Adiabatic limit, Witten deformation and analytic torsion forms
aa r X i v : . [ m a t h . DG ] S e p ADIABATIC LIMIT, WITTEN DEFORMATIONAND ANALYTIC TORSION FORMS
MARTIN PUCHOL, YEPING ZHANG, AND JIALIN ZHUA
BSTRACT . We consider a smooth fibration equipped with a flat complex vector bundleand a hypersurface cutting the fibration into two pieces. Our main result is a gluingformula relating the Bismut-Lott analytic torsion form of the whole fibration to that ofeach piece. This result confirms a conjecture proposed in a conference in G¨ottingenin 2003. Our approach combines an adiabatic limit along the normal direction of thehypersurface and a Witten type deformation on the flat vector bundle. C ONTENTS
0. Introduction 21. Preliminaries 121.1. Finite dimensional Hodge theory and some estimates 121.2. Torsion forms and some estimates 141.3. Analytic torsion forms 192. Finite dimensional model 212.1. Chain complexes from a pair of linear maps 222.2. A flat family of complexes 233. Gluing formula for analytic torsion forms 233.1. A two-parameter deformation 243.2. Several intermediate results 254. One-dimensional Witten type deformation 294.1. Hodge theory for an interval 294.2. Witten type deformation on an interval 304.3. Witten type deformation on a cylinder 325. Adiabatic limit and Witten type deformation 355.1. Kernel of D Z R T D Z R T associated with small eigenvalues 425.3. De Rham operator on E [ − , ,R,T L -metric on E [ − , ,R,T T k vert ,R,T and T k hor ,R,T Date : September 30, 2020.
References 740. I
NTRODUCTION
We consider a unitarily flat complex vector bundle ( F, ∇ F ) over a compact smoothmanifold X without boundary whose cohomology with coefficients in F vanishes,i.e., H • ( X, F ) = 0 . Franz [22], Reidemeister [53] and de Rham [19] constructed atopological invariant associated with ( F, ∇ F ) , known as Reidemeister-Franz topolog-ical torsion (RF-torsion). RF-torsion is the first algebraic-topological invariant whichcan distinguish the homeomorphism types of certain homotopy-equivalent topologicalspaces [22, 53]. RF-torsion could be extended to the case H • ( X, F ) = 0 [19, 42, 58].Both the original construction of RF-torsion and its extensions are based on a complexof simplicial chains in X with values in F .By replacing the complex of simplicial chains by the de Rham complex, Ray andSinger [52] obtained an analytic version of RF-torsion, known as Ray-Singer analytictorsion (RS-torsion). In the same paper, Ray and Singer conjectured that RF-torsionand RS-torsion are equivalent.Ray-Singer conjecture was proved independently by Cheeger [18] and M¨uller [45].Their result is now known as Cheeger-M¨uller theorem. Bismut, Zhang and M¨ullersimultaneously considered its extension. M¨uller [46] extended Cheeger-M¨uller theo-rem to the unimodular case, i.e., the induced metric on the determinant line bundle det F is flat. Bismut and Zhang [12] extended Cheeger-M¨uller theorem to arbitraryflat vector bundle. There are also various extensions to equivariant cases [13, 36, 37].Wagoner [57] conjectured that RF-torsion and RS-torsion can be extended to invari-ants of a fiber bundle, i.e., a fibration π : M → S together with a flat complex vectorbundle ( F, ∇ F ) over M . Bismut and Lott [11] confirmed the analytic part of Wagoner’sconjecture by constructing analytic torsion forms (BL-torsion), which are even differ-ential forms on S . Inspired by the work of Bismut and Lott, Igusa [33] constructedhigher topological torsions, known as Igusa-Klein torsion (IK-torsion). Goette, Igusaand Williams [27, 26] used IK-torsion to detect the exotic smooth structure of fiberbundles. Dwyer, Weiss and Williams [21] constructed another version of higher topo-logical torsion (DWW-torsion). Then a natural and important problem is to understandthe relation among these higher torsion invariants.Bismut and Goette [8] established a higher version of Cheeger-M¨uller/Bismut-Zhangtheorem under the assumption that there exist a fiberwise Morse function f : M → R and a fiberwise Riemannian metric such that the fiberwise gradient of f is Morse-Smale [54]. Goette [23, 24] extended the results in [8] to fiberwise Morse functionswhose gradient vector fields are not necessarily Morse-Smale. Bismut and Goette [8]also extended BL-torsion to the equivariant case. And there are related works [16, 9].We refer to the survey by Goette [25] for an overview on higher torsions.Igusa [34] axiomatized higher torsion invariants. His axiomatization consists of twoaxioms: additivity axiom and transfer axiom. Igusa proved that IK-torsion satisfies hisaxioms. Moreover, any higher torsion invariant satisfying Igusa’s axioms is a linearcombination of IK-torsion and the higher Miller-Morita-Mumford class [48, 43, 41].Badzioch, Dorabiala, Klein and Williams [2] showed that DWW-torsion satisfies Igusa’s NALYTIC TORSION FORMS 3 axioms. Ma [39] studied the behavior of BL-torsion under the composition of sub-mersions. The result of Ma implies that BL-torsion satisfies the transfer axiom. Theadditivity of BL-torsion was proposed as an open problem in a conference on highertorsion invariants in 2003 .Igusa’s theory begins with higher torsion invariants for fibrations with closed fibers.In the additivity axiom [34, (3.1)], Igusa used fiberwise double to avoid consideringfibrations with boundaries. Assuming that the torsion invariant in question is alsodefined for fibrations with boundaries, Igusa [34, §
5] stated a gluing axiom equivalentto the additivity axiom. More precisely, given a hypersurface cutting the fibrationinto two pieces, the gluing axiom basically says that the torsion invariant of the totalfibration should be the sum of that of each piece.The gluing formula for BL-torsion was first precisely formulated by Zhu [61]. Zhuconstructed analytic torsion forms for fibrations with boundaries and formulated aprecise gluing formula for BL-torsion. This gluing formula, once proved, will lead tothe conclusion that BL-torsion satisfies the gluing axiom.Now we briefly recall previous works on the gluing formula for RS-torsion and BL-torsion. The gluing formula for RS-torsion associated with unitarily flat vector bundleswas proved by L¨uck [37]. The proof is based on Cheege-M¨uller theorem and the workof Lott and Rothenberg [36]. Vishik [56] gave an alternative proof without usingCheege-M¨uller theorem or the work of Lott and Rothenberg. The gluing formula forRS-torsion was proved by Br¨uning and Ma [15] in full generality. The proof is basedon the work of Bismut and Zhang [13], which is the equivariant version of [12], andthe work of Br¨uning and Ma [14]. In our earlier paper [50] (announced in [51]),we gave another proof by means of adiabatic limit along the normal direction of thehypersurface, which is also one of the key tools in the present paper. There are alsorelated works [28, 35, 44]. Zhu [61] proved the gluing formula for BL-torsion underthe same assumption as in [8]. Zhu [62] also proved the gluing formula for BL-torsionunder the assumption that the fiberwise cohomology of the hypersurface vanishes.This vanishing condition yields a uniform spectral gap of the fiberwise Hodge de Rhamoperator as the metric on the normal direction tends to infinity, which considerablysimplifies the analysis involved.The method used in [50] cannot be directly generalized to the family case. In otherwords, it does not lead to a proof of the gluing formula for BL-torsion in full generality.The main reason is the lack of a good interpretation of the limit of the analytic torsionforms when the metric on the normal direction of the hypersurface tends to infinity.The purpose of this paper is to prove a gluing formula for BL-torsion in full gen-erality, i.e., to solve the problem proposed in the conference on higher torsion in-variants mentioned above. The technical core of this paper consists of two analytictools: the adiabatic limit [20, 47, 17, 49] along the normal direction of hypersurface,which is exactly the same as in our earlier paper [50], and a Witten type deformation[59, 32, 12, 13, 60] on the flat vector bundle. By introducing the Witten type defor-mation, we overcome the difficulties mentioned in the previous paragraph. We willgive a more detailed explanation by the end of this introduction. NALYTIC TORSION FORMS 4
Now we briefly recall previous works on the two analytic tools used in this paper.The adiabatic limit of η -invariant first appeared in the work of Bismut and Freed [6]and in the work of Bismut and Cheeger [5]. The adiabatic limit used in our paper firstappeared in the work of Douglas and Wojciechowski [20] and was further developedin [47, 17, 49]. We refer to the introduction of [50] for more details on previousworks on the adiabatic limit. The Witten deformation was introduced by Witten [59]in the language of physics. In a series of works [29, 30, 31, 32], Helffer and Sj¨ostrandshowed that the Witten instanton complex, which arises from Witten deformation, isisomorphic to the Thom-Smale complex. Bismut and Zhang [12, §
8] extended theresult of Helffer and Sj¨ostrand to arbitrary flat vector bundles. Later they gave asimple proof in [13, §
6] (cf. [60, § Bismut-Lott’s Riemann-Roch-Grothendieck type formula and analytic torsion forms.
Let M be a smooth manifold. Let ( F, ∇ F ) be a flat complex vector bundle over M withflat connection ∇ F , i.e., (cid:0) ∇ F (cid:1) = 0 . Let h F be a Hermitian metric on F . Let F ∗ bethe bundle of antilinear functionals on F . We will view h F as a map from F to F ∗ .Following [12, (4.1)] and [11, (1.31)], set(0.1) ω ( F, h F ) = (cid:0) h F (cid:1) − ∇ F h F ∈ Ω ( M, End( F )) . Let f be an odd polynomial, i.e., f ( − x ) = − f ( x ) . We fix a square root of i , which wedenote by i / . In what follows, the choice of square root will be irrelevant. Following[11, (1.34)], set(0.2) f ( ∇ F , h F ) = (2 πi ) / Tr h f (cid:16)
12 (2 πi ) − / ω ( F, h F ) (cid:17)i ∈ Ω odd ( M ) . Bismut and Lott [11, §
1] showed that f ( ∇ F , h F ) is closed and its de Rham cohomolgyclass(0.3) f ( ∇ F ) := (cid:2) f ( ∇ F , h F ) (cid:3) ∈ H odd ( M ) is independent of h F . For a Z -graded flat complex vector bundle (cid:0) F • = L k F k , ∇ F • = L k ∇ F k (cid:1) and a Hermitian metric h F • = L k h F k on F • , we denote f ( ∇ F • , h F • ) = X k ( − k f ( ∇ F k , h F k ) ∈ Ω odd ( M ) ,f ( ∇ F • ) = X k ( − k f ( ∇ F k ) ∈ H odd ( M ) . (0.4)If f is an odd formal power series, the constructions still make sense. In the sequel,we take(0.5) f ( x ) = xe x . Now let π : M → S be a fibration with compact fiber Z . Let o ( T Z ) be the orientationline of the fiberwise tangent bundle T Z . Let e ( T Z ) ∈ H dim Z ( M, o ( T Z )) be the Eulerclass of T Z (cf. [12, (3.17)]). Let H • ( Z, F ) be the fiberwise de Rham cohomologyof Z with coefficients in F . Then H • ( Z, F ) is a Z -graded complex vector bundle over NALYTIC TORSION FORMS 5 S equipped with a canonical flat connection ∇ H • ( Z,F ) (see [11, Def. 2.4]). Bismutand Lott [11, Thm. 3.17] established the following Riemann-Roch-Grothendieck typeformula(0.6) f (cid:0) ∇ H • ( Z,F ) (cid:1) = Z Z e ( T Z ) f ( ∇ F ) ∈ H odd ( S ) . Bismut and Lott [11] refined equation (0.6). We consider a connection of the fibra-tion, i.e., a splitting(0.7)
T M = T H M ⊕ T Z , a metric g T Z on T Z and a Hermitian metric h F on F . Let ∇ T Z be the Bismut connectionassociated with T H M and g T Z [3, Def. 1.6]. Let e ( T Z, ∇ T Z ) ∈ Ω dim Z (cid:0) M, o ( T Z ) (cid:1) bethe Euler form (cf. [12, (3.17)]). Let h H • ( Z,F ) be the L -metric on H • ( Z, F ) inducedby the Hodge theory. Let Q S be the vector space of real even differential forms on S .Bismut and Lott [11, Def. 3.22] constructed a differential form T ∈ Q S depending on (cid:0) T H M, g
T Z , h F (cid:1) and showed that(0.8) d T = Z Z e ( T Z, ∇ T Z ) f ( ∇ F , h F ) − f (cid:0) ∇ H • ( Z,F ) , h H • ( Z,F ) (cid:1) . The differential form T is called the analytic torsion form of Bismut-Lott. Now weexplain the setup of our gluing formula for analytic torsion forms of Bismut-Lott. Gluing formula.
Let N ⊆ M be a hypersurface transversal to Z . We suppose that π (cid:12)(cid:12) N : N → S is surjective. Then π (cid:12)(cid:12) N is a fibration over S with fiber Y := N ∩ Z . Wesuppose that N cuts M into two pieces, which we denote by M ′ and M ′ . We identifya tubular neighborhood of N with(0.9) IN := [ − , × N such that(0.10) IN ∩ M ′ = [ − , × N , IN ∩ M ′ = [0 , × N .
Set π = π (cid:12)(cid:12) IN : IN → S . Then π is a fibration over S with fiber(0.11) IY := [ − , × Y .
For j = 1 , , set M j = M ′ j ∪ IN . Let π j : M j → S be the restriction of π . Then π j is afibration over S with fiber Z j := M j ∩ Z . For convenience, we denote(0.12) π = π , M = M, Z = Z , M = IN , Z = IY .
Then, for j = 0 , , , , we have a fibration π j : M j → S with fiber Z j .Let ( u, y ) ∈ [ − , × Y be coordinates on IY . We suppose that the splitting (0.7) on IN is the pullback of a splitting(0.13) T N = T H N ⊕ T Y .
In particular, we have(0.14) T H M (cid:12)(cid:12) IN = p ∗ (cid:0) T H N (cid:12)(cid:12) N (cid:1) , NALYTIC TORSION FORMS 6 F IGURE
1. from the top to bottom: Z = Z , Z , Z and Z = IY where p : IN → N is the canonical projection. Let g T Y be the metric on
T Y inducedby the canonical embedding
Y ֒ → Z . We suppose that the metric g T Z is product on IN , i.e.,(0.15) g T Z (cid:12)(cid:12) { u }× Y = du + g T Y . We trivialize F (cid:12)(cid:12) IN using the parallel transport along the curve [ − , ∋ u ( u, y ) with respect to ∇ F . Since ∇ F is flat, we have(0.16) ( F, ∇ F ) (cid:12)(cid:12) IN = p ∗ (cid:0) F (cid:12)(cid:12) N , ∇ F (cid:12)(cid:12) N (cid:1) , where p : IN → N is the canonical projection. We assume that(0.17) h F (cid:12)(cid:12) IN = p ∗ (cid:0) h F (cid:12)(cid:12) N (cid:1) . For j = 0 , , , , let d Z j be the fiberwise de Rham operator on Z j with values in F .Let d Z j , ∗ be the formal adjoint of d Z j with respect to the L -product (see (0.51)). TheHodge de Rham operator is defined as(0.18) D Z j = d Z j + d Z j , ∗ . We identify the normal bundle n of ∂Z j with the orthogonal complement of T ( ∂Z j ) ⊆ T Z j (cid:12)(cid:12) ∂Z j . We denote by e n the inward pointing unit normal vector field on ∂Z j . Let e n be the dual vector field. We denote by i · (resp. ∧· ) the interior (resp. exterior) multi-plication. Following [15, (1.11),(1.12)] and [50, (1.4),(1.5))], we denote Ω • abs ( Z j , F ) = n ω ∈ Ω • ( Z j , F ) : i e n ω (cid:12)(cid:12) ∂Z j = 0 o , Ω • abs ,D ( Z j , F ) = n ω ∈ Ω • ( Z j , F ) : i e n ω (cid:12)(cid:12) ∂Z j = i e n ( d Z j ω ) (cid:12)(cid:12) ∂Z j = 0 o . (0.19)The self-adjoint extensions of D Z j and D Z j , with domains(0.20) Dom (cid:0) D Z j (cid:1) = Ω • abs ( Z j , F ) , Dom (cid:0) D Z j , (cid:1) = Ω • abs ,D ( Z j , F ) , NALYTIC TORSION FORMS 7 will also be denoted by D Z j and D Z j , . In the sequel, the boundary condition as abovewill be called absolute boundary condition. By the Hodge theorem (cf. [15, Thm.1.1]), we have an isomorphism(0.21) H • ( Z j , F ) ≃ Ker (cid:0) D Z j , (cid:1) ⊆ Ω • ( Z j , F ) . Let h H • ( Z j ,F ) be the Hermitian metric on H • ( Z j , F ) induced by the L -metric on Ω • ( Z j , F ) via the identification (0.21).We have a Mayer-Vietoris exact sequence of flat complex vector bundles over S ,(0.22) · · · → H k ( Z, F ) → H k ( Z , F ) ⊕ H k ( Z , F ) → H k ( IY, F ) → · · · . Let T H ∈ Q S be the torsion form ([11, Def. 2.20], cf. § (cid:0) h H • ( Z j ,F ) (cid:1) j =0 , , , . By [11,Def. 2.22], we have(0.23) d T H = X j =0 ( − j ( j − / f (cid:0) ∇ H • ( Z j ,F ) , h H • ( Z j ,F ) (cid:1) . We put the absolute boundary condition on the boundary of Z j (see (0.19) and(0.20)). The analytic torsion form for fibration with boundary equipped with absoluteboundary condition was constructed by Zhu [61, Def. 2.18]. For j = 0 , , , , let T j ∈ Q S be the analytic torsion form associated with(0.24) (cid:0) π j , T H M (cid:12)(cid:12) M j , g T Z (cid:12)(cid:12) M j , F (cid:12)(cid:12) M j , ∇ F (cid:12)(cid:12) M j , h F (cid:12)(cid:12) M j (cid:1) . We denote(0.25) [ ∂Z j : Y ] = if j = 0;1 if j = 1 , if j = 3 . In other words, ∂Z j consists of [ ∂Z j : Y ] copies of Y . Let ∇ T Y be the Bismut connectionon
T Y associated with T H N and g T Y [3, Def. 1.6]. By [61, Thm. 2.19], we have d T j = Z Z j e ( T Z, ∇ T Z ) f ( ∇ F , h F ) + [ ∂Z j : Y ]2 Z Y e ( T Y, ∇ T Y ) f ( ∇ F , h F ) − f (cid:0) ∇ H • ( Z j ,F ) , h H • ( Z j ,F ) (cid:1) . (0.26)Let Q S, ⊆ Q S be the vector subspace of exact real even differential forms on S .The main result in this paper is the following theorem. Theorem 0.1.
The following equation holds, (0.27) T − T − T + T + T H ∈ Q S, . For any closed oriented submanifold O ⊆ S , the following map(0.28) Z O : Q S → R may be viewed as a linear function on Q S /Q S, . By the Stokes’ formula and the deRham theorem (cf. [15, Thm. 1.1 (d)]), these linear functions separate the elementsof Q S /Q S, . As a consequence, to prove Theorem 0.1, it is sufficient to show that theintegration of the left hand side of (0.27) on each O vanishes. Hence, without loss NALYTIC TORSION FORMS 8 of generality, we may and we will assume that S is a compact manifold withoutboundary in the whole paper.We note that in Theorem 0.1, we only use the absolute boundary condition, whereasthe relative boundary condition appears in the gluing formula for the RS-tosrion in[15, 50] and in the Zhu’s formulation of the gluing formula for the BL-torsion [61]. Infact, Theorem 0.1 implies Zhu’s formula. In order to keep this paper to a reasonablelength, this will be proved in a subsequent paper, in which we will also discuss moreprecisely the link between BL-torsion and IK-torsion resulting from the work of Igusa[34] combined with [39] and Theorem 0.1.Now we briefly describe the strategy of our proof. A two-parameter deformation and anomaly formulas.
