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CONVERGENCE OF EINSTEIN YANG-MILLS SYSTEMS HONGLIANGSHAO 2 1 0 2 n Abstract. Inthispaper,weproveaconvergencetheoremforsequences a ofEinsteinYang-MillssystemsonU(1)-bundlesoverclosedn-manifolds J with some bounds for volumes, diameters, L2-norms of bundle curva- 1 turesandLn2-normsofcurvaturetensors. Thisresultisageneralization of earlier compactness theorems for Einstein manifolds. ] G 1. Introduction D . A Riemannian metric g on a smooth manifold M is called an Einstein h metric with Einstein constant λ, if g has constant Ricci curvature λ, i.e., t a m Ric = λg [ (cf. [5]). Einstein metrics are considered as the nicest metrics on mani- 1 folds. Nonetheless, it is well-known that some manifolds wouldn’t admit v any Einstein metric (cf. [5], [12]). In [16] and [18], the notion of Einstein 8 Yang-Mills system is introduced as a generalization of Einstein metrics by 3 couplingEinstein equations with Yang-Mills equations. Let L bea principal 3 0 U(1)-bundle over a smooth manifold M. If a Riemannian metric g on M . and a connection A of L satisfy the following equations 1 0 Ric− 1η = λg 2 (1) 2 1 d∗F = 0 , (cid:26) : v where F is the curvature of A and η = gklF F , then (M,L,g,F,λ) is ij ik jl i X calledanEinsteinYang-Mills(EYMforshort)systemandλiscalledtheEin- stein Yang-Mills constant. Besides Einstein metrics with flat U(1)-bundles, r a other solutions of (1) are obtained in Section 2 of [16]. The parabolic version of (1), so called Ricci Yang-Mills flow, is studied for solving (1) in [16] and [18]. A solution of (1) also solves Einstein-Maxwell equations which are studied in the literatures of both physics and mathematics (cf. [13] and ref- erences in it). In this paper, we study the compactness of families of EYM systems. Theconvergence of Einstein manifolds in theGromov-Hausdorff topology has been studied by many authors (cf. [1], [2], [4], [7], [17] etc). Gromov’s pre-compactness theorem says that, if (M ,g ) is a family of Riemannian n- i i manifolds with diameters boundedfrom above and Ricci curvaturebounded from below, a subsequence of (M ,g ) converges to a compact length space i i (X,d ) in the Gromov-Hausdorff sense (cf. [9]). In addition, if the Ricci X curvatures |Ric(g )| < µ, the volumes Vol > v > 0 for constants µ and v i Mi 1 2 HONGLIANGSHAO n independent of i, and the L2-norms of the curvature tensors of gi have an uniform bound, then it is shown in [2] that (X,d ) is an orbifold with finite X singular points {p }N . Furthermore, d is induced by a C1,α-metric g k k=1 X on the regular part X\{p }N , and by passing to a subsequence, g C1,α∞- k k=1 i convergestog intheCheeger-Gromovsense. Ifg areEinsteinmetricswith i ∞ bounded Einstein constants, then g is an Einstein metric and g converge i ∞ to g in the C -topology (cf. [1], [4], [17]). In the present paper, we prove ∞ ∞ an analogue convergence theorem for EYM systems. Providing (M,L,g,F,λ) is an Einstein Yang-Mills system, for any m ∈ N+, (M,Lm,m2g,mF, λ ) also satisfies (1). By choosing an appropriate m, m2 we can normalize the Einstein Yang-Mills system such that the EYM constant | λ | ≤ 1. Thus, we