|dAαuα|2dx =∫

D(R)

|dAαvα|2dx.

On the other hand, we have

‖FAα‖2

L2(D(R)) = ρ2α‖FAα‖2

L2(D(Rρα)) → 0.

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480 C. Song

By Theorem 2.2, when ρα is small enough

‖Aα‖W 1,2(D(R)) ≤ C‖FAα‖2

L2(D(R)) → 0. (5.3)

Moreover, we deduce from Eq. (3.4) of uα that vα satisfies

−∆vα − Γ(vα)(dvα, dvα) = 2(α− 1)〈d2vα − Aαdvα, dAα

vα〉dAαvα

ρ2α + |dAα

vα|2

+ 2Aαdvα + A2αvα + ρ2

α

(µ(vα) − c)∇µ(vα)α(1 + |dAαuα|2)α−1

. (5.4)

and Aα satisfies

−∆Aα − [Aα, dAα] − [Aα, [Aα, Aα]] = ρ3α(1 + |dAαuα|2)α−1〈dAαuα, uα〉. (5.5)

By the same analysis as in the proof of Lemma 5.2, and by a diagonal argument onR → ∞ we can prove

Aα → 0 in C∞(C)

vα → v1 in C∞(C\x1, . . . , xd).Moreover, by letting α→ 1 in (5.4), v1 satisfies the equation of harmonic map

∆v1 − Γ(v1)(dv1, dv1) = 0.

By the removable singularity theorem for harmonic maps [17], v1 extends to aharmonic map on S2. Obviously, the energy of v1 satisfies

limα→1

∫Dr

|dAαφα|2 ≥∫

S2|dv1|2 +

d∑i=1

limr′→0

limα→1

∫D(xi,r′)

|dvα|2

for any r > 0.Then we can repeat the blow-up procedure at the concentration points of vα,

and so on. By Sacks and Uhlenbeck [17], if the harmonic spheres are not trivial,then the energy is bounded from blow by a constant ε1. Because the total energy isbounded by B, there are at most finitely many harmonic spheres and the processwill stop after finite steps, as desired.

6. Energy Identity

In this section, we prove the last part of the main Theorem 1.1, which is the so-calledenergy identity.

Let us first recall the energy identity for harmonic maps from surface. Li andWang [7], and Lamm [6] discovered a weak energy identity for Sacks–Uhlenbeck-type approximation for harmonic maps from surfaces. More precisely, they provedthe following:

Theorem 6.1. Let Σ be a smooth, compact Riemannian surface and let F be asmooth and compact Riemannian manifold, which we assume to be isometrically

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Critical Points of Yang–Mills–Higgs Functional 481

embedding into Rn. Let uα ∈ C∞(Σ, F )(α → 1) be a sequence of critical points of

Eα with uniformly bounded energy. Moreover we assume that uα satisfies

lim infα→1

(α− 1)∫

Σ

log(1 + |duα|2)(1 + |duα|2)αdV = 0. (6.1)

Then there exists a sub-sequence, still denoted by uα, and at most finitely manypoints x1, . . . , xd, such that

uα → u0 in C∞(Σ\x1, . . . , xd),where u0 : Σ → F is a smooth harmonic map.

Moreover, there exist at most finitely many harmonic spheres vi : S2 → F, 1 ≤i ≤ k such that

limα→1

Eα(uα) = E(u0) + vol(Σ) +k∑

i=1

E(vi).

For the proof, one can refer to [6]. We would like to mention that Li and Wanggave a very detailed analysis about the “neck” at blow-up point in [7].

Another key ingredient in proving the energy identity is to verify that Eα sat-isfies the requirement (6.1). This was first observed by Struwe [19]. Here we onlyneed to make a few modifications on his proof to make it compatible with ourfunctional Lα.

Lemma 6.2. Let α > 1 and let I be a compact parameter space without boundary,h0 : I → A1,2 ×S1,2α be a continuous map and H be the set of maps homotopic toh0. Define

βα := infh∈H

supt∈I

Lα(h(t)).

Then for almost every α there exists a critical pair (Aα, φα) ∈ A × S of Lα withLα(Aα, φα) = βα such that

lim infα→1

(α− 1)log(

1α− 1

) ∫Σ

log(1 + |dAαφα|2)(1 + |dAαφα|2)αdV = 0. (6.2)

Proof. We follow closely the work of Struwe [19] (see also [6]).First of all, since Lα satisfies the P.S. condition by Lemma 3.2, the standard

minimax principle (see, for example, [18]) guarantees the existence of a critical point(Aα, φα) ∈ A1,2×S1,2α of Lα for every α > 1, with Lα(Aα, φα) = βα. The difficultpart is to prove (6.2). Note that βα is differentiable almost everywhere with

lim infα→1

(α− 1)log(

1α− 1

)dβα

dα= 0.