For j = 1 , , set M ′′ j = M j \ IN . For R > , set(0.29) IN R = [ − R, R ] × N , M R = M ′′ ∪ N IN R ∪ N M ′′ , where we identify ∂M ′′ j = N with { ( − j R } × N ⊆ IN R for j = 1 , . Then M R is aclosed manifold. In particular, M R (cid:12)(cid:12) R =1 = M . We construct a smooth fibration(0.30) π R : M R → S as follows: π R (cid:12)(cid:12) M ′′ j = π (cid:12)(cid:12) M ′′ j for j = 1 , and π R (cid:12)(cid:12) IN R being the composition of the canoni-cal projection IN R → N and π (cid:12)(cid:12) N : N → S .For R > , let φ R : [ − , → [ − R, R ] be a smooth bijective map such that φ ′ R ( u ) > , φ ( − u ) = − φ ( u ) for u ∈ [ − , ,φ R ( u ) = u − R + 1 for u ∈ [ − , − / . (0.31)We construct a diffeomorphism ϕ R : M → M R as follows:(0.32) ϕ R (cid:12)(cid:12) M ′′ ∪ M ′′ = Id , ϕ R (cid:12)(cid:12) IN : ( u, y ) ( φ R ( u ) , y ) . Then the following diagram commutes(0.33) M ϕ R / / π (cid:15) (cid:15) M Rπ R } } ③③③③③③③③ S .
Let Z R be the fiber of π R . We construct a metric g T Z R on T Z R as follows:(0.34) g T Z R (cid:12)(cid:12) M ′′ ∪ M ′′ = g T Z (cid:12)(cid:12) M ′′ ∪ M ′′ , g T Z R (cid:12)(cid:12) IN R = du + g T Y . Set g T ZR = ϕ ∗ R (cid:0) g T Z R (cid:1) . It is obvious that (cid:0) π : M → S, g
T ZR (cid:1) and (cid:0) π R : M R → S, g
T Z R (cid:1) areisometric. We will work on one or another depending on the context.Let f ∞ : [ − , → R be a self-indexed Morse function such that (cid:8) u ∈ [ − ,
1] : f ′∞ ( u ) = 0 (cid:9) = (cid:8) − , , (cid:9) ,f ∞ ( −
1) = f ∞ (1) = 0 , f ∞ (0) = 1 . (0.35)We can construct a family smooth function (cid:0) f T : [ − , → R (cid:1) T > such that(0.36) f T ( u ) = 0 , for | u ± | e − T ; f ′ T ( u ) − f ′∞ ( u ) = O (cid:0) e − T (cid:1) . NALYTIC TORSION FORMS 9
We will view f T as a smooth function on M R as follows:(0.37) f T (cid:12)(cid:12) M \ IN = 0 , f T ( u, y ) = f T ( u/R ) for ( u, y ) ∈ IN R . Then ϕ ∗ R ( f T ) is a smooth function on M . Set(0.38) h FR,T = exp (cid:0) − T ϕ ∗ R ( f T ) (cid:1) h F . Replacing (cid:0) g T Z , h F (cid:1) by (cid:0) g T ZR , h
FR,T (cid:1) and proceeding in the same way as before, weget analytic torsion forms T j,R,T ∈ Q S ( j = 0 , , , ) and torsion form T H ,R,T ∈ Q S .By anomaly formulas [11, Thm. 3.24], [62, Thm. 1.5], the class(0.39) (cid:2) T R,T − T ,R,T − T ,R,T + T ,R,T + T H ,R,T (cid:3) ∈ Q S /Q S, is independent of R, T . As a consequence, to prove Theorem 0.1, it is sufficient toshow that (0.39) tends to zero as
R, T → + ∞ . Spectral gap and Witten type theorem.
For simplicity, the pushforward ϕ R, ∗ ( F, ∇ F , h F ) will also be denoted by ( F, ∇ F , h F ) . We construct a family of Hermitian metrics on F over Z R as follows:(0.40) h FT = e − T f T h F . Then we have h FT = ϕ R, ∗ (cid:0) h FR,T (cid:1) . Replacing (cid:0) g T Z j , h F (cid:1) by (cid:0) g T Z j R , h FR,T (cid:1) in the construc-tion of the Hodge de Rham operator D Z j and identifying ( Z j , g T Z j R ) with ( Z j,R , g T Z j,R ) via the isometry ϕ R (cid:12)(cid:12) Z , we obtain e D Z j,R T acting on Ω • abs ( Z j,R , F ) . The operator e D Z j,R T isself-adjoint with respect to the L -metric induced by g T Z j,R and h FT . For convenience,we consider the conjugated operator D Z j,R T = e − T f T e D Z j,R T e T f T , which is self-adjointwith respect to the L -metric induced by g T Z j,R and h F .We fix a constant κ ∈ ]0 , / . The following result is crucial (see Theorem 3.1):there exists α > such that for T = R κ ≫ , we have(0.41) Sp (cid:0) RD Z j,R T (cid:1) ⊆ ] − ∞ , − α √ T ] ∪ [ − , ∪ [ α √ T , + ∞ [ , where Sp( · ) is the spectrum. We call the eigenvalues of RD Z j,R T lying in [ − , (resp.out of [ − , ) small eigenvalues (resp. large eigenvalues). Let E [ − , j,R,T ⊆ Ω • ( Z j,R , F ) bethe eigenspace of RD Z j,R T associated with small eigenvalues. Set(0.42) d Z j,R T = e − T f T d Z j,R e T f T . Since d Z j,R T commutes with D Z j,R , T , we get a finite dimensional complex(0.43) (cid:0) E [ − , j,R,T , d Z j,R T (cid:1) . We will show that dim E [ − , j,R,T is independent of R for R ≫ and explicitly constructa complex ( C • , • j , ∂ ) and show that the complex (0.43) is ‘asymptotic’ to ( C • , • j , ∂ ) as T = R κ → + ∞ (see Theorem 3.3). For instance, taking j = 0 , we have C k, • = 0 for k = 0 , ,C , • = H • ( Z , F ) ⊕ H • ( Z , F ) , C , • = H • ( Y, F ) = H • ( IY, F ) (0.44) NALYTIC TORSION FORMS 10 with ∂ : H • ( Z , F ) ⊕ H • ( Z , F ) → H • ( IY, F ) being the same map as in (0.22). Thisresult may be viewed as a variation of the Witten deformation. Finite propagation speed.
By the finite propagation speed for solutions of hyper-bolic equations (cf. [55, § as T = R κ → + ∞ . On the other hand, we canexplicitly estimate the contribution of small eigenvalues by applying our Witten typetheorem (Theorem 3.3). These estimates will lead to the conclusion that (0.39) tendsto zero as T = R κ → + ∞ .If we take T = 0 and R → + ∞ , the situation on each fiber is exactly what wasstudied in our earlier paper [50]. We owe readers an explanation for introducing thesecond parameter T . Now we try to answer the following questions.- Why we cannot prove the gluing formula for analytic torsion forms by simplytaking T = 0 and R → + ∞ ?- How does the second parameter T improve the situation ?Both in [50] and in this paper, the contribution of large eigenvalues can be controlledby means of the finite propagation speed method. The difficulties come from thesmall eigenvalues. In [50], the small eigenvalues are handled in a rather brutal way:we estimate the contribution of each eigenvalue and take the sum of them. Sucha proof highly relies on the expression of the analytic torsion in terms of the zeta-function associated with the eigenvalues, which does not hold for analytic torsionforms. An alternative way is to build a model encoding the asymptotic limit of thesmall eigenvalues. However, with T = 0 and R → + ∞ , we find infinitely many smalleigenvalues. It seems hopeless to find a reasonable model. This problem is solvedby taking T = R κ → + ∞ . With the new parameter T introduced, there remainfinitely many small eigenvalues (see (0.43)). Moreover, for T = R κ large enough, thedimension of the eigenspace associated with small eigenvalues is a constant. And amodel ( C • , • j , ∂ ) is built accordingly (see (0.44)).Now we explain the model in more detail. Recall that the eigenspace associated withsmall eigenvalues is denoted by E [ − , j,R,T (see (0.43)). Since we work with a fibrationover S , both C • , • j and E [ − , j,R,T are vector bundles over S . The vague word ‘model’ shouldbe interpreted as follows: we construct a bijection (parameterized by R, T ) betweenvector bundles C • , • j → E [ − , j,R,T , which we denote by S j,R,T in this paper (see Theorem3.3). We denote(0.45) F E [ − , j,R,T = S j,R,T ( C , • j ) ⊆ E [ − , j,R,T . Then we have induced bijections(0.46) C , • j → E [ − , j,R,T /F E [ − , j,R,T , C , • j → F E [ − , j,R,T . There is a canonical way to equip C • , • j and E [ − , j,R,T with superconnections (parameter-ized by R, T ). As T = R κ → ∞ , the maps in (0.46) tend to be compatible with thesuperconnections in certain sense. Similar phenomena appeared in various works onthe analytic torsion forms (cf. [8, § § NALYTIC TORSION FORMS 11
The situation becomes more clear once we pass to the cohomology. We will constructa bijection (see (5.51), (5.56), (5.212) and (5.217))(0.47) (cid:2) S Hj,R,T (cid:3) T : H • (cid:0) C • , • j , ∂ (cid:1) → H • (cid:0) E [ − , j,R,T , d Z j,R T (cid:1) ≃ H • ( Z j , F ) , where the last isomorphism is induced by the Hodge theory. Here (cid:2) S Hj,R,T (cid:3) T is notdirectly induced by S j,R,T . Necessary modification is required since S j,R,T is not amap between complexes (see (3.34)). Both H • (cid:0) C • , • j , ∂ (cid:1) and H • ( Z j , F ) are flat vectorbundles over S . But (cid:2) S Hj,R,T (cid:3) T is not necessarily a map between flat vector bundles.To properly interpret the flatness of (cid:2) S Hj,R,T (cid:3) T , we consider the short exact sequenceinduced by (0.47),(0.48) → H (cid:0) C • , • j , ∂ (cid:1) → H • ( Z j , F ) → H (cid:0) C • , • j , ∂ (cid:1) → . This is indeed an exact sequence of flat vector bundles.This paper is organized as follows.In §
1, we establish several technical results concerning the finite dimensional Hodgetheory and torsion forms. We also recall the construction of analytic torsion forms.In §
2, we build up a finite dimensional model of the problem addressed in this paper.In §
3, we state several intermediate results and show that these results lead to The-orem 0.1. The proof of these results are delayed to §
5, 6, 7.In §
4, we study a one-dimensional Witten type deformation.In §
5, we establish the crucial spectral gap (0.41) and study the asymptotics of thecomplex (cid:0) E [ − , j,R,T , d Z j,R T (cid:1) .In §
6, we study the asymptotics of the analytic torsion forms T j,R,T as T = R κ → + ∞ .In §
7, we study the asymptotics of the torsion form T H ,R,T as T = R κ → + ∞ . Notations.
Hereby we summarize some frequently used notations and conventions.For a manifold X and a flat complex vector bundle ( F, ∇ F ) over X , we denote(0.49) Ω • ( X, F ) = C ∞ ( X, Λ • ( T ∗ X ) ⊗ F ) , the vector space of smooth differential forms on X with values in F . The de Rhamoperator on Ω • ( X, F ) is defined as follows:(0.50) d X : ω ⊗ s dω ⊗ s + ( − deg ω ω ∧ ∇ F s , for ω ∈ Ω • ( X ) , s ∈ C ∞ ( X, F ) . Then (cid:0) Ω • ( X, F ) , d X (cid:1) is the de Rham complex of smooth differential forms on X withvalues in F . Its cohomology is denoted by H • ( X, F ) .For a submanifold U ⊆ X and ω ∈ Ω • ( X, F ) , we denote by ω (cid:12)(cid:12) U ∈ C ∞ ( U, Λ • ( T ∗ X ) ⊗ F ) its restriction on U . Let j : U → X be the canonical embedding. For ω ∈ Ω • ( X, F ) closed, we denote [ ω ] (cid:12)(cid:12) U = j ∗ [ ω ] ∈ H • ( U, F ) . We remark that in general ω (cid:12)(cid:12) U / ∈ [ ω ] (cid:12)(cid:12) U ,unless dim U = dim X .If T X is equipped with a Riemannian metric g T X , and F is equipped with a Hermit-ian metric h F , we denote by (cid:13)(cid:13) · (cid:13)(cid:13) X (resp. (cid:10) · , · (cid:11) X ) the L -norm (resp. L -product) on Ω • ( X, F ) . More precisely, for ω, µ ∈ Ω • ( X, F ) , we have(0.51) (cid:10) ω, µ (cid:11) X = Z X (cid:10) ω x , µ x (cid:11) Λ • ( T ∗ x X ) ⊗ F x dv ( x ) , NALYTIC TORSION FORMS 12 where (cid:10) · , · (cid:11) Λ • ( T ∗ x X ) ⊗ F x is the scalar product on Λ • ( T ∗ x X ) ⊗ F x induced by g T Xx and h Fx ,and dv is the Riemannian volume form on ( X, g
T X ) . For a submanifold U ⊆ X , wedenote by (cid:13)(cid:13) · (cid:13)(cid:13) U (resp. (cid:10) · , · (cid:11) U ) the L -norm (resp. L -product) on C ∞ ( U, Λ • ( T ∗ X ) ⊗ F ) with respect to the induced Riemannian metric on T U . For simplicity, for ω, µ ∈ Ω • ( X, F ) , we will abuse the notations as follows,(0.52) (cid:13)(cid:13) ω (cid:13)(cid:13) U = (cid:13)(cid:13) ω (cid:12)(cid:12) U (cid:13)(cid:13) U , (cid:10) ω, µ (cid:11) U = (cid:10) ω (cid:12)(cid:12) U , µ (cid:12)(cid:12) U (cid:11) U . For any set X , we denote by Id X : X → X the identity map.For a self-adjoint operator A , we denote by Sp( A ) its spectrum. Acknowledgments.
We are grateful to Professor Xiaonan Ma for having raised thequestion which is solved in this paper. Y. Z. thanks his advisor Professor Jean-MichelBismut for helpful discussions. Y. Z. thanks Professor Weiping Zhang and ProfessorHuitao Feng for their warm reception in Chern Institute of Mathematics. J. Z. thanksShanghai Center for Mathematical Sciences for its working environment and supportduring the completion of this work.Y. Z. was supported by Japan Society for the Promotion of Science (JSPS) KAKENHIgrant No. JP17F17804 and Korea Institute for Advanced Study individual grant No.MG077401. J. Z. was supported by National Natural Science Foundation of China(NNSFC) grants No. 11601089 and No. 11571183.1. P
RELIMINARIES
This section is organized as follows. In § § § Finite dimensional Hodge theory and some estimates.
Let(1.1) ( W • , ∂ ) : 0 → W → · · · → W n → be a chain complex of finite dimensional complex vector spaces. Let H • ( W • , ∂ ) be thecohomology of ( W • , ∂ ) . Let h W • = L nk =0 h W k be a Hermitian metric on W • . Let ∂ ∗ bethe adjoint of ∂ . Set(1.2) D = ∂ + ∂ ∗ , which is self-adjoint with respect to h W • .Now we state the finite dimensional Hodge theorem without proof. Theorem 1.1.
The following orthogonal decomposition holds, (1.3) W • = Ker D ⊕ Im ∂ ⊕ Im ∂ ∗ . We have (1.4)
Ker D = Ker D = Ker ∂ ∩ Ker ∂ ∗ ⊆ W • . NALYTIC TORSION FORMS 13
Moreover, the induced map
Ker D → H • ( W • , ∂ ) w [ w ] (1.5) is an isomorphism. Let(1.6) W • = M λ > W • λ be the spectral decomposition with respect to D , i.e., D (cid:12)(cid:12) W • λ = λ Id . We denote(1.7) W • λ ′ = W • λ ∩ Ker ∂ , W • λ ′′ = W • λ ∩ Ker ∂ ∗ . The following orthogonal decomposition holds for λ > ,(1.8) W • λ = W • λ ′ ⊕ W • λ ′′ . Let (cid:13)(cid:13) · (cid:13)(cid:13) be the norm on W • induced by h W • . For w ′ ∈ W • λ ′ and w ′′ ∈ W • λ ′′ , we have(1.9) (cid:13)(cid:13) ∂ ∗ w ′ (cid:13)(cid:13) = λ (cid:13)(cid:13) w ′ (cid:13)(cid:13) , (cid:13)(cid:13) ∂w ′′ (cid:13)(cid:13) = λ (cid:13)(cid:13) w ′′ (cid:13)(cid:13) . For Λ ⊆ R , let(1.10) P Λ : W • → M λ ∈ Λ W • λ be the orthogonal projection.We state a naive estimate without proof. Proposition 1.2.
Let α, β > and w ∈ W • . If (cid:13)(cid:13) Dw (cid:13)(cid:13) αβ , then (cid:13)(cid:13) w − P [0 ,β ] w (cid:13)(cid:13) α . Now we establish a more sophisticated estimate.
Proposition 1.3.
Let α, β, γ > and w, v ∈ W • . If (1.11) (cid:13)(cid:13) ∂w (cid:13)(cid:13) αγ , (cid:13)(cid:13) ∂ ∗ v (cid:13)(cid:13) αγ , (cid:13)(cid:13) w − v (cid:13)(cid:13) β , then (1.12) (cid:13)(cid:13) w − P [0 ,γ ] w (cid:13)(cid:13) α + 2 β , (cid:13)(cid:13) v − P [0 ,γ ] v (cid:13)(cid:13) α + 2 β . Proof.
Let(1.13) w = X λ w λ , v = X λ v λ be the decompositions with respect to (1.6), i.e., w λ , v λ ∈ W • λ . For λ > , let(1.14) w λ = w ′ λ + w ′′ λ , v λ = v ′ λ + v ′′ λ be the decompositions with respect to (1.8), i.e., w ′ λ , v ′ λ ∈ W • λ ′ and w ′′ λ , v ′′ λ ∈ W • λ ′′ .By (1.9), (1.13) and (1.14), we have(1.15) (cid:13)(cid:13) ∂w (cid:13)(cid:13) = X λ> λ (cid:13)(cid:13) w ′′ λ (cid:13)(cid:13) , (cid:13)(cid:13) ∂ ∗ v (cid:13)(cid:13) = X λ> λ (cid:13)(cid:13) v ′ λ (cid:13)(cid:13) . NALYTIC TORSION FORMS 14
By (1.11) and (1.15), we have(1.16) (cid:13)(cid:13)(cid:13) X λ>γ w ′′ λ (cid:13)(cid:13)(cid:13) = X λ>γ (cid:13)(cid:13) w ′′ λ (cid:13)(cid:13) α , (cid:13)(cid:13)(cid:13) X λ>γ v ′ λ (cid:13)(cid:13)(cid:13) = X λ>γ (cid:13)(cid:13) v ′ λ (cid:13)(cid:13) α . On the other hand, by the third inequality in (1.11), (1.13) and (1.14), we have(1.17) (cid:13)(cid:13)(cid:13) X λ>γ w ′ λ − X λ>γ v ′ λ (cid:13)(cid:13)(cid:13) β . By the first identity in (1.14), (1.16) and (1.17), we have (cid:13)(cid:13)(cid:13) X λ>γ w λ (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) X λ>γ w ′′ λ (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) X λ>γ w ′ λ (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) X λ>γ w ′′ λ (cid:13)(cid:13)(cid:13) + 2 (cid:18)(cid:13)(cid:13)(cid:13) X λ>γ v ′ λ (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) X λ>γ w ′ λ − X λ>γ v ′ λ (cid:13)(cid:13)(cid:13) (cid:19) α + 2 β , (1.18)which implies the first inequality in (1.12). The second inequality in (1.12) can beproved in the same way. This completes the proof of Proposition 1.3. (cid:3) For w ∈ W • , we define (cid:13)(cid:13) w (cid:13)(cid:13) = (cid:13)(cid:13) w (cid:13)(cid:13) + (cid:13)(cid:13) Dw (cid:13)(cid:13) . Corollary 1.4.