only consider EYM systems with EYM constants m2 belonging to [−1,1] in this paper. Theorem 1. Let {(M ,L ,g ,F ,λ )} be EYM systems with EYM constants i i i i i λ ∈ [−1,1], where {M } is a family of connected closed n-manifolds. As- i i sume that there are constants Ω > 0, v > 0, D > 0, C > 0 and c > 0 0 0 independent of i such that (i) Vol ≥ v > 0, and diam ≤ D, Mi Mi (ii) |F |2dµ ≤ Ω, i i ZMi (iii) b (M )≤ c , for n = 4, or 2 i 0 n |Rm(g )|2 dµ ≤ C for n > 4. i i 0 ZMi Then asubsequence of (M ,g ) converges, without changing the subscripts, i i intheGromov-Hausdorff sense, toaconnectedRiemannianorbifold(M ,g ) with finite singular points {p }N , each having a neighborhood homeo∞mor∞- k k=1 phictothe coneC Sn 1/Γ ,withΓ afinitesubgroupof O(n).Themetric − k k g is a C0 Riemannian orbifold metric on M , which is smooth off the sin- ∞ (cid:0) (cid:1) ∞ gular points. Furthermore, there is a U(1)-bundle L on the regular part Mo = M \{p }N , a Yang-Mills connection A o∞f L with curvature k k=1 F ∞, and a∞constant λ ∈ [−1,1] such that (Mo∞,L ,g∞,F ,λ ) is an E∞YM system. And, fo∞r any compact subset K ⊂∞⊂ M∞o ,∞ther∞e ar∞e embeddings ΦiK :K → Mi such that ΦKi,−1Li ∼= L |K for i ≫∞1, ∞ i, i, ΦK∗gi → g , ΦK∗Fi → F , and λi → λ , ∞ ∞ ∞ when i→ ∞ in the C -sense. ∞ Remark 1. If we assume that λ > κ > 0 for a uniform positive constant i κ, then the Ricci curvature of g is bounded below by κ, since η is a non- i negative symmetric tensor. By Myers’ Theorem, the diameters of g are i uniformly bounded from above. Therefore, the condition of diameter bounds in Theorem 1 can be removed. CONVERGENCE OF EINSTEIN YANG-MILLS SYSTEMS 3 Remark 2. If M in Theorem 1 are odd-dimensional oriented manifolds, i by Corollary 2.8 in [2], there is no singular points in M , and the EYM ∞ systems {(M ,L ,g ,F ,λ )} smoothly converges to a smooth EYM system i i i i i (M ,L ,g ,F ,λ ). ∞ ∞ ∞ ∞ ∞ n Remark3. Ifwereplace theassumptionVol ≥ v > 0and |Rm|2 dµ ≤ Mi Mi C by the condition of injective radius bounded from below, we can obtain 0 R C subconvergence of the sequence {(M ,L ,g ,F ,λ )} to a smooth EYM ∞ i i i i i system (M ,L ,g ,F ,λ ), by Theorem 1.1 in [2] and the similar argu- ∞ ∞ ∞ ∞ ∞ ments in the proof of Theorem 1. From Theorem 1, one can see that orbifolds with orbifold metrics solving Einstein Yang-Mills equations on regular parts appear naturally as limits of sequences of EYM systems. We would like to construct such orbifolds which do not admit any Einstein orbifold metrics. Firstly, let’s recall an example of EYM systems from [16]. Let g be the standard metric with Gaussian 1 curvature1onS2,g bethestandardmetricwithGaussiancurvature−1on 2 asurfaceH withgenusgbiggerthan1,andω (resp. ω )bethevolumeform 1 2 ofg (resp. g ). IfUS2andUH denotetheunittangentbundlesofS2 andH 1 2 respectively,thenF = π ω +π ω isthecurvatureofL = π 1US2⊗π 1UH ∗1 1 ∗2 2 −1 −2 on M = S2 ×H, where π and π are standard projections from M to S2 1 2 and H. For any λ < 0, the Riemannian metric g = A π g +B π g and F λ ∗1 1 λ ∗2 2 solve the Einstein Yang-Mills equations (1), i.e. (M,L,g,F,λ) is an EYM system, where A = 1 √1 2λ and B = 1 √1 2λ. Now we assume that H