The main idea is to prove the existence of a sequence of critical points (Aα, φα)which satisfies

(∂αLα)(Aα, φα) ≤ dβα

dα+ 3.

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482 C. Song

Fix α > 1 a differentiable point of βα and let αk → α be a non-increasingsequence. Define on S1,2α′ a new functional

Lα′(φ) := Lα′(Aα, φ) for α′ > 1.

By a similar argument as in Lemma 5.2, we may assume that when αk is closeenough to α, Aαk

is close to Aα in C∞. Now define the set of sections

Ωk =φ ∈ S1,2αk

|βα − (αk−α) ≤ Lα(φ) ≤ Lαk(φ) ≤ βα +

(dβα

dα+ 2

)(αk − α)

.

We prove the lemma in several steps:

(1) Ωk = ∅.(2) For every φ ∈ Ωk we have

(∂αLα)(φ) ≤ dβα

dα+ 3.

(3) We have uniformly for all φ ∈ Ωk

sup|〈dLαk(φ), v〉 − 〈dLα(φ), v〉| : ‖v‖W 1,2αk (φ∗TFv) ≤ 1 → 0.

(4) There exists a sequence φk ∈ Ωk such that

‖dLαk(φk)‖(W 1,2αk (φ∗TFv))∗ → 0.

(5) φk → φα strongly in S1,2α.

Statements (1)–(4) can be proved one by one exactly in the same way as in [6].One only needs to keep in mind that the connection Aα and the moment map µ

are smooth, and hence the terms involving them are always under good control.Now we may proceed to prove (5). We first embed the total space F into a

Euclidean space RK and then extend Lα to a functional on W 1,2α(Σ,RK). To thisend, we can find a neighborhood N ⊂ RK of F such that for each point x ∈ N ,there exists a unique point y = P (x) ∈ F which achieves the distance from x to F ,i.e. |x−y| = dist(x,F). Moreover, one can prove the map P : N → F is smooth andso is the tangent map dP : TN → TF . Next we choose a cut-off function η withsupport in N which equals to 1 on F . Then we can define an extended “connection”on W 1,2α(Σ,RK) by

DAφ = η(φ) · πA dPφ dφand a map µ : RK → g∗ by

µ(x) = η(x) · µ P (x).

It is easy to see DA|F = dA, µ|F = µ and they vanish outside of N . Now we definea functional on W 1,2α(Σ,RK) by

Lα(φ) :=∫

Σ

(1 + |DAαφ|2)αdV + ‖FAα‖2L2 + ‖µ(φ) − c‖2

L2, ∀φ ∈W 1,2α(Σ,RK).

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Critical Points of Yang–Mills–Higgs Functional 483

Obviously, we have Lα|S 1,2α = Lα, hence dLα|S 1,2α = dLα.Following the proof of [21, Lemma 3.26] we get that

‖(id− πA dPφk)(φk − φl)‖W 1,2αk (Σ,RK) → 0, k, l → ∞.

Hence we get from Step (4) as in [21, Lemma 3.7] that

〈dLαk(φk), φk − φl〉 → 0, k, l → ∞.

Finally, we can deduce by the convexity of the functional as in [6] that φk actuallyconverges to φα strongly in W 1,2α(Σ,F), hence in S1,2α.

Obviously, (6.2) implies (6.1). Moreover, from the proof of Lemma 5.3, we knowthat after rescaling, the blow-up process is exactly the same as in the harmonicmap case. So we conclude that the energy identity in our main Theorem 1.1 holds.

Acknowledgment

The author would like to express his deep and sincere gratitude to Prof. Gang Tianfor his instructions and encouragement. He would also like to thank the referee forcareful reading and valuable advices.

The author was partially supported by 973 project of China, Grant No.2006CB805902.

Appendix. Background Knowledge

A.1. Connection on fiber bundles

Let G be a compact Lie group and Σ be a compact Riemann surface with a Rie-mannian metric g. Let P be a principal G-bundle on Σ with projection π : P → Σ.

Definition A.1. A connection A on a principal G-bundle P is a horizontal subdistribution H of the tangent bundle TP such that

(1) at every point p of P , the restriction map (π∗)p|Hp : Hp → Tπ(p)Σ is an iso-morphism;

(2) H is invariant under the left action of G, i.e. for every p ∈ P , g ∈ G, we have

(g∗)p(Hp) = Hg·p.