Propositions 1.2, 1.3 hold with (cid:13)(cid:13) · (cid:13)(cid:13) replaced by (cid:13)(cid:13) · (cid:13)(cid:13) .Proof. All the properties concerning (cid:13)(cid:13) · (cid:13)(cid:13) hold for (cid:13)(cid:13) · (cid:13)(cid:13) . In particular, the adjoint of ∂ with respect to (cid:13)(cid:13) · (cid:13)(cid:13) is still ∂ ∗ . (cid:3) Torsion forms and some estimates.
Let S be a compact manifold without bound-ary.Let(1.19) ( W • , ∂ ) : 0 → W → · · · → W n → be a chain complex of complex vector bundles over S , i.e., ∂ : W • → W • +1 is a linearmap between complex vector bundles satisfying(1.20) ∂ = 0 . We extend the action of ∂ to Ω • ( S, W • ) as follows: for τ ∈ Ω k ( S ) and w ∈ C ∞ ( S, W • ) ,(1.21) ∂ (cid:0) τ ⊗ w (cid:1) = ( − k τ ⊗ ∂w . Let ∇ W • = L nk =0 ∇ W k be a connection on W • . We extend the action of ∇ W • to Ω • ( S, W • ) in the same way as in (0.50). We assume that ∇ W • is a flat connection.Equivalently, we assume that(1.22) (cid:0) ∇ W • (cid:1) = 0 . Now we assume that ( W • , ∇ W • , ∂ ) is a chain complex of flat complex vector bundles.Equivalently, we assume that(1.23) ∂ ∇ W • + ∇ W • ∂ = 0 . By (1.23), ∂ is covariantly constant with respect to the connection ∇ W • . Thus thereis a Z -graded complex vector bundle H • over S whose fiber over s ∈ S is the coho-mology of (cid:0) W • s , ∂ (cid:12)(cid:12) W • s (cid:1) (see [11, p. 307]). Let ∇ H • be the connection on H • induced NALYTIC TORSION FORMS 15 by ∇ W • in the sense of [11, Def. 2.4]. By [11, Prop. 2.5], ( H • , ∇ H • ) is a Z -graded flatcomplex vector bundle.Recall that f ( z ) = ze z . Let f (cid:0) ∇ W • (cid:1) , f (cid:0) ∇ H • (cid:1) ∈ H odd ( S ) be as in (0.4). By [11, Thm.2.19], we have(1.24) f (cid:0) ∇ W • (cid:1) = f (cid:0) ∇ H • (cid:1) . Set(1.25) A ′′ = ∂ + ∇ W • . By (1.20), (1.22), (1.23) and (1.25), we have(1.26) (cid:0) A ′′ (cid:1) = 0 , i.e., A ′′ is a flat superconnection in the sense of [11, § h W • = L nk =0 h W k be a Hermitian metric on W • . Let ω W • ∈ Ω (cid:0) S, End( W • ) (cid:1) beas in (0.1) with ( ∇ F , h F ) replaced by ( ∇ W • , h W • ) , i.e.,(1.27) ω W • = (cid:0) h W • (cid:1) − ∇ W • h W • . Let ∂ ∗ be the adjoint of ∂ . Let A ′ be the adjoint superconnection of A ′′ in the sense of[11, § § A ′ = ∂ ∗ + ∇ W • + ω W • . Set(1.29) X = 12 ( A ′ − A ′′ ) = 12 ( ∂ ∗ − ∂ ) + 12 ω W • ∈ Ω • (cid:0) S, End( W • ) (cid:1) . Let N W • be the number operator on W • , i.e., N W • (cid:12)(cid:12) W k = k Id . For t > , set h W • t = t N W • h W • . Let X t be the operator X associated with h W • t . We have(1.30) X t = 12 ( t∂ ∗ − ∂ ) + 12 ω W • . We define ϕ : Ω even ( S ) → Ω even ( S ) as follows,(1.31) ϕω = (2 πi ) − k ω , for ω ∈ Ω k ( S ) . We remark that f ′ ( z ) = (1 + 2 z ) e z . Set(1.32) f ∧ ( A ′′ , h W • t ) = ϕ Tr (cid:20) ( − N W • N W • f ′ ( X t ) (cid:21) ∈ Ω even ( S ) . Set(1.33) X t = t N W • / X t t − N W • / = √ t ∂ ∗ − ∂ ) + 12 ω W • . We have an alternative definition,(1.34) f ∧ ( A ′′ , h W • t ) = ϕ Tr (cid:20) ( − N W • N W • f ′ ( X t ) (cid:21) . We denote(1.35) χ ′ ( W • ) = X k ( − k k rk (cid:0) W k (cid:1) , χ ′ ( H • ) = X k ( − k k rk (cid:0) H k (cid:1) . NALYTIC TORSION FORMS 16
The following definition is due to Bismut and Lott [11, Def. 2.20].
Definition 1.5.
The torsion form associated with ( ∇ W • , ∂, h W • ) is defined by T (cid:0) ∇ W • , ∂, h W • (cid:1) = − Z + ∞ (cid:20) f ∧ ( A ′′ , h W • t ) − χ ′ ( H • ) − (cid:0) χ ′ ( W • ) − χ ′ ( H • ) (cid:1) f ′ (cid:16) i √ t (cid:17)(cid:21) dtt . (1.36)By [11, Thm. 2.13, Prop. 2.18], the integrand in (1.36) is integrable.Let h H • be the Hermitian metric on H • induced by h W • via the identification H • ≃ Ker (cid:0) ( ∂ + ∂ ∗ ) (cid:1) ֒ → W • defined by (1.5). Let f (cid:0) ∇ W • , h W • (cid:1) , f (cid:0) ∇ H • , h H • (cid:1) ∈ Ω odd ( S ) beas in (0.4). By [11, Thm. 2.22], we have(1.37) d T (cid:0) ∇ W • , ∂, h W • (cid:1) = f (cid:0) ∇ W • , h W • (cid:1) − f (cid:0) ∇ H • , h H • (cid:1) . Let ( f W • = L nk =0 f W k , ∇ f W • , e ∂ ) be another chain complex of flat complex vector bun-dles over S . Let e H • be its cohomology. We assume that for k = 0 , · · · , n ,(1.38) rk (cid:0) W k (cid:1) = rk (cid:0)f W k (cid:1) , rk (cid:0) H k (cid:1) = rk (cid:0) e H k (cid:1) . Let h f W • = L nk =0 h f W k be a Hermitian metric on f W • .Let g T S be a Riemannian metric on
T S . Let (cid:12)(cid:12) · (cid:12)(cid:12) be the norm on T S induced by g T S .For ω ∈ Ω • ( S ) , we denote(1.39) (cid:12)(cid:12) ω (cid:12)(cid:12) = sup k ∈ N , x ∈ S, v , ··· ,v k ∈ T x S, | v | , ··· , | v k | (cid:12)(cid:12) ω ( v , · · · , v k ) (cid:12)(cid:12) . For an operator A on W • , we denote by (cid:13)(cid:13) A (cid:13)(cid:13) its operator norm with respect to h W • .For A ∈ Ω • ( S, End( W • )) , we denote(1.40) (cid:13)(cid:13) A (cid:13)(cid:13) = sup k ∈ N , x ∈ S, v , ··· ,v k ∈ T x S, | v | , ··· , | v k | (cid:13)(cid:13) A ( v , · · · , v k ) (cid:13)(cid:13) . Let < λ min λ max such that(1.41) Sp (cid:0) ( ∂ ∗ + ∂ ) (cid:1) ⊆ { } ∪ [ λ , λ ] . Let l > such that(1.42) (cid:13)(cid:13) ω W • (cid:13)(cid:13) l . Proposition 1.6.
There exists a function C : N × N × R + × R + → R + such that forany ( W • , ∇ W • , ∂, h W • ) , ( f W • , ∇ f W • , e ∂, h f W • ) , λ min , λ max and l as above, if there exist anisomorphism of graded complex vector bundles α : W • → f W • and < δ < − λ min λ − satisfying (1.43) (cid:13)(cid:13) α ∗ e ∂ − ∂ (cid:13)(cid:13) λ min δ , − δh W • α ∗ h f W • − h W • δh W • , (cid:13)(cid:13) α ∗ ω f W • − ω W • (cid:13)(cid:13) δ , then (1.44) (cid:12)(cid:12)(cid:12) T (cid:0) ∇ W • , ∂, h W • (cid:1) − T (cid:0) ∇ f W • , e ∂, h f W • (cid:1)(cid:12)(cid:12)(cid:12) C (cid:0) dim S, rk( W • ) , l, λ max /λ min (cid:1) δ / . NALYTIC TORSION FORMS 17
Proof.
Replacing ∂ by λ − ∂ and replacing e ∂ by λ − e ∂ , we may assume that λ min = 1 .Then (1.41) and the first inequality in (1.43) become(1.45) Sp (cid:0) ( ∂ ∗ + ∂ ) (cid:1) ⊆ { } ∪ [1 , λ ] , (cid:13)(cid:13) α ∗ e ∂ − ∂ (cid:13)(cid:13) δ . By (1.45), we have(1.46) (cid:13)(cid:13) α ∗ e ∂ (cid:13)(cid:13) δ + (cid:13)(cid:13) ∂ (cid:13)(cid:13) δ + λ max . Since (cid:13)(cid:13) A (cid:13)(cid:13) = (cid:13)(cid:13) A ∗ (cid:13)(cid:13) for any operator A on W • , we have(1.47) (cid:13)(cid:13)(cid:0) α ∗ e ∂ (cid:1) ∗ − ∂ ∗ (cid:13)(cid:13) = (cid:13)(cid:13) α ∗ e ∂ − ∂ (cid:13)(cid:13) . Let e ∂ ∗ be the adjoint of e ∂ with respect to h f W • . Note that α ∗ e ∂ ∗ is the adjoint of α ∗ e ∂ withrespect to α ∗ h f W • and (cid:0) α ∗ e ∂ (cid:1) ∗ is the adjoint of α ∗ e ∂ with respect to h W • , by the secondinequality in (1.43), we have(1.48) (cid:13)(cid:13) α ∗ e ∂ ∗ − (cid:0) α ∗ e ∂ (cid:1) ∗ (cid:13)(cid:13) δ (cid:13)(cid:13) α ∗ e ∂ (cid:13)(cid:13) . By (1.45)-(1.48) and the assumption < δ < − λ − , we have (cid:13)(cid:13) α ∗ e ∂ ∗ − ∂ ∗ (cid:13)(cid:13) (cid:13)(cid:13) α ∗ e ∂ ∗ − (cid:0) α ∗ e ∂ (cid:1) ∗ (cid:13)(cid:13) + (cid:13)(cid:13)(cid:0) α ∗ e ∂ (cid:1) ∗ − ∂ ∗ (cid:13)(cid:13) δ ( δ + λ max ) + δ δλ max . (1.49)By (1.45), (1.49) and the assumption < δ < − λ − , we have (cid:13)(cid:13) ∂ ∗ + ∂ − α ∗ ( e ∂ ∗ + e ∂ ) (cid:13)(cid:13) δλ max , (cid:13)(cid:13) ∂ ∗ − ∂ − α ∗ ( e ∂ ∗ − e ∂ ) (cid:13)(cid:13) δλ max . (1.50)By (1.45) and (1.50), we have(1.51) Sp (cid:0) ( e ∂ ∗ + e ∂ ) (cid:1) ⊆ h , i ∪ h , λ i . Moreover, the dimension of the eigenspace of ( e ∂ ∗ + e ∂ ) associated with eigenvalues in (cid:2) , (cid:3) equals the dimension of Ker (cid:0) ( ∂ ∗ + ∂ ) (cid:1) . On the other hand, by (1.5) and thesecond identity in (1.38), we have(1.52) dim Ker (cid:0) ( ∂ ∗ + ∂ ) (cid:1) = rk H • = rk e H • = dim Ker (cid:0) ( e ∂ ∗ + e ∂ ) (cid:1) . As a consequence, the only possible eigenvalue of ( e ∂ ∗ + e ∂ ) in (cid:2) , (cid:3) is zero, i.e.,(1.53) Sp (cid:0) ( e ∂ ∗ + e ∂ ) (cid:1) ⊆ (cid:8) (cid:9) ∪ h , λ i . In the sequel, we will use C , C , · · · to denote constants depending on dim S , rk W • , l and λ max /λ min .Let ω f W • be as in (1.27) with ( W • , ∇ W • , h W • ) replaced by ( f W • , ∇ f W • , h f W • ) . For t > ,we denote(1.54) X t = √ t ∂ ∗ − ∂ ) + 12 ω W • , e X t = α ∗ (cid:16) √ t e ∂ ∗ − e ∂ ) + 12 ω f W • (cid:17) . NALYTIC TORSION FORMS 18
Set U = (cid:8) λ ∈ C : − < Re( z ) < (cid:9) . By (1.42), the third inequality in (1.43), (1.45),(1.50) and (1.53), for λ ∈ ∂U and t > , we have(1.55) (cid:13)(cid:13) ( λ − X t ) − (cid:13)(cid:13) C , (cid:13)(cid:13) ( λ − e X t ) − (cid:13)(cid:13) C , (cid:13)(cid:13) X t − e X t (cid:13)(cid:13) C (1 + √ t ) δ . Let f ∧ (cid:0) e A ′′ , h f W • t (cid:1) be as in (1.32) with ( ∇ W • , ∂, h W • ) replaced by ( ∇ f W • , e ∂, h f W • ) . For t > , by (1.34), we have f ∧ (cid:0) A ′′ , h W • t (cid:1) − f ∧ (cid:0) e A ′′ , h f W • t (cid:1) = 12 πi Z ∂U ϕ Tr (cid:20) ( − N W • N W • (cid:16) ( λ − X t ) − − ( λ − e X t ) − (cid:17)(cid:21) f ′ ( λ ) dλ = 12 πi Z ∂U ϕ Tr (cid:20) ( − N W • N W • λ − X t ) − ( X t − e X t )( λ − e X t ) − (cid:21) f ′ ( λ ) dλ . (1.56)By (1.55) and (1.56), we have(1.57) (cid:12)(cid:12)(cid:12) f ∧ (cid:0) A ′′ , h W • t (cid:1) − f ∧ (cid:0) e A ′′ , h f W • t (cid:1)(cid:12)(cid:12)(cid:12) C (1 + √ t ) δ . Proceeding the same way as in the proofs of [11, Prop. 2.18, Thm. 2.13] and apply-ing (1.42), the third inequality in (1.43), (1.45) and (1.53), we obtain the followingestimates: for < t < ,(1.58) (cid:12)(cid:12)(cid:12) f ∧ (cid:0) A ′′ , h W • t (cid:1) − χ ′ ( W • ) (cid:12)(cid:12)(cid:12) C t , (cid:12)(cid:12)(cid:12) f ∧ (cid:0) e A ′′ , h f W • t (cid:1) − χ ′ ( W • ) (cid:12)(cid:12)(cid:12) C t ; for t > ,(1.59) (cid:12)(cid:12)(cid:12) f ∧ (cid:0) A ′′ , h W • t (cid:1) − χ ′ ( H • ) (cid:12)(cid:12)(cid:12) C √ t , (cid:12)(cid:12)(cid:12) f ∧ (cid:0) e A ′′ , h f W • t (cid:1) − χ ′ ( H • ) (cid:12)(cid:12)(cid:12) C √ t . By (1.36), (1.38) and (1.57)-(1.59), we have (cid:12)(cid:12)(cid:12) T (cid:0) ∇ W • , ∂, h W • (cid:1) − T (cid:0) ∇ f W • , e ∂, h f W • (cid:1)(cid:12)(cid:12)(cid:12) Z δ C t dtt + Z δ − δ C (1 + √ t ) δ dtt + 2 Z + ∞ δ − C √ t dtt C δ / . (1.60)This completes the proof of Proposition 1.6. (cid:3) Remark . Let µ ∈ Ω (cid:0) S, End( f W • ) (cid:1) . We assume that µ preserves the degree, i.e., µ (cid:16) C ∞ (cid:0) S, W k (cid:1)(cid:17) ⊆ Ω (cid:0) S, W k (cid:1) for k = 0 , · · · , n . Set(1.61) f ∧ ( e ∂, h f W • t , µ ) = ϕ Tr " ( − N f W • N f W • f ′ (cid:16) √ t e ∂ ∗ − e ∂ ) + 12 µ (cid:17) . Let T (cid:0) e ∂, h f W • , µ (cid:1) be as in (1.36) with f ∧ ( A ′′ , h W • t ) replaced by f ∧ ( e ∂, h f W • t , µ ) . ThenProposition 1.6 holds with ω f W • replaced by µ and T (cid:0) ∇ f W • , e ∂, h f W • (cid:1) replaced by T (cid:0) e ∂, h f W • , µ (cid:1) .Let ( F, ∇ F ) be a flat complex vector bundle over S . Let h F and h F be Hermitianmetrics on F . Let ω F (resp. ω F ) be as in (0.1) with h F replaced by h F (resp. h F ).We consider the chain complex F Id −→ F , where the first F is equipped with Hermitian NALYTIC TORSION FORMS 19 metric h F and the second F is equipped with Hermitian metric h F . Let T (cid:0) ∇ F , h F , h F (cid:1) be its torsion form.Let l > such that(1.62) (cid:13)(cid:13) ω F (cid:13)(cid:13) l . Corollary 1.8.
There exists a function C : N × N × R + → R + such that for any ( F, ∇ F , h F , h F ) and l as above, if there exists δ ∈ (0 , − ) satisfying (1.63) − δh F h F − h F δh F , (cid:13)(cid:13) ω F − ω F (cid:13)(cid:13) δ , then (1.64) (cid:12)(cid:12)(cid:12) T (cid:0) ∇ F , h F , h F (cid:1)(cid:12)(cid:12)(cid:12) C (cid:0) dim S, rk( F ) , l (cid:1) δ / . Proof.
Note that T (cid:0) ∇ F , h F , h F (cid:1) = 0 , the inequality (1.64) is a direct consequence ofProposition 1.6. (cid:3) Analytic torsion forms.
Let π : M → S be a smooth fibration with compact fiber Z . Let N = ∂M . We assume that π (cid:12)(cid:12) N : N → S is a smooth fibration with fiber Y . Thenwe have Y = ∂Z .We identify a tubular neighborhood of N ⊆ M with [ − , × N such that N isidentified with { } × N and the following diagram commutes,(1.65) [ − , × N (cid:31) (cid:127) / / pr (cid:15) (cid:15) M π (cid:15) (cid:15) N π | N / / S , where pr : [ − , × N → N is the projection to the second factor.Let T H M ⊆ T M be a horizontal sub bundle of
T M , i.e.,(1.66)
T M = T H M ⊕ T Z .
Then we have(1.67) Λ • ( T ∗ M ) = Λ • ( T H, ∗ M ) ⊗ Λ • ( T ∗ Z ) ≃ π ∗ (cid:0) Λ • ( T ∗ S ) (cid:1) ⊗ Λ • ( T ∗ Z ) . We assume that T H M is product on [ − , × N , i.e.,(1.68) T H M (cid:12)(cid:12) N ⊆ T N , T H M (cid:12)(cid:12) [ − , × N = pr ∗ (cid:0) T H M (cid:12)(cid:12) N (cid:1) . We remark that T H N := T H M (cid:12)(cid:12) N ⊆ T N is a horizontal sub bundle of
T N , i.e.,(1.69)
T N = T H N ⊕ T Y .