λ − 2λ− λ − −2λ− isahyperellipticRiemannsurface,i.e. H admitsaconformalinvolutionwith 2g+2 fixed points. Consider the involution ι of M obtained as the product of a 180 rotation of S2 around an axis and the hyperelliptic involution of ◦ H. It is clear that the Z -action induced by the involution preserves L, g 2 and F, i.e. ι 1L ∼= L, ι g = g and ι F = F. Thus the orbifold M/Z has − ∗ ∗ 2 4g+4 singular points, g induces an orbifold Riemannian metric on M/Z , 2 and (L,g,F,λ) induces an EYM system on the regular part of M/Z . We 2 claimthatM/Z wouldn’tadmitanyorbifoldEinsteinmetric. Notethatthe 2 orbifold Euler characteristic χ (M/Z ) = 1χ(M) = 2−2g < 0. However, orb 2 2 if there is an orbifold Einstein metric g on M/Z , then ′ 2 1 χ (M/Z )= |Rm(g )|2dµ ≥ 0, orb 2 8π2 ZM/Z2 ′ by (6.2) in [1], which is a contradiction. InSection 2,somepropertiesofEYMsystemsarestudied,and,inSection 3 Theorem 1 is proved. 2. Preliminary properties of EYM systems To start things off we present some basic preliminary properties of EYM systems which are used in the proof of Theorem 1. Firstly, we have the following lemma for an EYM system. 4 HONGLIANGSHAO Lemma 1. If (Mn,L,g,F,λ) is an EYM system, then the scalar curvature R of g and |F|2 are constants. Proof. By taking trace of the first equation of (1), we have 1 R− |F|2 = nλ. 2 Hence 1 0 = ∇ R− |F|2 k 2 (cid:18) (cid:19) 1 = 2gij∇ R − ∇ |F|2 i jk k 2 1 1 = 2gij∇ λg + η − ∇ |F|2 i jk 2 jk 2 k (cid:18) (cid:19) 1 = gij∇ η − ∇ |F|2 . i jk 2 k And on the other hand, gij∇ η = gij∇ (gpqF F ) i jk i jp kq = gijgpqF ∇ F +gijgpqF ∇ F jp i kq kq i jp = gijgpqF ∇ F −gpqF d F jp i kq kq ∗ p = −gijgpqF (∇ F +∇ F ) jp k qi q ik 1 = ∇ |F|2 , k 4 wherewe have usedthe Yang-Mills equation of the EYMsystem (1)and the Bianchi identity. Thus we have ∇ |F|2 = 0, and then ∇ R =0. We obtain k k the conclusion, i.e. R and |F|2 are constants. (cid:3) AsaconsequenceofLemma1, theW1,2p esitmateofthebundlecurvature F of an EYM system is easily obtained. Proposition 1. Let (M,L,g,F,λ) be an EYM system with |λ| ≤ 1. If the volume of the underlying n-manifold M is bounded from above by V, then for any 1 < p < +∞, 1 1 p 1 p |∇F|2pdµ ≤ 2max{1,2V p}|F|2 1+ |Rm|pdµ . (cid:18)ZM (cid:19) (cid:18)ZM (cid:19) ! Moreover, there exists a constant C = C(p,V,|F|2) > 0 such that kFk2 ≤ C(1+kRmk ). W1,2p Lp Proof. We know that |F|2 is a constant due to Lemma 1. Hence 0 = ∆|F|2 = 2h∆F,Fi+2|∇F|2 = 2h∆ F,Fi−2(2R F F −R F F −R F F )+2|∇F|2 d ijkl jk il ik kj ij jk ik ij = −4R F F +4hRic,ηi+2|∇F|2. ijkl jk il CONVERGENCE OF EINSTEIN YANG-MILLS SYSTEMS 5 It follows that |∇F|2 = 2|F|2|Rm|−2hRic,ηi 1 = 2|F|2|Rm|−2 η+λg,η 2 (cid:28) (cid:29) ≤ 2|F|2|Rm|−2λ|F|2 ≤ 2|F|2(1+|Rm|), by |λ| ≤ 1. So 1 1 p p p |∇F|2pdµ ≤ 2|F|2(1+|Rm|) dµ (cid:18)ZM (cid:19) (cid:18)ZM (cid:16) (cid:17) (cid:19) 1 1 p p ≤ 2|F|2 2pdµ + |Rm|pdµ   Z|Rm|≤1 ! Z|Rm|>1 !  