Another equivalent definition is to define a g-valued 1-form on P .

Definition A.2. A connection on a principal G-bundle P is a 1-form ω on P takingits value in the Lie algebra g of G, such that

(1) restriction of the form ω to the fiber G yields the canonical 1-form ω0 on G

defined as follows: if ξ is any left-invariant tangent vector field on G, thenω0(y, ξ(y)) = ξ, y ∈ G;

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484 C. Song

(2) under the left action of G we have

g∗ω = Ad(g) · ω.The above two definitions are equivalent in the sense that if ω is a connection

and we let

Hp = ξ ∈ TpP |ω(ξ) = 0, ∀ p ∈ P,

then H is a connection in the sense of Definition A.1. Conversely, every connectionH on P introduces a splitting

TP = H ⊕ V,

where V is the vertical distribution with

Vp = Tp(π−1(p)) = ker(π∗)p.

Then there is a projection πA : TP → V induced by this splitting. On the otherhand, for all ξ ∈ g, which can be viewed as a tangent vector at the unit elemente ∈ G, we can define an isomorphism ip : g → Vp such that

ip(ξ) =d

dt

∣∣∣t=0

expe(tξ) · p, ∀ p ∈ P.

Thus we get a g-valued 1-form

ω = i−1 πA,

which is a connection in the sense of Definition A.2.Let F be a symplectic manifold of dimension n with a symplectic form ωF .

Suppose there is an action of G on F on the left, then we can define an associatedfiber bundle F = P ×G F = (P × F )/∼, where the equivalent relation ∼ is definedas follows: we say (p1, f1), (p2, f2) ∈ P × F is equivalent if there is g ∈ G such thatb1 = b2 · g, f1 = g−1 · f2.

Any connection H on the principal bundle P induces a connection on the asso-ciated bundle F . More precisely, H induces a horizontal distribution H on F , hencea splitting TF = H⊕ TFv and a projection πA : TF → TFv. Then we can definethe covariant derivative of a section φ ∈ Γ(F) by

dAφ = πA dφ.Here dφ is the full derivative of φ and dAφ is a one form on Σ with values inthe pull-back bundle φ∗TFv. Alternatively, if ω is a connection on P , we can writeω = ω0+A where A is a g-valued 1-form on the base manifold Σ. Then the covariantderivative is given by

dA = d+A,

where in local charts A = Aαdxα and Aαφ is defined by

Aαφ =d

dt

∣∣∣t=0

expe(tAα) · φ.

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Critical Points of Yang–Mills–Higgs Functional 485

Suppose there is a compatible almost complex structure J and a symplectic formωF on F . Then we can define a Riemannian metric h by ωF (·, J ·). A connectionis called metric (or unitary under Hermitian metric) if it is compatible with h.Namely, assume X,Y ∈ TFv, then A satisfies

d(h(X,Y )) = h(dAX,Y ) + h(X, dAY ).

Equivalently, one can require that A is skew symmetric with respect to h, i.e.

h(AX, Y ) = h(X,A∗Y ) = −h(X,AY ), (A.1)

where A∗ acts on the T ∗Σ part of Y by contraction.

A.2. Sobolev completion

Let p ≥ 2 be a real number. Let A denote the affine space of smooth connections onthe principal bundle P . Note that though the connectionA ∈ A is not a tensor on Σ,the difference between two connections is, i.e. A1−A2 ∈ Ω1(P×Adg), ∀A1, A2 ∈ A .The Sobolev completion A1,p of A with respect to the W 1,p norm is defined byfixing a smooth connection A0 ∈ A and putting

A1,p = A0 + Ω1(P ×Ad g)W 1,p .

It is an affine space of the Banach manifold Ω1(P ×Ad g)W 1,p which is independentof the choice of A0.

Next, we define the Sobolev completion S1,p of the space of sections S = Γ(F)of the fiber bundle F . Let the total space F be embedded into an Euclidean spaceRK . We can define the distance of two sections φ1, φ2 ∈ S by the W 1,p norm ofthe difference between two maps from Σ to RK . This is a metric on S and gives acompletion S1,p. Moreover, the completion is independent of the embedding.

Finally, we consider the completion G2,p of the gauge group G = C∞(AutF)with respect to the W 2,p norm. The group G2,p is a Banach Lie group and it actssmoothly on A1,p and on S1,p. Its Lie algebra is Lie(G2,p) = Ω0(P ×Ad g)W 2,p .

The Sobolev embedding theorems all comply in the above Sobolev spaces.

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