Let g T Z be a Riemannian metric on
T Z . Let g T Y be the Riemannian metric on
T Y induced by g T Z via the embedding N = ∂M ֒ → M . Let ( u, y ) ∈ [ − , × N becoordinates. We assume that g T Z is product on [ − , × N , i.e.,(1.70) g T Z ( u,y ) = du + g T Yy . Let ( F, ∇ F ) be a flat complex vector bundle over M . We trivialize F (cid:12)(cid:12) [ − , × N alongthe curve [ − , ∋ u ( u, y ) using the parallel transport with respect to ∇ F . We have(1.71) ( F, ∇ F ) (cid:12)(cid:12) [ − , × N = pr ∗ ( F (cid:12)(cid:12) N , ∇ F (cid:12)(cid:12) N ) . NALYTIC TORSION FORMS 20
Let h F be a Hermitian metric on F . We assume that h F is product on [ − , × N ,i.e., under the identification (1.71), we have(1.72) h F (cid:12)(cid:12) [ − , × N = pr ∗ (cid:0) h F (cid:12)(cid:12) N (cid:1) . Set F = Ω • ( Z, F ) , which is a Z -graded complex vector bundle of infinite dimensionover S . By (1.67), we have the formal identity Ω • ( M, F ) = Ω • ( S, F ) .For U ∈ T S , let U H ∈ T H M be its horizontal lift, i.e., π ∗ U H = U . For U ∈ C ∞ ( S, T S ) , let L U H be the Lie differentiation operator acting on Ω • ( M, F ) . For U ∈ C ∞ ( S, T S ) and s ∈ Ω • ( S, F ) = Ω • ( M, F ) , we define(1.73) ∇ F U s = L U H s . Then ∇ F is a connection on F preserving the grading.Let P T Z : T M → T Z be the projection with respect to (1.66). For
U, V ∈ C ∞ ( S, T S ) ,set(1.74) T ( U, V ) = − P T Z [ U H , V H ] ∈ C ∞ ( M, T Z ) . Then
T ∈ C ∞ (cid:0) M, π ∗ (cid:0) Λ ( T ∗ S ) (cid:1) ⊗ T Z (cid:1) . Let i T ∈ C ∞ (cid:0) M, π ∗ (cid:0) Λ ( T ∗ S ) (cid:1) ⊗ End (cid:0) Λ • ( T ∗ Z ) (cid:1)(cid:1) be the interior multiplication by T in the vertical direction.The flat connection ∇ F (resp. ∇ F (cid:12)(cid:12) Z ) naturally extends to an exterior differentiationoperator on Ω • ( M, F ) (resp. Ω • ( Z, F ) = F ), which we denote by d M (resp. d Z ). Inthe sense of [11, § d M is a superconnection of total degree on F .By [11, Prop. 3.4], we have(1.75) d M = d Z + ∇ F + i T . Let T ∗ ∈ C ∞ (cid:0) M, π ∗ (cid:0) Λ ( T ∗ S ) (cid:1) ⊗ T ∗ Z (cid:1) be the dual of T with respect to g T Z .Let h F be the L -metric on F with respect to g T Z and h F . Let d M, ∗ , d Z, ∗ , ∇ F , ∗ be theformal adjoints of d M , d Z , ∇ F with respect to h F in the sense of [11, Def. 1.6]. By [11,Prop. 3.7], we have(1.76) d M, ∗ = d Z, ∗ + ∇ F , ∗ − T ∗ ∧ . Let N T Z be the number operator on Λ • ( T ∗ Z ) , i.e., N T Z (cid:12)(cid:12) Λ p ( T ∗ Z ) = p Id . Then N T Z acts on F in the obvious way. For t > , let d M, ∗ t be the formal adjoints of d M withrespect to h F t := t N TZ h F . We have(1.77) d M, ∗ t = td Z, ∗ + ∇ F , ∗ − t T ∗ ∧ . Set D t = t N TZ / (cid:0) d M, ∗ t − d M (cid:1) t − N TZ / = √ t (cid:0) d Z, ∗ − d Z (cid:1) + 12 (cid:0) ∇ F , ∗ − ∇ F (cid:1) − √ t (cid:0) T ∗ ∧ + i T (cid:1) . (1.78)We denote(1.79) ω F = ∇ F , ∗ − ∇ F ∈ Ω ( S, End( F )) . For X ∈ T Z , we denote by X ∗ ∈ T ∗ Z its dual with respect to g T Z . For X ∈ T Z , wedenote(1.80) ˆ c ( X ) = X ∗ ∧ + i X ∈ End(Λ • ( T ∗ Z )) . NALYTIC TORSION FORMS 21
By (1.78)-(1.80), we have(1.81) D t = √ t (cid:0) d Z, ∗ − d Z (cid:1) + 12 ω F − √ t ˆ c ( T ) . In particular,(1.82) D t = − t (cid:0) d Z, ∗ d Z + d Z d Z, ∗ (cid:1) + nilpotent operator , where d Z, ∗ d Z + d Z d Z, ∗ is the fiberwise Hodge Laplacian.By (1.82), the operator D t is fiberwise essentially self-adjoint with respect to theabsolute boundary condition (see (0.19)). Its self-adjoint extension with respect tothe absolute boundary condition will still be denoted by D t . Let End tr ( F ) ⊆ End( F ) be the sub vector bundle of trace class operators. Recall that f ′ ( z ) = (1 + 2 z ) e z . By(1.82), we have f ′ (cid:0) D t (cid:1) ∈ Ω • (cid:0) End tr ( F ) (cid:1) .Let Tr : End tr ( F ) → C be the trace map, which extends to Tr : End tr ( F ) ⊗ Λ • ( T ∗ S ) → Λ • ( T ∗ S ) . Let ϕ be as in (1.31).Let H • ( Z, F ) be the fiberwise singular cohomology of Z with coefficients in F . Then H • ( Z, F ) is a Z -graded complex vector bundle over S . We denote(1.83) χ ′ ( Z, F ) = dim Z X p =0 ( − p p rk (cid:0) H p ( Z, F ) (cid:1) . Now we recall the definition of analytic torsion forms [61, Def. 2.18], [11, Def.3.22].
Definition 1.9.
The analytic torsion form associated with ( T H M, g
T Z , h F ) is definedby T ( T H M, g
T Z , h F ) = − Z + ∞ (cid:26) ϕ Tr h ( − N TZ N T Z f ′ (cid:0) D t (cid:1)i − χ ′ ( Z, F )2 − (cid:16) dim Z rk( F ) χ ( Z )4 − χ ′ ( Z, F )2 (cid:17) f ′ (cid:16) i √ t (cid:17)(cid:27) dtt . (1.84)The convergence of the integral in (1.84) follows from the family local index theorem[11, Thm 3.21] [61, Thm 2.17]. And d T ( T H M, g
T Z , h F ) is given by (0.26) with j = 0 .Recall that Q S is the vector space of real even differential forms on S and Q S, ⊆ Q S is the sub vector space of exact forms. The analytic torsion form T ( T H M, g
T Z , h F ) isviewed as an element in Q S /Q S, .2. F INITE DIMENSIONAL MODEL
The construction in this section may be viewed as a model of the problem addressedin this paper, in which the fibration has zero-dimensional fibers. This section is or-ganized as follows. In § § § NALYTIC TORSION FORMS 22
Chain complexes from a pair of linear maps.
Let W , W and V be finite di-mensional complex vector spaces. Let τ : W → V and τ : W → V be linear maps.We define a chain complex (cid:0) C • ( τ , τ ) , ∂ (cid:1) as follows, → C ( τ , τ ) := W ⊕ W ∂ −→ C ( τ , τ ) := V → w , w ) τ ( w ) − τ ( w ) . (2.1)We denote(2.2) V = Im( τ ) ⊆ V , V = Im( τ ) ⊆ V .
We define a chain complex (cid:0) C • r ( τ , τ ) , ∂ (cid:1) as follows, → C ( τ , τ ) := V ⊕ V ∂ −→ C ( τ , τ ) := V → v , v ) v − v . (2.3)We denote(2.4) K = Ker( τ ) ⊆ W , K = Ker( τ ) ⊆ W . We have a short exact sequence of chain complexes,(2.5) / / / / C ( τ , τ ) Id / / C ( τ , τ ) / / / / K ⊕ K / / O O C ( τ , τ ) τ ⊕ τ / / ∂ O O C ( τ , τ ) / / ∂ O O , where K ⊕ K → C ( τ , τ ) = W ⊕ W is the direct sum of the embeddings in (2.4).Set(2.6) C • = C • ( τ , τ ) , C • = C • ( τ , Id V ) , C • = C • (Id V , τ ) , C • = C • (Id V , Id V ) . We define(2.7) α : C • → C • , α : C • → C • , β : C • → C • , β : C • → C • as follows, α (cid:12)(cid:12) C = Id W ⊕ τ , α (cid:12)(cid:12) C = τ ⊕ Id W , α (cid:12)(cid:12) C = α (cid:12)(cid:12) C = Id V ,β (cid:12)(cid:12) C = τ ⊕ Id V , β (cid:12)(cid:12) C = Id V ⊕ τ , β (cid:12)(cid:12) C = β (cid:12)(cid:12) C = Id V . (2.8)We have a short exact sequence of chain complexes,(2.9) / / C • α ⊕ α / / C • ⊕ C • β − β / / C • / / . For j = 0 , , , , let H k (cid:0) C • j , ∂ (cid:1) be the k -th cohomology group of (cid:0) C • j , ∂ (cid:1) , i.e.,(2.10) H k (cid:0) C • j , ∂ (cid:1) = Ker (cid:0) ∂ : C kj → C k +1 j (cid:1) Im (cid:0) ∂ : C k − j → C kj (cid:1) . From (2.9), we get a long exact sequence of cohomology groups,(2.11) · · · / / H k (cid:0) C • , ∂ (cid:1) / / H k (cid:0) C • ⊕ C • , ∂ (cid:1) / / H k (cid:0) C • , ∂ (cid:1) / / · · · . NALYTIC TORSION FORMS 23
We denote(2.12) W = n ( w , w ) ∈ W ⊕ W : τ ( w ) = τ ( w ) o . A direct calculation yields H (cid:0) C • , ∂ (cid:1) = W , H (cid:0) C • ⊕ C • , ∂ (cid:1) = W ⊕ W , H (cid:0) C • , ∂ (cid:1) = V ,H (cid:0) C • , ∂ (cid:1) = V / (cid:0) V + V (cid:1) , H (cid:0) C • ⊕ C • , ∂ (cid:1) = H (cid:0) C • , ∂ (cid:1) = 0 . (2.13)Thus the long exact sequence (2.11) is(2.14) / / W (cid:31) (cid:127) / / W ⊕ W τ − τ / / V / / / / V / (cid:0) V + V (cid:1) / / , where W ֒ → W ⊕ W is the direct sum of the obvious embeddings W ֒ → W and W ֒ → W .2.2. A flat family of complexes.
Now let W , W and V be flat complex vector bun-dles over a smooth manifold S . Let τ : W → V and τ : W → V be morphismsbetween flat complex vector bundles. Then the chain complexes (cid:0) C • j , ∂ (cid:1) ( j = 0 , , , )considered in § S .Let h W , h W and h V be Hermitian metrics on W , W and V . For j = 0 , , , , weconstruct a Hermitian metric h C • j = h C j ⊕ h C j on C • j as follows, h C = h W ⊕ h W ; h C = 12 h V ⊕ h V ; h C j = h W j ⊕ h V for j = 1 , h C j = h V for j = 0 , , , . (2.15)Recall that Q S, ⊆ Q S ⊆ Ω • ( S ) were defined in the paragraph containing (0.8). Let T j ∈ Q S be the torsion form (cf. § (cid:0) C • j , ∂, h C • j (cid:1) .The exact sequence (2.11) becomes an exact sequence of flat complex vector bun-dles. Let T H ∈ Q S be the torsion form (cf. § h C • j via (1.5).The following theorem is a consequence of [24, Thm. 7.37], which may be viewedas an analogue of a result of Ma on the Bott-Chern forms [38, Thm 1.2], and may alsobe viewed as a finite dimensional version of [39, Thm 0.1]. Theorem 2.1.
The following equation holds, (2.16) T − T − T + T + T H ∈ Q S, .
3. G
LUING FORMULA FOR ANALYTIC TORSION FORMS
This section is the heart of this paper. The central idea is to deform the metrics on
T Z and F such that the gluing formula considered in Theorem 0.1 degenerates to thegluing formula given in Theorem 2.1. This section is organized as follows. In § § §
5, 6, 7.
NALYTIC TORSION FORMS 24
A two-parameter deformation.
Recall that π R : M R → S was constructed in theparagraph containing (0.30). By the second identity in (0.29), we may view M ′′ , M ′′ and IN R as subsets of M R . Set(3.1) M ,R = M ′′ ∪ IN R , M ,R = M ′′ ∪ IN R , M ,R = IN R . For convenience, we denote M ,R = M R . For j = 0 , , , , set(3.2) π j,R = π R (cid:12)(cid:12) M j,R : M j,R → S .
Let Z j,R be the fiber of π j,R . We denote Z R = Z ,R .Recall that the diffeomorphism ϕ R : M → M R was constructed in (0.32). Recallthat T H M ⊆ T M was constructed in the paragraph containing (0.7). Set(3.3) T H M R = ϕ R, ∗ (cid:0) T H M (cid:1) ⊆ T M R . Then we have(3.4)
T M R = T H M R ⊕ T Z R . Recall that T H N ⊆ T N was constructed in the paragraph containing (0.13). By (0.14),(0.32) and (3.3), we have(3.5) T H M R (cid:12)(cid:12) IN R = pr ∗ (cid:0) T H N (cid:1) , where pr : IN R = [ − R, R ] × N → N is the projection to the second factor. For j = 0 , , , , set(3.6) T H M j,R = T H M R (cid:12)(cid:12) M j,R ⊆ T M j,R . Recall that the metric g T Z R on T Z R was constructed in (0.34). For j = 0 , , , , set(3.7) g T Z j,R = g T Z R (cid:12)(cid:12) M j,R . Recall that the flat complex vector bundle ( F, ∇ F ) over Z R and the Hermitian metric h F on F were constructed in the paragraph containing (0.40). We remark that (0.16)and (0.17) hold with IN replaced by IN R .Let f ∞ : [ − , → R be as in (0.35). We further assume that f ∞ ( s ) = f ∞ ( − s ) , (cid:12)(cid:12) f ′∞ ( s ) (cid:12)(cid:12) , for | s | f ∞ ( s ) = 1 − s / , for | s |
14 ; f ∞ ( s ) = ( s − b ) / , for b = ± , | s − b | . (3.8)Let χ : R → R be a smooth function such that(3.9) χ , χ (cid:12)(cid:12) ] −∞ , / = 0 , χ (cid:12)(cid:12) [1 / , + ∞ [ = 1 , χ ′ . As f ′∞ is an odd function, for T > , there exists a unique smooth function f T :[ − , → R satisfying(3.10) f T ( −
1) = f T (1) = 0 , f ′ T ( s ) = f ′∞ ( s ) χ (cid:0) e T (1 − | s | ) (cid:1) . By (3.8)-(3.10), the following uniform estimates hold,(3.11) f T ( s ) = f ∞ ( s ) + O (cid:0) e − T (cid:1) , f ′ T ( s ) = f ′∞ ( s ) + O (cid:0) e − T (cid:1) , f ′′ T ( s ) = O (cid:0) (cid:1) . NALYTIC TORSION FORMS 25
Moreover, we have supp( f T ) ⊆ (cid:2) − e − T / , − e − T / (cid:3) . We will view f T as a smoothfunction on M R in the sense of (0.37). Set(3.12) h FT = e − T f T h F . For j = 0 , , , , let(3.13) T j,R,T ∈ Q S be the analytic torsion form (cf. Definition 1.9) associated with(3.14) (cid:0) π j,R , T H M j,R , g T Z j,R , F (cid:12)(cid:12) M j,R , ∇ F (cid:12)(cid:12) M j,R , h FT (cid:12)(cid:12) M j,R (cid:1) . For j = 0 , , , , let d Z j,R be the de Rham operator on Ω • ( Z j,R , F ) . Let (cid:13)(cid:13) · (cid:13)(cid:13) Z j,R be the L -metric on Ω • ( Z j,R , F ) with respect to g T Z j,R and h F . Let d Z j,R , ∗ be the formal adjointof d Z j,R with respect to (cid:13)(cid:13) · (cid:13)(cid:13) Z j,R . Set d Z j,R T = e − T f T d Z j,R e T f T , d Z j,R , ∗ T = e T f T d Z j,R , ∗ e − T f T ,D Z j,R T = d Z j,R T + d Z j,R , ∗ T . (3.15)We remark that e T f T D Z j,R T e − T f T is the Hodge de Rham operator with respect to g T Z j,R and h FT . The self-adjoint extension of D Z j,R T with domain Dom (cid:0) D Z j,R T (cid:1) = Ω • abs ( Z j,R , F ) (cf. [50, (1.4)]) will still be denoted by D Z j,R T . By the Hodge theorem (cf. [14, Thm3.1] [50, Thm. 1.1]), the following map is bijective, Ker (cid:0) D Z j,R T (cid:1) → H • ( Z j,R , F ) ω (cid:2) e T f T ω (cid:3) . (3.16)For j = 0 , , , , let h H • ( Z j ,F ) R,T be the Hermitian metric on H • ( Z j , F ) = H • ( Z j,R , F ) induced by (cid:13)(cid:13) · (cid:13)(cid:13) Z j,R via the identification (3.16). Let(3.17) T H ,R,T ∈ Q S be the torsion form ([11, § § (cid:0) h H • ( Z j ,F ) R,T (cid:1) j =0 , , , .3.2. Several intermediate results.
We fix a constant < κ < / . Theorem 3.1.