1 1  p p ≤ 2|F|2 2pdµ + |Rm|pdµ "(cid:18)ZM (cid:19) (cid:18)ZM (cid:19) # 1 1 p ≤ 2max{1,2V p}|F|2 1+ |Rm|pdµ . (cid:18)ZM (cid:19) ! Combining with the fact that |F|2 is a constant, we have kFk2 ≤ C(1+kRmk ). W1,2p Lp (cid:3) After the classical Cheeger-Gromov’s theorem, many works have been done in the convergence of families of manifolds, and there are also many applications of these convergence results in solving geometric problems (cf. [3], [9] and [11] etc.). A key point in the proofs of these convergence results is to use the harmonic coordinates, that is, the corresponding coordinate functions are harmonic functions. One reason to use harmonic coordinates is that Ricci tensors are elliptic operators of metric tensors under such coordinates. Given tensors ξ and ζ, ξ ∗ζ denotes some linear combination of contrac- tions of ξ⊗ζ in this paper. Lemma 2. Let (M,L,g,F,λ) be an EYM system and u: U → Rn be a harmonic coordinate system of the underlying manifold M. Then in this coordinate, the EYM equations (1) are 1 ∂2g 1 (3) − gkl ij −Q (g,∂g)− gklF F −λg = 0, 2 ∂uk∂ul ij 2 ik jl ij ∂2F (4) gkl ij +P (g,∂g,∂F)+T (g,∂g,F) = 0, ∂uk∂ul ij ij 6 HONGLIANGSHAO where Q(g,∂g) = g−1 ∗2∗(∂g)∗2, (cid:0) (cid:1) P(g,∂g,∂F) = g−1 ∗2∗∂g∗∂F, and (cid:0) (cid:1) T(g,∂g,∂2g,F) = g−1 ∗3∗(∂g)∗2∗F + g−2 ∗2∗∂2g∗F. Proof. The Ricci tensor un(cid:0)der t(cid:1)he harmonic coor(cid:0)dinat(cid:1)e system is given by (cf. [15]) 1 ∂2g R = − gkl ij −Q (g,∂g) ij 2 ∂uk∂ul ij with Qij(g,∂g) = g−1 ∗2∗(∂g)∗2. The first equation of EYM equation(cid:0)s bec(cid:1)omes 1 ∂2g 1 − gkl ij −Q (g,∂g)− gklF F −λg = 0. 2 ∂uk∂ul ij 2 ik jl ij Note that the Yang-Mills equation d F = 0 is equivalent to ∆ F = 0 on ∗ d compact manifolds, i.e. F is harmonic. We use the Bochner formula of 2-forms, (∆ F) = (∆F) +2gjpgkqR F −gklR F −gklR F . d ij ij ijkl pq ik lj jk il Observethat ∆uk = 0, which is equivalent to gijΓk = 0 for all k = 1,··· ,n. ij Then in this coordinate chart, ∂2F (∆ F) = gkl ij +gkpglqR F d ij ∂uk∂ul ijkl pq ∂F ∂F ∂gpq ∂gpq −2gpq{Γl lj +Γl il}+ Γl F + Γl F pi ∂uq pj ∂uq ∂ui pq lj ∂uj pq il +2gpq{ΓmΓl F +ΓmΓl F +ΓmΓl F }. iq mp lj jq mp il jq ip lm And the Yang-Mills equation becomes ∂2F gkl ij +P (g,∂g,∂F)+T (g,∂g,∂2g,F) = 0, ∂uk∂ul ij ij where Pij(g,∂g,∂F) = g−1 ∗2∗∂g∗∂F, and (cid:0) (cid:1) Tij(g,∂g,∂2g,F) = g−1 ∗3∗(∂g)∗2∗F + g−2 ∗2∗∂2g∗F. (cid:3) (cid:0) (cid:1) (cid:0) (cid:1) CONVERGENCE OF EINSTEIN YANG-MILLS SYSTEMS 7 3. Convergence of EYM metrics Thissectionisdevoted totheproofofTheorem1. Webeginwithareview of harmonic coordinates which play essential roles in the Cheeger-Gromov convergence of Riemannian manifolds. A compact Riemannian n-manifold (M,g) is said to have an (r,σ,Cl,α) for 0 < α < 1 (resp. (r,σ,Wl,p) for 1 < p < ∞) adapted harmonic atlas, if there is a covering {B (r)}σ of M by geodesic r-balls, for which the xk k=1 balls B r also cover M andthe balls B r are disjoint, such that each xk 2 xk 4 B (10r) has a harmonic coordinate chart u: B (10r) → U ⊂ Rn, and xk (cid:0) (cid:1) (cid:0) (cid:1) xk the metric tensor in these coordinates are Cl,α (resp. Wl,p) bounded, i.e. if g = g ∂ , ∂ on B (10r), then ij ∂ui ∂uj xk (cid:16) (cid:17) C 1δ ≤ g ≤ Cδ , − ij ij ij and kg k ≤ C, (resp. kg k ≤ C,) ij Cl,α ij Wl,p for some constant C > 1, where the norms are taken with respect to the coordinates uj on B (10r). xk The following is a version of the local Cheeger-Gromov compactness the- (cid:8) (cid:9) orem (cf. Theorem 2.2 in [1], Lemma 