There exists α > such that for j = 0 , , , and T = R κ ≫ , we have (3.18) Sp (cid:16) RD Z j,R T (cid:17) ⊆ (cid:3) − ∞ , − α √ T (cid:3) ∪ (cid:2) − , (cid:3) ∪ (cid:2) α √ T , + ∞ (cid:2) . Let E [ − , j,R,T ⊆ Ω • ( Z j,R , F ) be eigenspace of RD Z j,R T associated with eigenvalues in [ − , . Since d Z j,R T commutes with D Z j,R , T , we have a finite dimensional complex(3.19) (cid:0) E [ − , j,R,T , d Z j,R T (cid:1) . Recall that the chain complexes of flat complex vector bundles (cid:0) C • j , ∂ (cid:1) with j =0 , , , were constructed in § τ : W → V and τ : W → V . In the sequel, we take(3.20) V = H • ( Y, F ) , W j = H • ( Z j , F ) , τ j (cid:0) [ α ] (cid:1) = [ α ] (cid:12)(cid:12) Y for j = 1 , . NALYTIC TORSION FORMS 26
Then W , W and V are Z -graded. We will use the notations W • , W • , V • and (cid:0) C • , • j , ∂ (cid:1) to emphasis the grading, i.e., C ,k = W k ⊕ W k , C ,k = V k , etc.Now we construct a Hermitian metric on C • , • j . Let D Y be the Hodge de Rhamoperator on Ω • ( Y, F ) with respect to g T Y and h F (cid:12)(cid:12) N . We denote H • ( Y, F ) = Ker (cid:0) D Y (cid:1) .For j = 1 , , let(3.21) H • abs ( Z j, ∞ , F ) ⊆ n ω ∈ Ω • ( Z j, ∞ , F ) : d Z j, ∞ ω = d Z j, ∞ , ∗ ω = 0 o × H • ( Y, F ) be as in [50, (2.52)]. By [50, Prop. 3.16, Thm. 3.19], the map H • abs ( Z j, ∞ , F ) → H • ( Z j, ∞ , F ) = W • j ( ω, ˆ ω ) [ ω ] (3.22)is bijective. By [50, (2.39), (2.52)], the following diagram commutes,(3.23) H • abs ( Z j, ∞ , F ) / / ( ω, ˆ ω ) ˆ ω (cid:15) (cid:15) W • j [ ω ] [ ω ] | Y (cid:15) (cid:15) H • ( Y, F ) ˆ ω [ˆ ω ] / / V • . Let D Z j, ∞ be the Hodge de Rham operator on Ω • ( Z j, ∞ , F ) with respect to g T Z j, ∞ and h F . By [50, (2.40)], we have(3.24) W • j = K • j ⊕ K • , ⊥ j with K • j = n [ ω ] : ( ω, ˆ ω ) ∈ H • abs ( Z j, ∞ , F ) , ˆ ω = 0 o ,K • , ⊥ j = n [ ω ] : ( ω, ˆ ω ) ∈ H • abs ( Z j, ∞ , F ) ,ω generalized eigensection of D Z j, ∞ associated with o . (3.25) Remark . As a convention, a generalized eigenvalue (resp. eigensection) is alwaysassociated with the absolutely continuous spectrum. In other words, a generalizedeigenvalue (resp. eigensection) is not an eigenvalue (resp. eigensection).By (3.23), the definition of K • j in (3.25) is compatible with (2.4). We construct aHermitian metric h K • j on K • j as follows: for ( ω, ˆ ω ) ∈ H • abs ( Z j, ∞ , F ) with ˆ ω = 0 ,(3.26) h K • j (cid:0) [ ω ] , [ ω ] (cid:1) = (cid:13)(cid:13) ω (cid:13)(cid:13) Z j, ∞ . By [50, (2.53)], we have (cid:13)(cid:13) ω (cid:13)(cid:13) Z j, ∞ < + ∞ . Hence h K • j is well-defined. We constructa Hermitian metric h K • , ⊥ j on K • , ⊥ j as follows: for ( ω, ˆ ω ) ∈ H • abs ( Z j, ∞ , F ) with ω ageneralized eigensection of D Z j, ∞ ,(3.27) h K • , ⊥ j (cid:0) [ ω ] , [ ω ] (cid:1) = (cid:13)(cid:13) ˆ ω (cid:13)(cid:13) Y . Set(3.28) h W • j R,T = h K • j ⊕ √ π RT − / h K • , ⊥ j . NALYTIC TORSION FORMS 27
Let h V • be the Hermitian metric on V • induced by (cid:13)(cid:13) · (cid:13)(cid:13) Y via the identification V • = H • ( Y, F ) induced by the Hodge theory. Set(3.29) h V • R,T = √ πRT − / h V • . We construct a Hermitian metric h C • , • j R,T on C • , • j as follows, h C , • R,T = h W • R,T ⊕ h W • R,T ,h C , • j R,T = h W • j R,T ⊕ h V • R,T for j = 1 , ,h C , • R,T = 12 h V • R,T ⊕ h V • R,T ,h C , • j R,T = h V • R,T for j = 0 , , , . (3.30)For a positive function G ( R, T ) on R, T and a two-parameter family of operators A R,T ∈ End (cid:0) C • , • j (cid:1) , we denote A R,T = O R,T (cid:0) G ( R, T ) (cid:1) if there exists C > such that theoperator norm of A R,T with respect to h C • , • j R,T is bounded by CG ( R, T ) . Theorem 3.3.
There exist linear maps (3.31) S j,R,T : C • , • j → E [ − , j,R,T with j = 0 , , , such that - the map S j,R,T preserves the grading, i.e., (3.32) S j,R,T (cid:0) C p,qj (cid:1) ⊆ E [ − , j,R,T ∩ Ω p + q ( Z j,R , F ) ; - for T = R κ ≫ , the map S j,R,T is bijective (as a consequence, dim E [ − , j,R,T isindependent of T = R κ ); - for T = R κ ≫ and σ ∈ C • , • j , we have (cid:13)(cid:13)(cid:13) S j,R,T ( σ ) (cid:13)(cid:13)(cid:13) Z j,R = h C • , • j R,T ( σ, σ ) (cid:16) O (cid:0) R − / κ/ (cid:1)(cid:17) ; (3.33)- for T = R κ ≫ , we have (3.34) S − j,R,T ◦ d Z j,R T ◦ S j,R,T = π − / R − T / e − T (cid:16) ∂ + O R,T (cid:0) R − / κ/ (cid:1)(cid:17) . For ease of notations, we denote(3.35) ∂ T = π − / T / e − T ∂ : C , • j → C , • j . Let c T kj,R,T ∈ Q S be the torsion form (cf. § (cid:0) C • ,kj , R − ∂ T , h C • ,kj R,T (cid:1) . Weview (cid:0) C • , • j , R − ∂ T (cid:1) as a complex, whose component of degree k is given by L p + q = k C p,qj .Let c T j,R,T ∈ Q S be the torsion form associated with (cid:0) C • , • j , R − ∂ T , h C • , • j R,T (cid:1) . The followingidentity is a consequence of [24, Thm. 7.37], which may be viewed as an analogue ofa result of Ma on the Bott-Chern forms [38, Thm 1.2], and may also be viewed a finitedimensional version of [39, Thm 0.1],(3.36) c T j,R,T = dim Z X k =0 ( − k c T kj,R,T . NALYTIC TORSION FORMS 28
For G ( R, T ) a positive function on R, T > and (cid:0) τ R,T (cid:1)
R,T > a family of differentialforms on S with values in a Hermitian vector bundle (cid:0) E, (cid:13)(cid:13) · (cid:13)(cid:13) E (cid:1) , we write(3.37) τ R,T = O E (cid:0) G ( R, T ) (cid:1) , if there exists C > such that the C -norm of τ R,T is dominated by CG ( R, T ) for R, T > . We remark that O E ( · ) is independent of the norm (cid:13)(cid:13) · (cid:13)(cid:13) E . If E is a trivial linebundle, we abbreviate (3.37) as τ R,T = O (cid:0) G ( R, T ) (cid:1) .We equip Q S with the C -norm. Then, by the de Rham theorem (cf. [15, Thm.1.1 (d)]), Q S, ⊆ Q S is closed. We equip Q S /Q S, with quotient norm. For a family ofelements in Q S /Q S, parameterized by R, T > , we use the same notation as in (3.37)with the C -norm replaced by the quotient norm. Theorem 3.4.
For T = R κ ≫ , the following identity holds in Q S /Q S, , (3.38) X j =0 ( − j ( j − / T j,R,T = X j =0 ( − j ( j − / c T j,R,T + O (cid:0) R − κ/ (cid:1) . We have a Mayer-Vietoris exact sequence of flat complex vector bundles over S , → H (cid:0) C • ,k , R − ∂ T (cid:1) → H (cid:0) C • ,k ⊕ C • ,k , R − ∂ T (cid:1) → H (cid:0) C • ,k , R − ∂ T (cid:1) → H (cid:0) C • ,k , R − ∂ T (cid:1) → , (3.39)which is induced by (2.9) with ∂ replaced by R − ∂ T . We equip the cohomology groupsin (3.39) with Hermitian metrics induced by h C • , • j R,T via (1.5). Let c T k H ,R,T ∈ Q S be thetorsion form (cf. § c T H ,R,T = X k ( − k c T k H ,R,T ∈ Q S . Theorem 3.5.
For T = R κ ≫ , the following identity holds in Q S /Q S, , (3.41) T H ,R,T = c T H ,R,T + O (cid:0) R − / κ/ (cid:1) . Proof of Theorem 0.1.
By Theorem 2.1, (3.36) and (3.40), the following identity holdsin Q S /Q S, ,(3.42) X j =0 ( − j ( j − / c T j,R,T + c T H ,R,T = 0 . By Theorems 3.4, 3.5 and (3.42), the following identity holds in Q S /Q S, as T = R κ → + ∞ ,(3.43) X j =0 ( − j ( j − / T j,R,T + T H ,R,T = O (cid:0) R − κ/ (cid:1) . On the other hand, using the anomaly formula [11, Thm. 3.24] [62, Thm. 1.5] in thesame way as in [62, § R and T . Hence, for any R > and T > , we have(3.44) X j =0 ( − j ( j − / T j,R,T + T H ,R,T ∈ Q S, . NALYTIC TORSION FORMS 29
Taking R = 1 and T = 0 in (3.44), we obtain (0.27). This completes the proof ofTheorem 0.1. (cid:3)
4. O NE - DIMENSIONAL W ITTEN TYPE DEFORMATION
The construction in this section may be viewed as a model of the problem addressedin this paper, in which the fibration has one-dimensional fibers. This one-dimensionalmodel and the zero-dimensional model constructed in § T → + ∞ . This section is organized as follows. In § V on [ − , and establish a Hodge theorem for V . In § § § Hodge theory for an interval.
We denote I = [ − , . Let u ∈ I be the coordi-nate. Let V be a finite dimensional complex vector space. Let V , V ⊆ V be vectorsubspaces. We construct a sheaf V on I as follows: for any open subset U ⊆ I , V ( U ) = n locally constant function α : U → V : α ( − ∈ V if − ∈ U , α (1) ∈ V if ∈ U o . (4.1)We construct sheaves (cid:0) R k (cid:1) k =0 , on I as follows: for any open subset U ⊆ I , R ( U ) = n s ∈ C ∞ ( U, V ) : s ( − ∈ V if − ∈ U , s (1) ∈ V if ∈ U o , R ( U ) = Ω ( U, V ) . (4.2)Let i : V → R is the obvious injection. Let d : R → R be the de Rham operator.Then(4.3) V i / / R d / / R is a resolution of V by fine sheaves. Let H • ( I, V ) be the sheaf theoretic cohomologyof I with coefficients in V . We have(4.4) H • ( I, V ) = H • ( R • ( I ) , d ) . Then a direct calculation yields(4.5) H ( I, V ) = V ∩ V , H ( I, V ) = V / ( V + V ) . Let h V be a Hermitian metric on V . We denote V [ du ] = V ⊕ V du = V ⊗ Λ • ( T ∗ I ) .Let (cid:13)(cid:13) · (cid:13)(cid:13) V [ du ] be the norm on V [ du ] induced by h V and the metric on Λ • ( T ∗ I ) such that (cid:12)(cid:12) du (cid:12)(cid:12) = 1 . We introduce the following Clifford actions on V [ du ] ,(4.6) c = du ∧ − i ∂∂u , ˆ c = du ∧ + i ∂∂u . Then c (resp. ˆ c ) is skew-adjoint (resp. self-adjoint) with respect to (cid:13)(cid:13) · (cid:13)(cid:13) V [ du ] . Moreover,(4.7) c = − , ˆ c = 1 , c ˆ c + ˆ cc = 0 . NALYTIC TORSION FORMS 30
For u ∈ I and ω ∈ Ω • ( I, V ) = C ∞ ( I, V [ du ]) , we denote by ω u ∈ V [ du ] the value of ω at u . Let (cid:13)(cid:13) · (cid:13)(cid:13) [ − , be the L -norm on Ω • ( I, V ) , i.e.,(4.8) (cid:13)(cid:13) ω (cid:13)(cid:13) − , = Z − (cid:13)(cid:13) ω u (cid:13)(cid:13) V [ du ] du . Let d V be the de Rham operator on Ω • ( I, V ) . Let d V, ∗ be its formal adjoint. Set(4.9) D V = d V + d V, ∗ . Then, by (4.6), we have(4.10) D V = c ∂∂u , D V, = − ∂ ∂u . For j = 1 , , let V ⊥ j ⊆ V be the orthogonal complement of V j ⊆ V with respect to h V . Set(4.11) Ω • bd ( I, V ) = n ω ∈ Ω • ( I, V ) : ω − ∈ V ⊕ V ⊥ du , ω ∈ V ⊕ V ⊥ du o . Let D V bd be the self-adjoint extension of D V with domain Dom (cid:0) D V bd (cid:1) = Ω • bd ( I, V ) . Wewill also consider D V, bd with domain(4.12) Dom (cid:0) D V, bd (cid:1) = n ω ∈ Ω • bd ( I, V ) : D V ω ∈ Ω • bd ( I, V ) o . We have(4.13)
Ker (cid:0) D V, bd (cid:1)(cid:12)(cid:12)(cid:12) Ω bd ( I,V ) = V ∩ V , Ker (cid:0) D V, bd (cid:1)(cid:12)(cid:12)(cid:12) Ω bd ( I,V ) = (cid:0) V ⊥ ∩ V ⊥ (cid:1) du , where the right hand sides are viewed as constant functions on I with values in V [ du ] .From (4.5) and (4.13), we get a natural isomorphism(4.14) Ker (cid:0) D V, bd (cid:1) ≃ H • ( I, V ) . Witten type deformation on an interval.
For T > , set(4.15) d VT = e − T f T d V e T f T , d V, ∗ T = e T f T d V, ∗ e − T f T , D VT = d VT + d V, ∗ T , where f T was defined by (3.10). The operator D VT is formally self-adjoint with respectto (cid:13)(cid:13) · (cid:13)(cid:13) [ − , . We have(4.16) D VT = D V + T f ′ T ˆ c = c ∂∂u + T f ′ T ˆ c , D V, T = − ∂ ∂u + T f ′′ T c ˆ c + T | f ′ T | . Let D VT, bd be the self-adjoint extension of D VT with domain Dom (cid:0) D VT, bd (cid:1) = Ω • bd ( I, V ) . Theorem 4.1.
There exist β > α > such that for T ≫ , we have (4.17) Sp (cid:0) D VT, bd (cid:1) ⊆ (cid:3) − ∞ , − α √ T (cid:3) ∪ (cid:2) − β √ T e − T , β √ T e − T (cid:3) ∪ (cid:2) α √ T , + ∞ (cid:2) . Proof.
Recall that f ∞ was defined by (3.8). For T > , set(4.18) e D VT = D V + T f ′∞ ˆ c . NALYTIC TORSION FORMS 31
Let e D VT, bd be the self-adjoint extension of e D VT with domain Dom (cid:0) e D VT, bd (cid:1) = Ω • bd ( I, V ) .By (3.11), (4.16) and (4.18), the operator norm of D VT − e D VT is bounded by O (cid:0) T e − T (cid:1) .Hence it is sufficient to show that there exist β > α > such that(4.19) Sp (cid:16) e D VT, bd (cid:17) ⊆ (cid:3) − ∞ , − α √ T (cid:3) ∪ (cid:2) − β √ T e − T / , β √ T e − T / (cid:3) ∪ (cid:2) α √ T , + ∞ (cid:2) . Recall that χ was defined by (3.9). For T > , we construct smooth functions φ ,T , φ ,T , φ ,T : I → R as follows, φ ,T ( u ) = φ ,T ( − u ) = (cid:0) − χ (4 u + 4) (cid:1) exp (cid:0) − T ( u + 1) / (cid:1) ,φ ,T ( u ) = (cid:0) − χ (4 | u | ) (cid:1) exp (cid:0) − T u / (cid:1) , for u ∈ I . (4.20)Let (cid:0) C • r , ∂ (cid:1) be the complex in (2.3) associated with V , V ⊆ V . For T > , we constructa linear map J T : C • r → Ω • ( I, V ) as follows,for ( v , v ) ∈ V ⊕ V = C , J T ( v , v ) = φ ,T v + φ ,T v ∈ C ∞ ( I, V ) ; for v ∈ V = C , J T ( v ) = φ ,T du ⊗ v ∈ Ω ( I, V ) . (4.21)Proceeding in the same way as in [13, §
6] with J T in [13, Def. 6.5] replaced by the J T constructed in (4.21), we obtain (4.19). This completes the proof of Theorem 4.1. (cid:3) For Λ ⊆ R , we denote by E Λ T the eigenspace of D VT, bd associated with eigenvalues in Λ . Theorem 4.2.
For T ≫ , we have dim E [ − , T = dim C • r .Proof. Let e D VT, bd be as in the proof of Theorem 4.1. Let e E [ − , T be the eigenspace of e D VT, bd associated with eigenvalues in [ − , . Proceeding in the same way as in [13, §
6] with J T in [13, Def. 6.5] replaced by the J T constructed in (4.21), we obtain dim e E [ − , T = dim C • r . On the other hand, by the proof of Theorem 4.1, we have dim E [ − , T = dim e E [ − , T . This completes the proof of Theorem 4.2. (cid:3) For j = 1 , , let(4.22) P j : V [ du ] → V j ⊕ V ⊥ j du be orthogonal projections with respect to (cid:13)(cid:13) · (cid:13)(cid:13) V [ du ] . We denote P ⊥ j = Id − P j . Let(4.23) P Λ T : Ω • ( I, V ) → E Λ T be the orthogonal projection with respect to (cid:13)(cid:13) · (cid:13)(cid:13) [ − , . For ω ∈ Ω • ( I, V ) , we denote(4.24) (cid:13)(cid:13) ω (cid:13)(cid:13) V [ du ] , max = max n(cid:13)(cid:13) ω u (cid:13)(cid:13) V [ du ] : u ∈ [ − , o . Proposition 4.3.
For T ≫ , ε > , < ǫ < √ T , −√ T < λ < √ T and ω ∈ Ω • ( I, V ) satisfying (4.25) D VT ω = λω , (cid:13)(cid:13) P ⊥ ω − (cid:13)(cid:13) V [ du ] + (cid:13)(cid:13) P ⊥ ω (cid:13)(cid:13) V [ du ] ǫε (cid:13)(cid:13) ω (cid:13)(cid:13) V [ du ] , max , we have (4.26) (cid:13)(cid:13)(cid:13) ω − P [ λ − ǫ,λ + ǫ ] T ω (cid:13)(cid:13)(cid:13) [ − , = O (cid:0) T / (cid:1) ε (cid:13)(cid:13) ω (cid:13)(cid:13) [ − , . NALYTIC TORSION FORMS 32
Proof.
Set χ T ( u ) = χ ( T u − T + 1) , where χ was constructed in (3.9). We construct ω ′ ∈ Ω • bd ( I, V ) as follows, ω ′ u = (cid:26) ω u − χ T ( − u ) P ⊥ ω − if u < ,ω u − χ T ( u ) P ⊥ ω if u > . (4.27)By the inequality in (4.25), (4.27) and the assumption < ǫ < √ T , we have(4.28) (cid:13)(cid:13) ω − ω ′ (cid:13)(cid:13) [ − , = O (cid:0) T − / (cid:1) ǫε (cid:13)(cid:13) ω (cid:13)(cid:13) V [ du ] , max = O (cid:0) (cid:1) ε (cid:13)(cid:13) ω (cid:13)(cid:13) V [ du ] , max . A direct calculation yields (cid:16)(cid:0) D VT, bd − λ (cid:1) ω ′ (cid:17) u = (cid:26) χ ′ T ( − u ) cP ⊥ ω − + (cid:0) λ − T f ′ T ˆ c (cid:1) χ T ( − u ) P ⊥ ω − if u < , − χ ′ T ( u ) cP ⊥ ω + (cid:0) λ − T f ′ T ˆ c (cid:1) χ T ( u ) P ⊥ ω if u > . (4.29)By the inequality in (4.25), (4.29) and the construction of χ T , we get(4.30) (cid:13)(cid:13)(cid:13)(cid:0) D VT, bd − λ (cid:1) ω ′ (cid:13)(cid:13)(cid:13) [ − , = O (cid:0) √ T (cid:1) ǫε (cid:13)(cid:13) ω (cid:13)(cid:13) V [ du ] , max . By Proposition 1.2 and (4.30), we have(4.31) (cid:13)(cid:13)(cid:13) ω ′ − P [ λ − ǫ,λ + ǫ ] T ω ′ (cid:13)(cid:13)(cid:13) [ − , = O (cid:0) √ T (cid:1) ε (cid:13)(cid:13) ω (cid:13)(cid:13) V [ du ] , max . By (4.7), (4.16) and the identity in (4.25), we have(4.32) ∂∂u ω = (cid:0) T f ′ T c ˆ c − λc (cid:1) ω . Let (cid:13)(cid:13) · (cid:13)(cid:13) H , [ − , be the H -norm on Ω • ( I, V ) . By Sobolev inequality and (4.32),(4.33) (cid:13)(cid:13) ω (cid:13)(cid:13) V [ du ] , max = O (cid:0) (cid:1)(cid:13)(cid:13) ω (cid:13)(cid:13) H , [ − , = O (cid:0) T (cid:1)(cid:13)(cid:13) ω (cid:13)(cid:13) [ − , . From (4.28), (4.31) and (4.33), we obtain (4.26). This completes the proof ofProposition 4.3. (cid:3)
Witten type deformation on a cylinder.
Let ( Y, g
T Y ) be a closed Riemannianmanifold. For R > , we denote I R = [ − R, R ] and IY R = I R × Y . Let ( u, y ) ∈ [ − R, R ] × Y be the coordinates. We will also use the coordinates ( s, y ) = ( u/R, y ) ∈ [ − , × Y .We equip T ( IY R ) with the Riemannian metric du + g T Y .Let F be a flat complex vector bundle over Y . Let h F be a Hermitian metric on F . The pull-back of F (resp. h F ) via the canonical projection IY R → Y will still bedenoted by F (resp. h F ). Let D Y be the Hodge de Rham operator on Ω • ( Y, F ) . Underthe identification Ω • ( I R , Ω • ( Y, F )) → Ω • ( IY R , F ) σ + du ⊗ τ σ + du ∧ τ , for σ, τ ∈ C ∞ ( I R , Ω • ( Y, F )) , (4.34)the Hodge de Rham operator on Ω • ( IY R , F ) is given by(4.35) D IY R = ˆ ccD Y + c ∂∂u = R − (cid:16) ˆ ccRD Y + c ∂∂s (cid:17) , NALYTIC TORSION FORMS 33 where the term ˆ cc = i ∂∂u du ∧ − du ∧ i ∂∂u comes from the fact that D Y anti-commuteswith du ∧ .Recall that the function f T : I → R was constructed in (3.10). We will view f T as afunction on IY R , i.e., f T ( s, y ) = f T ( s ) , f T ( u, y ) = f T ( u/R ) . We denote(4.36) f ′ T = ∂∂s f T = R ∂∂u f T . For T > , let e D IY R T be the Hodge de Rham operator with respect to du + g T Y and h FT := e − T f T h F . Set D IY R T = e − T f T e D IY R T e T f T . We have RD IY R T = ˆ ccRD Y + c ∂∂s + T f ′ T ˆ c ,R D IY R , T = R D Y, − ∂ ∂s + T f ′′ T c ˆ c + T | f ′ T | . (4.37)Let(4.38) Ω • ( Y, F ) = M µ E µ ( Y, F ) be the spectral decomposition with respect to D Y , i.e., D Y (cid:12)(cid:12) E µ ( Y,F ) = µ Id . We denote H • ( Y, F ) = Ker (cid:0) D Y (cid:1) = E ( Y, F ) . We have the formal decomposition(4.39) Ω • ( IY R , F ) = M µ Ω • (cid:0) I R , E µ ( Y, F ) (cid:1) . Let D E µ ( Y,F ) T be the operator D VT in § V replaced by E µ ( Y, F ) . We have(4.40) RD IY R T (cid:12)(cid:12)(cid:12) Ω • ( I R , E µ ( Y,F )) = Rµ ˆ cc + D E µ ( Y,F ) T = Rµ ˆ cc + c ∂∂s + T f ′ T ˆ c . As a consequence, we have(4.41) R D IY R , T (cid:12)(cid:12)(cid:12) Ω • ( I R , E µ ( Y,F )) = R µ − ∂ ∂s + T | f ′ T | + T f ′′ T c ˆ c . In particular, we have(4.42) RD IY R T (cid:12)(cid:12)(cid:12) Ω • ( I R , H • ( Y,F )) = D H • ( Y,F ) T = c ∂∂s + T f ′ T ˆ c . For α > , we define C α : R + × R + × [ − , → R as follows,(4.43) C α ( a, b, s ) = (cid:0) − e − α (cid:1) − (cid:16)(cid:0) a − be − α (cid:1) e α ( − s − + (cid:0) b − ae − α (cid:1) e α ( s − (cid:17) . Then the following identities hold,(4.44) C α ( a, b, −
1) = a , C α ( a, b,
1) = b , (cid:18) ∂ ∂s − α (cid:19) C α ( a, b, s ) = 0 . Lemma 4.4.
There exists α > such that for T = R κ ≫ , µ ∈ Sp (cid:0) D Y (cid:1) \{ } , ω ∈ Ω • (cid:0) I R , E µ ( Y, F ) (cid:1) and −√ R λ √ R satisfying (4.45) (cid:16) Rµ ˆ cc + D E µ ( Y,F ) T (cid:17) ω = λω , NALYTIC TORSION FORMS 34 we have (4.46) (cid:13)(cid:13) ω s (cid:13)(cid:13) Y C αR (cid:16)(cid:13)(cid:13) ω − (cid:13)(cid:13) Y , (cid:13)(cid:13) ω (cid:13)(cid:13) Y , s (cid:17) , for s ∈ [ − , . Proof.
We assert that for g ∈ C ∞ ([ − , , R + ) satisfying (cid:16) ∂ ∂s − α R (cid:17) g > , we have g ( s ) C αR (cid:0) g ( − , g (1) , s (cid:1) for s ∈ [ − , . To prove the assertion, we take s ∈ [ − , such that(4.47) h := g − C αR (cid:0) g ( − , g (1) , · (cid:1) ∈ C ∞ ([ − , , R ) reaches its maximum value at s . If h ( s ) > , then s = ± and h ′′ ( s ) > , which is acontradiction.Now it remains to show that(4.48) (cid:18) ∂ ∂s − α R (cid:19) (cid:13)(cid:13) ω s (cid:13)(cid:13) Y > . By (4.40), (4.41) and (4.45), we have(4.49) ∂ ∂s ω s = (cid:16) R µ + T f ′′ T ( s ) c ˆ c + T | f ′ T | ( s ) − λ (cid:17) ω s . Set α = min n | µ | : µ ∈ Sp (cid:0) D Y (cid:1) \{ } o . For R, T, µ, λ satisfying the hypothesis of Lemma4.4, we have(4.50) R µ + T f ′′ T ( s ) c ˆ c + T | f ′ T | ( s ) − λ > α R / . From (4.49), (4.50) and the obvious identity(4.51) ∂ ∂s (cid:13)(cid:13) ω s (cid:13)(cid:13) Y = D ∂ ∂s ω s , ω s E Y + D ω s , ∂ ∂s ω s E Y + 2 (cid:13)(cid:13)(cid:13) ∂∂s ω s (cid:13)(cid:13)(cid:13) Y , we obtain (4.48). This completes the proof of Lemma 4.4. (cid:3) Lemma 4.5.
For ω ∈ Ω • (cid:0) I, H • ( Y, F ) (cid:1) and λ ∈ R satisfying (4.52) D H • ( Y,F ) T ω = λω , we have (4.53) ∂∂s (cid:10) c ω s , ω s (cid:11) Y = 0 , for s ∈ [ − , . Proof.
By (4.42) and (4.52), we have(4.54) ∂∂s ω s = (cid:0) T f ′ T ( s ) c ˆ c − λc (cid:1) ω s . Note that c is skew-adjoint and that ˆ c is self-adjoint, equation (4.53) follows from (4.7)and (4.54). This completes the proof of Lemma 4.5. (cid:3) NALYTIC TORSION FORMS 35
5. A
DIABATIC LIMIT AND W ITTEN TYPE DEFORMATION
The purpose of this section is to prove Theorems 3.1 and 3.3.First we summarize the proof of Theorem 3.1 given in this section. Let λ be areasonably small eigenvalue of RD Z R T . Let ω be an eigensection associated with λ .First, in Lemma 5.7, we show that ω zm (the zero-mode of ω , see (5.5)) is the principalcontributor to the norm of ω . Second, in Lemma 5.8, we show that ω zm almost liesin the domain of D H • ( Y,F ) T, bd , which is the operator in (4.16) with V = H • ( Y, F ) :=Ker( D Y ) . Combining the results above, we show that λ is very close to an eigenvalueof D H • ( Y,F ) T, bd . On the other hand, by Theorem 4.1, D H • ( Y,F ) T, bd satisfies the desired spectralgap. Hence so does RD Z R T .Concerning the proof of Theorem 3.3, the idea is to explicitly construct G + R,T : C , • → Ω • ( Z R , F ) and I + R,T : C , • → Ω • ( Z R , F ) (see (5.119) and (5.127)). The map S R,T : C • , • → Ω • ( Z R , F ) is then defined by composing G + R,T ⊕ I + R,T with the orthog-onal projection to the eigenspace of D Z R T associated with small eigenvalues. In theproof of the injectivity of S R,T , the most subtle part is the injectivity of S R,T (cid:12)(cid:12) C , • . Thisis obtained by constructing an auxiliary map F + R,T : C , • → Ω • ( Z R , F ) and applyingProposition 1.3 with w = F + R,T and v = G + R,T (see Proposition 5.9). The proof of thesurjectivity of S R,T highly relies on the results mentioned in the last paragraph: wereduce the problem to D H • ( Y,F ) T, bd whose spectrum is studied in § D Z R T needs to be studied separately. Here the strat-egy is exactly the same as in the last paragraph. We explicitly construct F R,T , G
R,T : H ( C • , • , ∂ ) → Ω • ( Z R , F ) and I R,T , J
R,T : H ( C • , • , ∂ ) → Ω • ( Z R , F ) (see (5.20) and(5.44)). We construct a bijection S HR,T : H • ( C • , • , ∂ ) → Ker (cid:0) D Z R T (cid:1) by composing F R,T ⊕ I R,T with the orthogonal projection to the kernel of D Z R T . To show the injectivityof S HR,T , we apply Proposition 1.3 with w = F R,T ⊕ I R,T and v = G R,T ⊕ J R,T .This section is organized as follows. In § § § § L -metric on the eigenspace associated withsmall eigenvalues. Theorem 3.3 will be proved in this subsection.5.1. Kernel of D Z R T . Recall that D Z R T was defined in (3.15). For convenience, wedenote D Z R = D Z R T (cid:12)(cid:12) T =0 . By elliptic estimate (see the proof of [50, Prop. 3.4]), we maydefine the H -norm on Ω • ( Z R , F ) as follows: for ω ∈ Ω • ( Z R , F ) ,(5.1) (cid:13)(cid:13) ω (cid:13)(cid:13) H ,Z R = (cid:13)(cid:13) ω (cid:13)(cid:13) Z R + (cid:13)(cid:13) D Z R ω (cid:13)(cid:13) Z R . We fix κ ∈ ]0 , / as in (0.41). Proposition 5.1.
For T = R κ ≫ and ω ∈ Ω • ( Z R , F ) , we have (5.2) (cid:13)(cid:13) ω (cid:13)(cid:13) H ,Z R (cid:13)(cid:13) ω (cid:13)(cid:13) Z R + (cid:13)(cid:13) D Z R T ω (cid:13)(cid:13) Z R . NALYTIC TORSION FORMS 36
Proof.
By (3.11), (4.37) and the assumption T = R κ , we have(5.3) D Z R , T + Id > D Z R , . From (5.1) and (5.3), we obtain (5.2). This completes the proof of Proposition 5.1. (cid:3)
We will always use the canonical isometric embeddings(5.4) IY R ⊆ Z j,R ⊆ Z j, ∞ , Z j,R ⊆ Z R , for j = 1 , . Recall that the vector subspaces H • ( Y, F ) ⊆ Ω • ( Y, F ) and E µ ( Y, F ) ⊆ Ω • ( Y, F ) weredefined in the paragraph containing (4.38). For ω ∈ Ω • ( Z j,R , F ) with j = 0 , , , , wehave the orthogonal decomposition(5.5) ω (cid:12)(cid:12) IY R = ω zm + ω nz , with(5.6) ω zm ∈ Ω • (cid:0) I R , H • ( Y, F ) (cid:1) , ω nz ∈ M µ =0 Ω • (cid:0) I R , E µ ( Y, F ) (cid:1) . We call ω zm (resp. ω nz ) the zero-mode (resp. non-zero-mode) of ω .Recall that H • abs ( Z j, ∞ , F ) ⊆ H • ( Z j, ∞ , F ) was defined in (3.21). Let(5.7) R d F , R d F, ∗ : H • abs ( Z , ∞ , F ) → Ω • ([0 , + ∞ ) × Y, F ) be as in [50, (2.44)] with X ∞ replaced by Z , ∞ and H • ( X ∞ , F ) replaced by H • abs ( Z , ∞ , F ) .Here we recall their construction. We identify Z , ∞ with Z , ∪ [0 , + ∞ [ × Y . By [50,Prop. 2.5] and [50, (2.10)], for ( ω, ˆ ω ) ∈ H • abs ( Z , ∞ , F ) , we have(5.8) ω (cid:12)(cid:12) [0 , + ∞ [ × Y = ˆ ω + ω nz = ˆ ω + X µ =0 , µ ∈ Sp( D Y ) e −| µ | u (cid:0) τ µ, − du ∧ τ µ, (cid:1) , where ˆ ω ∈ H • ( Y, F )[ du ] is viewed as a constant section in(5.9) C ∞ (cid:0) [0 , + ∞ [ , H • ( Y, F )[ du ] (cid:1) = Ω • (cid:0) [0 , + ∞ [ , H • ( Y, F ) (cid:1) ⊆ Ω • (cid:0) [0 , + ∞ [ × Y, F (cid:1) , and τ µ, , τ µ, ∈ Ω • ( Y, F ) satisfy(5.10) d Y τ µ, = d Y, ∗ τ µ, = 0 , d Y, ∗ τ µ, = | µ | τ µ, , d Y τ µ, = | µ | τ µ, . We define R d F ( ω, ˆ ω ) = X µ =0 , µ ∈ Sp( D Y ) | µ | e −| µ | u τ µ, , R d F, ∗ ( ω, ˆ ω ) = X µ =0 , µ ∈ Sp( D Y ) | µ | e −| µ | u du ∧ τ µ, . (5.11)Let(5.12) Ω • ([0 , + ∞ ) × Y, F ) → Ω • ( IY R , F ) be induced by the isometric identifications IY R = [ − R, R ] × Y ≃ [0 , R ] × Y ֒ → [0 , + ∞ ) × Y . Composing (5.7) and (5.12), we get(5.13) R d F , R d F, ∗ : H • ( Z , ∞ , F ) → Ω • ( IY R , F ) . We construct(5.14) R d F , R d F, ∗ : H • ( Z , ∞ , F ) → Ω • ( IY R , F ) NALYTIC TORSION FORMS 37 in the same way. For j = 1 , and ( ω, ˆ ω ) ∈ H • ( Z j, ∞ , F ) , we have d Z R R d F ( ω, ˆ ω ) = d Z R , ∗ R d F, ∗ ( ω, ˆ ω ) = ω nz ,d Z R , ∗ R d F ( ω, ˆ ω ) = d Z R R d F, ∗ ( ω, ˆ ω ) = 0 ,i ∂∂u R d F ( ω, ˆ ω ) = du ∧ R d F, ∗ ( ω, ˆ ω ) = 0 . (5.15)Set(5.16) H • abs ( Z , ∞ , F ) = n ( ω , ω , ˆ ω ) : ( ω j , ˆ ω ) ∈ H • abs ( Z j, ∞ , F ) for j = 1 , o . Recall that the smooth function χ : R → R was defined in (3.9). Set(5.17) χ ( s ) = 1 − χ (cid:0) s + 1) (cid:1) , χ ( s ) = 1 − χ (cid:0) − s ) (cid:1) . We will view χ j ( j = 1 , ) as functions on IY R , i.e.,(5.18) χ j ( s, y ) = χ j ( s ) , χ j ( u, y ) = χ j ( u/R ) . Following [50, (3.26)], we define(5.19) F R,T , G
R,T : H • abs ( Z , ∞ , F ) → Ω • ( Z R , F ) as follows: for ( ω , ω , ˆ ω ) ∈ H • abs ( Z , ∞ , F ) , F R,T ( ω , ω , ˆ ω ) (cid:12)(cid:12) Z j, = G R,T ( ω , ω , ˆ ω ) (cid:12)(cid:12) Z j, = ω j , for j = 1 , ,F R,T ( ω , ω , ˆ ω ) (cid:12)(cid:12) IY R = e − T f T ˆ ω + e − T f T d Z R (cid:16) χ R d F ( ω , ˆ ω ) + χ R d F ( ω , ˆ ω ) (cid:17) ,G R,T ( ω , ω , ˆ ω ) (cid:12)(cid:12) IY R = e − T f T ˆ ω + e T f T d Z R , ∗ (cid:16) χ R d F, ∗ ( ω , ˆ ω ) + χ R d F, ∗ ( ω , ˆ ω ) (cid:17) , (5.20)where we use the identifications in (5.4). By (5.15), F R,T and G R,T are well-defined.Similarly to [50, (3.28)], by (3.15) and the identities D Y ˆ ω j = i ∂∂u ˆ ω j = 0 for j = 1 , ,we have(5.21) d Z R T F R,T ( ω , ω , ˆ ω ) = d Z R , ∗ T G R,T ( ω , ω , ˆ ω ) = 0 . Let P R,T : Ω • ( Z R , F ) → Ker (cid:0) D Z R T (cid:1) be the orthogonal projection with respect the L -metric induced by g T Z R and h F . Proposition 5.2.
For T = R κ ≫ and ( ω , ω , ˆ ω ) ∈ H • abs ( Z , ∞ , F ) , we have (5.22) (cid:13)(cid:13)(cid:13)(cid:0) Id − P R,T (cid:1) F R,T ( ω , ω , ˆ ω ) (cid:13)(cid:13)(cid:13) H ,Z R = O (cid:0) R − κ (cid:1)(cid:16)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , + (cid:13)(cid:13) ω (cid:13)(cid:13) Z , (cid:17) . Proof.
The proof follows closely the proof of [50, Prop. 3.5]. It consists of severalsteps.