2.1 in [2] and [11]). Proposition 2. Let V be a sequence of domains in closed C Riemannian i ∞ manifolds (M ,g ) such that V admits an adapted harmonic atlas δ,σ,Cl,α i i i foraconstant C > 1. Thenthere isasubsequencewhichconvergesuniformly on compact subsets in the Cl,α′ topology, α < α, to a Cl,α R(cid:0)iemannian(cid:1) ′ manifold V . ∞ In [2], Anderson shows a local existence of an adapted atlas. Proposition 3. (Proposition 2.5, Lemma 2.2 and Remarks 2.3 in [2]) Let (M,g) be a Riemannian manifold with |Ric | ≤ Λ, diam ≤ D and Vol ≥ v > 0 M M B(r) 0 for a ball B(r). Then, for any C > 1, there exist positive constants σ = σ(Λ,v ,n,D), ǫ = ǫ(Λ,v ,n,C) and δ = δ(Λ,v ,n,C), such that for any 0 0 0 1 < p < ∞, one can obtain a δ,σ,W2,p adapted atlas on the union U of those balls B(r) satisfying (cid:0) (cid:1) n |Rm|2 dµ ≤ ǫ. ZB(4r) More precisely, on any B(10δ) ⊂ U, there is a harmonic coordinate chart such that for any 1< p < ∞, C 1δ ≤ g ≤ Cδ , − ij ij ij and kg k ≤ C. ij W2,p 8 HONGLIANGSHAO Now we need the following lemma before proving Theorem 1. Lemma 3. Let (M,L,g,F,λ) be an EYM system with λ ∈ [−1,1]. Assume that Vol ≥ v > 0, and |F|2dµ ≤ Ω, M ZM then there is a constant C = C (v,Ω,n) such that 1 1 |Ric(g)| ≤ C . 1 Furthermore, if M is a 4-manifold and diam ≤ D, b (M)≤ c , M 2 0 then there is a constant C = C (v,Ω,D,c ) such that 2 2 0 |Rm(g)|2dµ ≤ C . 2 ZM Proof. Since (M,L,g,F,λ) is an EYM system, by Lemma 1, |F| is a constant, and |F|2Vol = |F|2dµ ≤Ω. M ZM By the assumption Vol ≥ v > 0, we have M Ω |F|2 ≤ . v Then it is clear that 2 1 |Ric(g)|2 = η(g,F)+λg 2 (cid:12) (cid:12) (cid:12)1 (cid:12) ≤ (cid:12) |F|4+λ|F|2(cid:12)+nλ2 (cid:12) (cid:12) 4 ≤ C (Ω,v,n). 1 From now on, we assume that the dimension of M is 4, diam ≤ D, and M b (M)≤ c . The Bishop comparison theorem implies that the volume of M 2 0 is bounded above by a constant c(Ω,v,D,n), i.e. Vol ≤ c(Ω,v,D,n). M By the standard Gauss-Bonnet-Chern formula (cf. [5]), the Euler characteristic χ(M) of an oriented compact 4-manifold M can be expressed as follows, 1 χ(M) = |Rm|2−4|Ric|2+R2 dµ. 8π2 ZM (cid:16) (cid:17) Thus 1 1 |Rm(g)|2dµ = χ(M)+ 4|Ric(g)|2−R2(g) dµ 8π2 8π2 ZM ZM (cid:16) (cid:17) max |Ric(g)|2 M ≤ 2+b (M)+ Vol 2 2π2 M ≤ C (v,Ω,D,c ). 2 0 CONVERGENCE OF EINSTEIN YANG-MILLS SYSTEMS 9 (cid:3) Now we are ready to prove the main theorem of this paper. Proof of Theorem 1. By Lemma 3 we know that under the assumption of n Theorem 1 the Ricci curvatures and the L2-norms of the curvature tensors of g have uniform bounds. By Theorem 2.6 in [2], a subsequence of (M ,g ) i i i converges to a Riemannian orbifold (M ,g ) with finite isolated singular points in the Gromov-Hausdorff sense. ∞Fur∞thermore, g C1,α-converges to i g on the regular part in the Cheeger-Gromov sense. Now we improve the ∞ convergence to the C -sense, and study the convergence of F . ∞ i For agiven r > 0, let {Bi (r)} bea family of metric balls of radius r such xk that {Bi (r)} covers (M ,g ), and Bi (r) are disjoint. Denote xk i i xk 2 G (r)= ∪ Bi (r) |Rm(g )|2dµ ≤ ǫ , i xk i i ( (cid:12)ZBxik(4r) ) (cid:12) where ǫ = ǫ(Ω,v,D,n,C ) > 0 is obtained in Proposition 3 for a constant 3 