Step 1 . We calculate ( F R,T − G R,T )( ω , ω , ˆ ω ) and D Z R T ( F R,T − G R,T )( ω , ω , ˆ ω ) .Recall that IY R = [ − R, R ] × Y . By (5.17) and (5.18), we have(5.23) χ (cid:12)(cid:12) [ − R, × Y = 0 . By (5.15), (5.20) and (5.23), we have ( F R,T − G R,T )( ω , ω , ˆ ω ) (cid:12)(cid:12)(cid:12) [ − R, × Y = χ (cid:16) e − T f T − e T f T (cid:17) ω nz1 + ∂χ ∂u (cid:16) e − T f T du ∧ R d F ( ω , ˆ ω ) + e T f T i ∂∂u R d F, ∗ ( ω , ˆ ω ) (cid:17) . (5.24) NALYTIC TORSION FORMS 38
By (5.8), (5.10), (5.11) and (5.15), we have d Z R (cid:16) du ∧ R d F ( ω , ˆ ω ) (cid:17) = − du ∧ ω nz1 , d Z R , ∗ (cid:16) i ∂∂u R d F, ∗ ( ω , ˆ ω ) (cid:17) = − i ∂∂u ω nz1 ,d Z R , ∗ (cid:16) du ∧ R d F ( ω , ˆ ω ) (cid:17) = − i ∂∂u ω nz1 , d Z R (cid:16) i ∂∂u R d F, ∗ ( ω , ˆ ω ) (cid:17) = − du ∧ ω nz1 . (5.25)By (3.15), the third identity in (5.15) and (5.25), we have D Z R T (cid:16) du ∧ R d F ( ω , ˆ ω ) (cid:17) = − du ∧ ω nz1 − i ∂∂u ω nz1 + T ∂f T ∂u R d F ( ω , ˆ ω ) ,D Z R T (cid:16) i ∂∂u R d F, ∗ ( ω , ˆ ω ) (cid:17) = − du ∧ ω nz1 − i ∂∂u ω nz1 + T ∂f T ∂u R d F, ∗ ( ω , ˆ ω ) . (5.26)By (5.24) and (5.26), we have D Z R T ( F R,T − G R,T )( ω , ω , ˆ ω ) (cid:12)(cid:12)(cid:12) [ − R, × Y = 2 T ∂f T ∂u χ (cid:16) e − T f T i ∂∂u − e T f T du ∧ (cid:17) ω nz1 − ∂χ ∂u (cid:16) e − T f T i ∂∂u + e T f T du ∧ (cid:17) ω nz1 + 2 T ∂f T ∂u ∂χ ∂u (cid:16) e − T f T R d F ( ω , ˆ ω ) + e T f T R d F, ∗ ( ω , ˆ ω ) (cid:17) + ∂ χ ∂u (cid:16) − e − T f T R d F ( ω , ˆ ω ) + e T f T R d F, ∗ ( ω , ˆ ω ) (cid:17) . (5.27) Step 2 . We estimate (cid:13)(cid:13)(cid:13) ( F R,T − G R,T )( ω , ω , ˆ ω ) (cid:13)(cid:13)(cid:13) Z R and (cid:13)(cid:13)(cid:13) D Z R T ( F R,T − G R,T )( ω , ω , ˆ ω ) (cid:13)(cid:13)(cid:13) Z R .For(5.28) τ ∈ n ω nz1 , R d F ( ω , ˆ ω ) , R d F, ∗ ( ω , ˆ ω ) o and − R u , by (5.8) and (5.11), we have(5.29) (cid:13)(cid:13) τ (cid:13)(cid:13) { u }× Y = O (cid:0) e − a ( R + u ) (cid:1)(cid:13)(cid:13) ω nz1 (cid:13)(cid:13) ∂Z , = O (cid:0) e − a ( R + u ) (cid:1)(cid:13)(cid:13) ω (cid:13)(cid:13) ∂Z , , where a > is a universal constant. Since ω ∈ Ker (cid:0) D Z , ∞ (cid:1) , by the Trace theorem forSobolev spaces, we have(5.30) (cid:13)(cid:13) ω (cid:13)(cid:13) ∂Z , = O (cid:0) (cid:1)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , . By (5.29) and (5.30), for − R u , we have(5.31) (cid:13)(cid:13) τ (cid:13)(cid:13) { u }× Y = O (cid:0) e − a ( R + u ) (cid:1)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , . By (3.10), (5.17) and (5.18), for − R u , we have (cid:12)(cid:12) f T (cid:12)(cid:12) = O (cid:0) R − (cid:1) ( R + u ) , (cid:12)(cid:12)(cid:12)(cid:12) ∂f T ∂u (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:0) R − (cid:1) ( R + u ) , (cid:12)(cid:12) χ (cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) ∂χ ∂u (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:0) R − (cid:1) , (cid:12)(cid:12)(cid:12)(cid:12) ∂ χ ∂u (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:0) R − (cid:1) . (5.32) NALYTIC TORSION FORMS 39
By (5.24), (5.27), (5.31), (5.32) and the assumption T = R κ , we have (cid:13)(cid:13)(cid:13) ( F R,T − G R,T )( ω , ω , ˆ ω ) (cid:13)(cid:13)(cid:13) − R, × Y + (cid:13)(cid:13)(cid:13) D Z R T ( F R,T − G R,T )( ω , ω , ˆ ω ) (cid:13)(cid:13)(cid:13) − R, × Y = O (cid:0) R − κ (cid:1)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , . (5.33)The same argument also shows that (5.33) holds with [ − R, × Y replaced by [0 , R ] × Y and (cid:13)(cid:13) ω (cid:13)(cid:13) Z , replaced by (cid:13)(cid:13) ω (cid:13)(cid:13) Z , . On the other hand, by (5.20), we have(5.34) ( F R,T − G R,T )( ω , ω , ˆ ω ) (cid:12)(cid:12)(cid:12) Z , ∪ Z , = 0 . By (5.33) and (5.34), we have (cid:13)(cid:13)(cid:13) ( F R,T − G R,T )( ω , ω , ˆ ω ) (cid:13)(cid:13)(cid:13) Z R + (cid:13)(cid:13)(cid:13) D Z R T ( F R,T − G R,T )( ω , ω , ˆ ω ) (cid:13)(cid:13)(cid:13) Z R = O (cid:0) R − κ (cid:1)(cid:16)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , + (cid:13)(cid:13) ω (cid:13)(cid:13) Z , (cid:17) . (5.35)The estimates in § (cid:0) W • , ∂, (cid:13)(cid:13) · (cid:13)(cid:13)(cid:1) replaced by (cid:0) Ω • ( Z R , F ) , d Z R T , (cid:13)(cid:13) · (cid:13)(cid:13) Z R (cid:1) .Applying (5.21), (5.35) and Corollary 1.4 with γ = 0 , w = F R,T ( ω , ω , ˆ ω ) and v = G R,T ( ω , ω , ˆ ω ) , we get (cid:13)(cid:13)(cid:13)(cid:0) Id − P R,T (cid:1) F R,T ( ω , ω , ˆ ω ) (cid:13)(cid:13)(cid:13) Z R + (cid:13)(cid:13)(cid:13) D Z R T (cid:0) Id − P R,T (cid:1) F R,T ( ω , ω , ˆ ω ) (cid:13)(cid:13)(cid:13) Z R = O (cid:0) R − κ (cid:1)(cid:16)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , + (cid:13)(cid:13) ω (cid:13)(cid:13) Z , (cid:17) . (5.36)From Proposition 5.1 and (5.36), we obtain (5.22). This completes the proof of Propo-sition 5.2. (cid:3) For j = 1 , , let(5.37) L • j, abs ⊆ H • ( Y, F ) , L • j, rel ⊆ H • ( Y, F ) du , L • j = L • j, abs ⊕ L • j, rel be as in [50, (2.47),(2.49)] with X ∞ replaced by Z j, ∞ . More precisely, under theidentification H • ( Y, F ) = H • ( Y, F ) , we have(5.38) L • j, abs = Im (cid:0) H • ( Z j , F ) → H • ( Y, F ) (cid:1) , where the map is induced by Y = ∂Z j ֒ → Z j , and(5.39) L • +1 j, rel = du L • , ⊥ j, abs , where L • , ⊥ j, abs ⊆ H • ( Y, F ) is the orthogonal complement of L • j, abs with respect to (cid:13)(cid:13) · (cid:13)(cid:13) Y .Hence we have(5.40) L • +11 , rel ∩ L • +12 , rel = du (cid:0) L • , ⊥ , abs ∩ L • , ⊥ , abs (cid:1) . Let H • rel ( Z j, ∞ , F ) be as in [50, (2.52)] with X ∞ replaced by Z j, ∞ . For ˆ ω ∈ L • , ⊥ , abs ∩ L • , ⊥ , abs ,let(5.41) ( ω , du ∧ ˆ ω ) ∈ H • +1rel ( Z , ∞ , F ) , ( ω , du ∧ ˆ ω ) ∈ H • +1rel ( Z , ∞ , F ) be the unique element such that ω (resp. ω ) is a generalized eigensection of D Z , ∞ (resp. D Z , ∞ ). The existence and uniqueness are guaranteed by [50, (2.40)]. NALYTIC TORSION FORMS 40
Similarly to (5.17), set(5.42) χ ( s ) = 1 − χ (cid:0) | s | (cid:1) . We will view χ as a function on IY R in the same way as χ , χ in (5.18). We define(5.43) I R,T , J
R,T : L • , ⊥ , abs ∩ L • , ⊥ , abs → Ω • +1 ( Z R , F ) as follows: for ˆ ω ∈ L • , ⊥ , abs ∩ L • , ⊥ , abs , I R,T (ˆ ω ) (cid:12)(cid:12) Z j, = 0 , J R,T (ˆ ω ) (cid:12)(cid:12) Z j, = e − T ω j , for j = 1 , ,I R,T (ˆ ω ) (cid:12)(cid:12) IY R = χ e T f T − T du ∧ ˆ ω ,J R,T (ˆ ω ) (cid:12)(cid:12) IY R = e T f T − T du ∧ ˆ ω + e T f T − T d Z R , ∗ (cid:16) χ R d F, ∗ ( ω , du ∧ ˆ ω ) + χ R d F, ∗ ( ω , du ∧ ˆ ω ) (cid:17) . (5.44)By (3.10), (3.15) and (5.15), I R,T and J R,T are well-defined. Moreover, we have(5.45) d Z R T I R,T (ˆ ω ) = d Z R , ∗ T J R,T (ˆ ω ) = 0 . Proposition 5.3.
There exists a > such that for T = R κ ≫ and ˆ ω ∈ L • , ⊥ , abs ∩ L • , ⊥ , abs ,we have (5.46) (cid:13)(cid:13)(cid:13)(cid:0) Id − P R,T (cid:1) I R,T (ˆ ω ) (cid:13)(cid:13)(cid:13) H ,Z R = O (cid:0) e − aT (cid:1)(cid:13)(cid:13) ˆ ω (cid:13)(cid:13) Y . Proof.
We proceed in the same way as in the proof of Proposition 5.2. The map I R,T (resp. J R,T ) plays the role of F R,T (resp. G R,T ). (cid:3) By (2.12), (3.20), (3.23) and (5.16), we have(5.47) W • ≃ H • abs ( Z , ∞ , F ) . By (2.2), (3.20), (3.23) and (5.38), we have(5.48) V • j ≃ L • j, abs , for j = 1 , . As a consequence, we have(5.49) V • / ( V • + V • ) ≃ L • , ⊥ , abs ∩ L • , ⊥ , abs . Recall that the complex ( C • , • , ∂ ) was defined by (2.1) and (3.20). By (2.13), (5.47)and (5.49), we have(5.50) H ( C • , • , ∂ ) ≃ H • abs ( Z , ∞ , F ) , H ( C • , • , ∂ ) ≃ L • , ⊥ , abs ∩ L • , ⊥ , abs . We define a map S HR,T : H • ( C • , • , ∂ ) → Ker (cid:0) D Z R T (cid:1) , S HR,T (cid:12)(cid:12)(cid:12) H ( C • , • ,∂ ) = P R,T F R,T , S HR,T (cid:12)(cid:12)(cid:12) H ( C • , • ,∂ ) = P R,T I R,T . (5.51) Theorem 5.4.
For T = R κ ≫ , the map S HR,T is bijective.
NALYTIC TORSION FORMS 41
Proof.
By (5.19), (5.20), (5.43), (5.44) and the fact that χ χ = χ χ = 0 , for ( ω , ω , ˆ ω ) ∈ H • abs ( Z , ∞ , F ) and ˆ τ ∈ L • , ⊥ , abs ∩ L • , ⊥ , abs , we have (cid:13)(cid:13)(cid:13) F R,T ( ω , ω , ˆ ω ) (cid:13)(cid:13)(cid:13) Z R > (cid:13)(cid:13) ω (cid:13)(cid:13) Z , + (cid:13)(cid:13) ω (cid:13)(cid:13) Z , , (cid:13)(cid:13)(cid:13) I R,T (ˆ τ ) (cid:13)(cid:13)(cid:13) Z R > (cid:13)(cid:13) ˆ τ (cid:13)(cid:13) Y , D F R,T ( ω , ω , ˆ ω ) , I R,T (ˆ τ ) E Z R = 0 . (5.52)By Propositions 5.2, 5.3 and (5.52), we have(5.53) (cid:13)(cid:13)(cid:13) P R,T F R,T ( ω , ω , ˆ ω ) + P R,T I R,T (ˆ τ ) (cid:13)(cid:13)(cid:13) Z R > (cid:16)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , + (cid:13)(cid:13) ω (cid:13)(cid:13) Z , + (cid:13)(cid:13) ˆ τ (cid:13)(cid:13) Y (cid:17) . By (5.50)-(5.51) and (5.53), the map S HR,T is injective. On the other hand, by theexactness of (0.22), the construction of ( C • , • , ∂ ) and the Hodge theorem, we have(5.54) dim H • ( C • , • , ∂ ) = dim H • ( Z, F ) = dim Ker (cid:0) D Z R T (cid:1) . Hence the map S HR,T is bijective. This completes the proof of Theorem 5.4. (cid:3)
For ω ∈ Ω • ( Z R , F ) satisfying d Z R T ω = 0 , i.e., d Z R (cid:0) e T f T ω (cid:1) = 0 , we denote(5.55) (cid:2) ω (cid:3) T = (cid:2) e T f T ω (cid:3) ∈ H • ( Z R , F ) = H • ( Z, F ) . Corollary 5.5.
For T = R κ ≫ , the map (cid:2) S HR,T (cid:3) T : H • ( C • , • , ∂ ) → H • ( Z, F ) σ (cid:2) S HR,T ( σ ) (cid:3) T (5.56) is bijective.Proof. This is a direct consequence of the Hodge theorem and Theorem 5.4. (cid:3)
Remark . By Corollary 5.5 and (5.50), we have a bijection(5.57) H • abs ( Z , ∞ , F ) ⊕ (cid:16) L • , ⊥ , abs ∩ L • , ⊥ , abs (cid:17) ∼ −→ H • ( Z, F ) . Set(5.58) H • ( Z , ∞ , F ) = n ( ω , ω , ˆ ω ) : ( ω j , ˆ ω ) ∈ H • ( Z j, ∞ , F ) for j = 1 , o . By [50, Thm. 3.7] and the Hodge theorem, we have a bijection(5.59) H • ( Z , ∞ , F ) ∼ −→ H • ( Z, F ) . The vector spaces in (5.57) and (5.59) are linked by the short exact sequence(5.60) → H • abs ( Z , ∞ , F ) → H • ( Z , ∞ , F ) → L • , ⊥ , abs ∩ L • , ⊥ , abs → , which follows from (5.16), (5.58) and [50, (2.49)-(2.53)]. NALYTIC TORSION FORMS 42
Eigenspace of D Z R T associated with small eigenvalues. For ω ∈ Ω • ( IY R , F ) = C ∞ ([ − R, R ] , H • ( Y, F )[ du ]) , we denote(5.61) (cid:13)(cid:13) ω (cid:13)(cid:13) Y, max = max n(cid:13)(cid:13) ω u (cid:13)(cid:13) Y : u ∈ [ − R, R ] o . Lemma 5.7.
For T = R κ ≫ and ω ∈ Ω • ( Z R , F ) an eigensection of RD Z R T associatedwith eigenvalue λ ∈ (cid:2) − √ R, √ R (cid:3) \{ } , we have (5.62) (cid:13)(cid:13) ω (cid:13)(cid:13) Z , ∪ Z , = O (cid:0) (cid:1)(cid:13)(cid:13) ω zm (cid:13)(cid:13) Y, max . Proof.
We will follow [50, Lemma 3.10]. Suppose, on the contrary, that there exist R i → ∞ , T i = R κi , λ i ∈ [ −√ R i , √ R i ] \{ } and ω i ∈ Ω • ( Z R i , F ) such that(5.63) R i D Z Ri T i ω i = λ i ω i , (cid:13)(cid:13) ω i (cid:13)(cid:13) Z , ∪ Z , = 1 , lim i →∞ (cid:13)(cid:13) ω zm i (cid:13)(cid:13) Y, max = 0 . Without loss of generality, we may assume that(5.64) lim inf i →∞ (cid:13)(cid:13) ω i (cid:13)(cid:13) Z , > . Step 1 . We extract a convergent subsequence of (cid:0) ω i (cid:1) i .By the Trace theorem for Sobolev spaces and (5.63), we have(5.65) (cid:13)(cid:13) ω i (cid:13)(cid:13) ∂Z , = O (cid:0) (cid:1) , (cid:13)(cid:13) ω i (cid:13)(cid:13) ∂Z , = O (cid:0) (cid:1) . By Lemma 4.4, (5.5) and (5.65), we have(5.66) (cid:13)(cid:13) ω nz i (cid:13)(cid:13) Y, max = O (cid:0) (cid:1) . By (5.63) and (5.66), we have(5.67) (cid:13)(cid:13) ω i (cid:13)(cid:13) Y, max (cid:13)(cid:13) ω zm i (cid:13)(cid:13) Y, max + (cid:13)(cid:13) ω nz i (cid:13)(cid:13) Y, max = O (cid:0) (cid:1) . For r ∈ N and R > r , let IY r ⊆ Z ,r ⊆ Z , ∞ and Z ,r ⊆ Z R be the canonical isometricembeddings. By (4.36) and (4.37), we have(5.68) RD Z R T (cid:12)(cid:12) Z , = RD Z , ∞ (cid:12)(cid:12) Z , , RD Z R T (cid:12)(cid:12) IY r = RD Z , ∞ (cid:12)(cid:12) IY r + T f ′ T ˆ c (cid:12)(cid:12) IY r . By (5.63) and (5.68), we have(5.69) D Z , ∞ ω i (cid:12)(cid:12) Z , = λ i R − i ω i (cid:12)(cid:12) Z , , D Z , ∞ ω i (cid:12)(cid:12) IY r = (cid:16) λ i R − i − R − i T i f ′ T i ˆ c (cid:17) ω i (cid:12)(cid:12) IY r . Since λ i R − i → and R − i T i → , by the second identity in (5.32), (5.63), (5.67)and (5.69), the series (cid:0) ω i (cid:12)(cid:12) Z ,r (cid:1) i is H -bounded. Using Rellich’s lemma, by extractinga subsequence, we may suppose that ω i (cid:12)(cid:12) Z ,r is L -convergent. Applying (5.69) onceagain, we see that (cid:0) ω i (cid:12)(cid:12) Z ,r (cid:1) i is H -Cauchy. Let ω ∞ ,r be the limit of (cid:0) ω i (cid:12)(cid:12) Z ,r (cid:1) i , which isat least a H -current on Z ,r with values in F . Taking the limit of (5.69), we get(5.70) D Z , ∞ ω ∞ ,r (cid:12)(cid:12) Z ,r = 0 . Since D Z , ∞ is elliptic, equation (5.70) implies ω ∞ ,r ∈ Ω • ( Z ,r , F ) . NALYTIC TORSION FORMS 43
The standard diagonal argument allows us to extract a subsequence (cid:0) ω i j (cid:1) j of (cid:0) ω i (cid:1) i such that for any r ∈ N , (cid:0) ω i j (cid:12)(cid:12) Z ,r (cid:1) j converges to ω r, ∞ in H -norm. Now we replace (cid:0) ω i (cid:1) i by (cid:0) ω i j (cid:1) j . There exists ω ∞ ∈ Ω • ( Z , ∞ , F ) such that for any r ∈ N ,(5.71) ω i (cid:12)(cid:12) Z ,r → ω ∞ (cid:12)(cid:12) Z ,r in H -norm . Since (5.70) holds for all r ∈ N , we have(5.72) D Z , ∞ ω ∞ = 0 . Step 2 . We show that ω ∞ is L -integrable.By the Trace theorem for Sobolev spaces, (5.67) and (5.71), we have(5.73) (cid:13)(cid:13) ω ∞ (cid:13)(cid:13) Y, max < + ∞ . By (5.63) and (5.71), we have(5.74) ω zm ∞ = 0 . By [50, (2.10)] and (5.72)-(5.74), there exists a > such that for r > , we have(5.75) (cid:13)(cid:13) ω ∞ (cid:13)(cid:13) ∂Z ,r = O (cid:0) e − ar (cid:1) . In particular, ω ∞ is L -integrable. Step 3 . We look for a contradiction.By (3.21), (5.16), (5.72) and (5.75), we have(5.76) ( ω ∞ , , ∈ H • abs ( Z , ∞ , F ) . Recall that P R,T F R,T : H • abs ( Z , ∞ , F ) → Ker (cid:0) D Z R T (cid:1) was constructed in § µ i = P R i ,T i F R i ,T i ( ω ∞ , , ∈ Ker (cid:0) D Z Ri T i (cid:1) ⊆ Ω • ( Z R i , F ) . We decompose Z R i into two pieces Z R i = Z ,R i ∪ Z , . We have(5.78) (cid:10) µ i , ω i (cid:11) Z Ri − (cid:10) ω ∞ , ω i (cid:11) Z ,Ri = (cid:10) µ i − ω ∞ , ω i (cid:11) Z ,Ri + (cid:10) µ i , ω i (cid:11) Z , . By (5.63) and (5.67), we have(5.79) (cid:13)(cid:13) ω i (cid:13)(cid:13) Z ,Ri + (cid:13)(cid:13) ω i (cid:13)(cid:13) Z , = (cid:13)(cid:13) ω i (cid:13)(cid:13) Z Ri = O (cid:0) R i (cid:1) . By Proposition 5.2 and (5.77), we have(5.80) (cid:13)(cid:13) µ i − F R i ,T i ( ω ∞ , , (cid:13)(cid:13) Z Ri = O (cid:0) R − κi (cid:1) . By (5.15), (5.17), (5.18), (5.20), (5.31) and (5.32), we have(5.81) (cid:13)(cid:13) F R i ,T i ( ω ∞ , , − ω ∞ (cid:13)(cid:13) Z ,Ri = O (cid:0) R − κi (cid:1) , (cid:13)(cid:13) F R i ,T i ( ω ∞ , , (cid:13)(cid:13) Z , = 0 . By (5.80) and (5.81), we have(5.82) (cid:13)(cid:13) µ i − ω ∞ (cid:13)(cid:13) Z ,Ri = O (cid:0) R − κi (cid:1) , (cid:13)(cid:13) µ i (cid:13)(cid:13) Z , = O (cid:0) R − κi (cid:1) . By (5.78), (5.79) and (5.82), we have(5.83) (cid:10) µ i , ω i (cid:11) Z Ri − (cid:10) ω ∞ , ω i (cid:11) Z ,Ri = O (cid:0) R − / κ/ i (cid:1) . NALYTIC TORSION FORMS 44
By the dominated convergence theorem, (5.67), (5.71) and (5.75), we have(5.84) lim i → + ∞ (cid:10) ω ∞ , ω i (cid:11) Z ,Ri = (cid:13)(cid:13) ω ∞ (cid:13)(cid:13) Z , ∞ . Since κ ∈ ]0 , / , by (5.83) and (5.84), we have(5.85) lim i → + ∞ (cid:10) µ i , ω i (cid:11) Z Ri = (cid:13)(cid:13) ω ∞ (cid:13)(cid:13) Z , ∞ . By (5.64) and (5.71), we have (cid:13)(cid:13) ω ∞ (cid:13)(cid:13) Z , ∞ > . Thus (cid:10) µ i , ω i (cid:11) Z Ri = 0 for i large enough.But, by (5.63), (5.77) and the assumption λ i = 0 , we have (cid:10) µ i , ω i (cid:11) Z Ri = 0 . Contradic-tion. This completes the proof of Lemma 5.7. (cid:3) Recall that the operator c was defined in (4.6). For σ ∈ H • ( Y, F )[ du ] , we denote σ = σ + + σ − such that cσ ± = ∓ iσ ± .For j = 1 , , let C j ( λ ) ∈ End (cid:0) H • ( Y, F )[ du ] (cid:1) be the scattering matrix as in [50,(2.32)] with X ∞ replaced by Z j, ∞ . By [50, (2.35)], we have(5.86) c C j ( λ ) = − C j ( λ ) c . Lemma 5.8.