C > 1. SoG (r) are covered by a δ,σ,W2,p (for any 1< p < ∞) adapted 3 i atlas with the harmonic radius uniformly bounded from below by virtue of (cid:0) (cid:1) Proposition 3. In these coordinates we have W2,p bounds for the metrics, i.e. C 1δ ≤ g ≤ C δ , 3− jk i,jk 3 jk and kg k ≤ C . i W2,p 3 Since the curvature tensor Rm(g) = ∂2g+∂g∗∂g, for any 1 < p < ∞, we have kRm(g )k ≤ C(n,kgk ). i Lp W2,2p By Proposition 1 and Lemma 3, we get an uniform W1,2p estimate for the 2-forms F , i kF k2 ≤ C(1+kRm(g )k ) i W1,2p i Lp ≤ C(p,n,Ω,v,C ). 3 By the Sobolev embbeding theorem we know that for all 0 < α ≤ 1− n, 2p kg k ≤C , i C1,α 4 kF k2 ≤ C , i Cα 5 where C and C > 0 are two constants. 4 5 Note that the EYM equations are elliptic equations (3) and (4) in harmonic coordinates by Lemma 2. Now we apply the Schauder estimate for elliptic partial differential equations (cf. [10]) to (3), and obtain kg k ≤ C(n,kg k ,kF k2 )≤ C , i C2,α i C1,α i Cα 6 10 HONGLIANGSHAO foraconstant C > 0independentofi. By applyingLp-estimates forelliptic 6 differential equations (cf. [10]) to (4), kF k ≤ C(n,kg k ,kF k2 ) ≤C , i W2,2p i C2,α i L2p 7 where C > 0 is a constant independent of i. Then 7 kF k ≤ C , i C1,α 8 by the Sobolev embedding theorem. By standard elliptic theory (cf. [10]), for any l ≥ 0 and 0 < α < 1, we obtain uniform Cl,α bounds for the metrics g and the bundle curvatures F , i i kg k ≤ C(l,α,n,Ω,v,C ), i Cl,α 3 kF k ≤ C(l,α,n,Ω,v,C ). i Cl−1,α 3 Hence all the covariant derivatives of the curvature tensor have uniform bounds. By Proposition 2, there is a subsequence of G (r) converges in i the C topology, to an open manifold G with a smooth metric g and a ∞ r r harmonic 2-form F . r n Because |Rm(g )|2 dµ is uniformly bounded, there are finitely many Mi i i of disjoint balls B(r) on which |Rm|2dµ > ǫ. And the number of such R B(4r) disjoint balls are independent of r. R Now choose r → 0, with r ≤ 1r . Let j j+1 2 j j G = {x ∈ M|x ∈ G (r ) for some m ≤ j}, i i m then G1 ⊂G2 ⊂ G3 ⊂ ··· ⊂ M . i i i i j Byusingthediagonalargument,asubsequenceof G ,g ,F ,λ ,converges, i i i i in the Cl,α topology for all l ≥ 0, to an open ma(cid:16)nifold Mo w(cid:17)ith a smooth Riemannianmetric g , aharmonic2-form F andaconsta∞nt λ ∈[−1,1], ∞ ∞ ∞ suchthat(g ,F ,λ )satisfiestheEYMequations. Moreprecisely, forany compact sub∞set ∞K ⊂∞⊂ Mo , there are embeddings Φi : K → M such that b K i ∞ b i, Cl,α i, Cl,α (5) ΦK∗gi → g , ΦK∗Fi → F , and λi → λ , ∞ ∞ ∞ when i → ∞ for any l ≥ 0 and 0 < α < 1. For a compact subsebt K ⊂⊂ Mo , we fix a decomposition H2(K;Z) ∼= H2(K;Z)⊕H2(K;Z) where H2(K∞;Z) is the torsion part and H2(K;Z) is F T T F the free part, which is a lattice in H2(K;R). Then the first Chern class c1(ΦKi,−1Li) = cF1,i + cT1,i, where cF1,i ∈ HF2(K;Z), cT1,i ∈ HT2(K;Z), and √2−π1ΦiK,∗Fi representscF1,i bythestandardChern-Weiltheory,i.e. √2−π1ΦiK,∗Fi = cF1,i ∈ HF2(K;Z) ⊂ HF2(K;R). By (5), cF1,i = √2−π1ΦiK,∗Fi cohnverges toi cF1, = √2−π1F in HF2(K;R), which implies thhat, by pasising to a sub- ∞ ∞ sequenceh, cF =icF for i ≫ 1. Since there are only finite elements in 1,i 1, H2(K;Z), by passi∞ng to a subsequence, we obtain that cT ≡ cT in T 1,i 1, ∞