For T = R κ ≫ and ω ∈ Ω • ( Z R , F ) an eigensection of RD Z R T associatedwith eigenvalue λ ∈ (cid:2) − √ R, √ R (cid:3) \{ } , we have (5.87) (cid:13)(cid:13)(cid:13) ω zm , + − C j (cid:0) λ/R (cid:1) ω zm , − (cid:13)(cid:13)(cid:13) ∂Z j, = O (cid:0) R − κ (cid:1)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , ∪ Z , , for j = 1 , . Proof.
We will follow [50, Lemma 3.12]. We only prove (5.87) for j = 1 .Let ω ′ ∈ Ω • ( Z , ∞ , F ) be the unique generalized eigensection of D Z , ∞ associatedwith eigenvalue λ/R satisfying(5.88) ω ′ zm , − (cid:12)(cid:12) ∂Z , = ω zm , − (cid:12)(cid:12) ∂Z , ∈ H • ( Y, F )[ du ] . The existence and uniqueness of ω ′ are guaranteed by [50, Prop. 2.4]. Moreover, by[50, Prop. 2.4] and (5.88), we have(5.89) ω ′ zm , + (cid:12)(cid:12) ∂Z , = C (cid:0) λ/R (cid:1) ω ′ zm , − (cid:12)(cid:12) ∂Z , = C (cid:0) λ/R (cid:1) ω zm , − (cid:12)(cid:12) ∂Z , . By the theory of ordinary differential equation, there exists ω ′′ ∈ Ω • ( Z ,R , F ) satisfying(5.90) ω ′′ (cid:12)(cid:12) Z , = ω ′ (cid:12)(cid:12) Z , , ω ′′ nz = ω ′ nz , RD Z R T (cid:12)(cid:12) IY R ω ′′ zm = λω ′′ zm . Set(5.91) µ = ω (cid:12)(cid:12) Z ,R − ω ′′ (cid:12)(cid:12) Z ,R . By (5.88)-(5.91), we have(5.92) µ zm (cid:12)(cid:12) ∂Z , = (cid:0) ω zm , + − C (cid:0) λ/R (cid:1) ω zm , − (cid:1) (cid:12)(cid:12) ∂Z , . By the construction of µ , we have RD Z R T (cid:12)(cid:12) IY R µ zm = λµ zm . By Lemma 4.5, (5.86) and(5.92), we have (cid:10) cµ zm , µ zm (cid:11) ∂Z ,R/ = (cid:10) cµ zm , µ zm (cid:11) ∂Z , = − i (cid:13)(cid:13) µ zm (cid:13)(cid:13) ∂Z , = − i (cid:13)(cid:13)(cid:13) ω zm , + − C (cid:0) λ/R (cid:1) ω zm , − (cid:13)(cid:13)(cid:13) ∂Z , . (5.93) NALYTIC TORSION FORMS 45
By the Trace theorem for Sobolev spaces and Proposition 5.1, we have(5.94) (cid:13)(cid:13) ω (cid:13)(cid:13) ∂Z , = O (cid:0) (cid:1)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , , (cid:13)(cid:13) ω (cid:13)(cid:13) ∂Z , = O (cid:0) (cid:1)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , . Applying Lemma 4.4 to ω nz , there exists a universal constant a > such that(5.95) (cid:13)(cid:13) ω nz (cid:13)(cid:13) ∂Z ,r = O (cid:0) e − ar (cid:1)(cid:16)(cid:13)(cid:13) ω nz (cid:13)(cid:13) ∂Z , + (cid:13)(cid:13) ω nz (cid:13)(cid:13) ∂Z , (cid:17) , for r R/ . By [50, Prop. 2.4], we have (cid:13)(cid:13) ω ′ nz (cid:13)(cid:13) Z , ∞ \ Z , < + ∞ . Moreover, by [50, (2.10),(2.38)],there exists a universal constant a > such that(5.96) (cid:13)(cid:13) ω ′ nz (cid:13)(cid:13) ∂Z ,r = O (cid:0) e − ar (cid:1)(cid:13)(cid:13) ω ′ nz (cid:13)(cid:13) Z , ∞ \ Z , = O (cid:0) e − ar (cid:1)(cid:13)(cid:13) ω ′ zm (cid:13)(cid:13) ∂Z , , for r > . Since C (cid:0) λ/R (cid:1) is unitary, (5.88) and (5.89) imply(5.97) (cid:13)(cid:13) ω ′ zm (cid:13)(cid:13) ∂Z , = (cid:13)(cid:13) ω ′ zm , − (cid:13)(cid:13) ∂Z , + (cid:13)(cid:13) ω ′ zm , + (cid:13)(cid:13) ∂Z , = 2 (cid:13)(cid:13) ω zm , − (cid:13)(cid:13) ∂Z , . By (5.94)-(5.97), we have(5.98) (cid:13)(cid:13) ω nz (cid:13)(cid:13) ∂Z ,r = O (cid:0) e − ar (cid:1)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , ∪ Z , , (cid:13)(cid:13) ω ′ nz (cid:13)(cid:13) ∂Z ,r = O (cid:0) e − ar (cid:1)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , ∪ Z , . By the second identity in (5.90), (5.91) and (5.98), we have(5.99) (cid:13)(cid:13) µ nz (cid:13)(cid:13) ∂Z ,r = O (cid:0) e − ar (cid:1)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , ∪ Z , . We identify IY R ⊆ Z ,R with [0 , R ] × Y . By the construction of µ and (5.68), wehave(5.100) RD Z R T µ (cid:12)(cid:12) Z , = λµ , RD Z R T µ (cid:12)(cid:12) [0 , R ] × Y = λµ − T f ′ T ˆ cω ′ nz . By (5.5) and (5.100), we have(5.101) (cid:10) RD Z R T µ, µ (cid:11) Z ,R/ − (cid:10) µ, RD Z R T µ (cid:11) Z ,R/ = 2 iT Im (cid:10) µ nz , f ′ T ˆ cω ′ nz (cid:11) [0 ,R ] × Y . On the other hand, by Green’s formula (cf. [50, (2.8)]), we have(5.102) (cid:10) RD Z R T µ, µ (cid:11) Z ,R/ − (cid:10) µ, RD Z R T µ (cid:11) Z ,R/ = R (cid:10) cµ, µ (cid:11) ∂Z ,R/ . By (5.101), (5.102) and the assumption T = R κ , we have (cid:12)(cid:12)(cid:12)(cid:10) cµ zm , µ zm (cid:11) ∂Z ,R/ + (cid:10) cµ nz , µ nz (cid:11) ∂Z ,R/ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:10) cµ, µ (cid:11) ∂Z ,R/ (cid:12)(cid:12)(cid:12) R − κ (cid:12)(cid:12)(cid:12)(cid:10) µ nz , f ′ T ˆ cω ′ nz (cid:11) [0 ,R ] × Y (cid:12)(cid:12)(cid:12) . (5.103)By (3.8) and (3.11), we have(5.104) (cid:12)(cid:12) f ′ T (cid:12)(cid:12) ∂Z ,r = O (cid:0) R − (cid:1) r . By (5.98) (5.99) and (5.104), we have (cid:12)(cid:12)(cid:12)(cid:10) µ nz , f ′ T ˆ cω ′ nz (cid:11) [0 ,R ] × Y (cid:12)(cid:12)(cid:12) Z R (cid:12)(cid:12) f ′ T (cid:12)(cid:12) ∂Z ,r (cid:13)(cid:13) µ nz (cid:13)(cid:13) ∂Z ,r (cid:13)(cid:13) ω ′ nz (cid:13)(cid:13) ∂Z ,r dr = O (cid:0) R − (cid:1)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , ∪ Z , . (5.105)By (5.99), (5.103) and (5.105), we have(5.106) (cid:12)(cid:12)(cid:12)(cid:10) cµ zm , µ zm (cid:11) ∂Z ,R/ (cid:12)(cid:12)(cid:12) = O (cid:0) R − κ (cid:1)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , ∪ Z , . NALYTIC TORSION FORMS 46
From (5.93) and (5.106), we obtain (5.87) with j = 1 . This completes the proof ofLemma 5.8. (cid:3) Proof of Theorem 3.1.
First we consider the case j = 0 .Let ω ∈ Ω • ( Z R , F ) be an eigensection of RD Z R T associated with eigenvalue λ ∈ [ −√ T , √ T ] \{ } . By Lemmas 5.7, 5.8, we have ω zm = 0 and(5.107) (cid:13)(cid:13) ω zm , + − C j (cid:0) λ/R (cid:1) ω zm , − (cid:13)(cid:13) ∂Z j, = O (cid:0) R − κ/ (cid:1)(cid:13)(cid:13) ω zm (cid:13)(cid:13) Y, max , for j = 1 , . Since λ C j ( λ ) is analytic (cf. [47, §
4] [50, Prop. 2.3]), by (5.107) and the assump-tion | λ | T / = R κ/ , we have(5.108) (cid:13)(cid:13) ω zm , + − C j (cid:0) (cid:1) ω zm , − (cid:13)(cid:13) ∂Z j, = O (cid:0) R − κ/ (cid:1)(cid:13)(cid:13) ω zm (cid:13)(cid:13) Y, max , for j = 1 , . Moreover, as C j (0) is unitary and (cid:0) C j (0) (cid:1) = Id (cf. [50, Prop. 2.3]), we have(5.109) (cid:13)(cid:13) ω zm − C j (cid:0) (cid:1) ω zm (cid:13)(cid:13) ∂Z j, = O (cid:0) R − κ/ (cid:1)(cid:13)(cid:13) ω zm (cid:13)(cid:13) Y, max , for j = 1 , . By [50, (2.48)] and (5.37), we have(5.110) L • j = Ker (cid:0) Id − C j (0) (cid:1) . For j = 1 , , let(5.111) P j : H • ( Y, F )[ du ] → L • j be orthogonal projections with respect to (cid:13)(cid:13) · (cid:13)(cid:13) Y . We denote(5.112) P ⊥ j = Id − P j . By (5.110), the estimate (5.109) is equivalent to the follows,(5.113) (cid:13)(cid:13) P ⊥ j ω zm (cid:13)(cid:13) ∂Z j, = O (cid:0) R − κ/ (cid:1)(cid:13)(cid:13) ω zm (cid:13)(cid:13) Y, max , for j = 1 , . Let D H • ( Y,F ) T, bd be the operator D VT, bd in (4.15) with(5.114) V = H • ( Y, F ) , V j = L • j, abs , for j = 1 , . Applying Proposition 4.3 to (5.113) with ǫ = R − κ and using the assumption T = R κ ,we get(5.115) (cid:13)(cid:13) ω zm − P [ λ − ǫ,λ + ǫ ] T ω zm (cid:13)(cid:13) IY R = O (cid:0) R − κ/ (cid:1)(cid:13)(cid:13) ω zm (cid:13)(cid:13) IY R . By (5.115), for T = R κ ≫ , we have P [ λ − ǫ,λ + ǫ ] T ω zm = 0 . As a consequence,(5.116) [ λ − ǫ, λ + ǫ ] ∩ Sp (cid:0) D H • ( Y,F ) T, bd (cid:1) = ∅ . From Theorem 4.1 and (5.116), we obtain (3.18) with j = 0 .We turn to the cases j = 1 , , . Proceeding in the same way as in [50, § Z j,R by its ’double’, which is a compact manifold without boundary. Thenwe apply (3.18) with j = 0 . This completes the proof of Theorem 3.1. (cid:3) NALYTIC TORSION FORMS 47
For convenience, we denote(5.117) H • abs ( Z , ∞ ⊔ Z , ∞ , F ) = H • abs ( Z , ∞ , F ) ⊕ H • abs ( Z , ∞ , F ) . Similarly to the constructions of F R,T and G R,T in § F + R,T , G + R,T : H • abs ( Z , ∞ ⊔ Z , ∞ , F ) → Ω • ( Z R , F ) as follows: for ( ω , ˆ ω , ω , ˆ ω ) ∈ H • abs ( Z , ∞ ⊔ Z , ∞ , F ) , F + R,T ( ω , ˆ ω , ω , ˆ ω ) (cid:12)(cid:12) Z j, = G + R,T ( ω , ˆ ω , ω , ˆ ω ) (cid:12)(cid:12) Z j, = ω j , for j = 1 , ,F + R,T ( ω , ˆ ω , ω , ˆ ω ) (cid:12)(cid:12) IY R = e − T f T (cid:16) χ ˆ ω + χ ˆ ω (cid:17) + e − T f T d Z R (cid:16) χ R d F ( ω , ˆ ω ) + χ R d F ( ω , ˆ ω ) (cid:17) ,G + R,T ( ω , ˆ ω , ω , ˆ ω ) (cid:12)(cid:12) IY R = e − T f T (cid:16) χ ˆ ω + χ ˆ ω (cid:17) + e T f T d Z R , ∗ (cid:16) χ R d F, ∗ ( ω , ˆ ω ) + χ R d F, ∗ ( ω , ˆ ω ) (cid:17) . (5.119)By [50, (2.52)], (3.21) and (5.117), we have d Y, ∗ ˆ ω j = i ∂∂u ˆ ω j = 0 for j = 1 , . Then, by(3.15), we have(5.120) d Z R , ∗ T e − T f T (cid:16) χ ˆ ω + χ ˆ ω (cid:17) = e T f T (cid:16) d Y, ∗ − i ∂∂u ∂∂u (cid:17) e − T f T (cid:16) χ ˆ ω + χ ˆ ω (cid:17) = 0 . By (5.119) and (5.120), we have(5.121) d Z R , ∗ T G + R,T ( ω , ˆ ω , ω , ˆ ω ) = 0 . Let P [ − , R,T : Ω • ( Z R , F ) → E [ − , ,R,T be the orthogonal projection with respect to (cid:13)(cid:13) · (cid:13)(cid:13) Z R ,where E [ − , ,R,T ⊆ Ω • ( Z R , F ) was defined in (3.19). Proposition 5.9.
For T = R κ ≫ and ( ω , ˆ ω , ω , ˆ ω ) ∈ H • abs ( Z , ∞ ⊔ Z , ∞ , F ) , we have (5.122) (cid:13)(cid:13)(cid:13)(cid:0) Id − P [ − , R,T (cid:1) G + R,T ( ω , ˆ ω , ω , ˆ ω ) (cid:13)(cid:13)(cid:13) H ,Z R = O (cid:0) R − κ (cid:1)(cid:16)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , + (cid:13)(cid:13) ω (cid:13)(cid:13) Z , (cid:17) . Proof.
Though the constructions of F + R,T and G + R,T are different from the constructionsof F R,T and G R,T in (5.20), we can directly verify that ( F + R,T − G + R,T )( ω , ˆ ω , ω , ˆ ω ) satisfies (5.24). Then, similarly to (5.35), we have (cid:13)(cid:13)(cid:13) ( F + R,T − G + R,T )( ω , ˆ ω , ω , ˆ ω ) (cid:13)(cid:13)(cid:13) Z R = O (cid:0) R − κ (cid:1)(cid:16)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , + (cid:13)(cid:13) ω (cid:13)(cid:13) Z , (cid:17) , (cid:13)(cid:13)(cid:13) D Z R T ( F + R,T − G + R,T )( ω , ˆ ω , ω , ˆ ω ) (cid:13)(cid:13)(cid:13) Z R = O (cid:0) R − κ (cid:1)(cid:16)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , + (cid:13)(cid:13) ω (cid:13)(cid:13) Z , (cid:17) . (5.123)By (3.8), (3.11), (3.15), (5.17), (5.18), (5.119) and the identities D Y ˆ ω j = 0 for j = 1 , , we have (cid:13)(cid:13)(cid:13) d Z R T F + R,T ( ω , ˆ ω , ω , ˆ ω ) (cid:13)(cid:13)(cid:13) Z R = O (cid:0) e − aT (cid:1)(cid:16)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , + (cid:13)(cid:13) ω (cid:13)(cid:13) Z , (cid:17) , (cid:13)(cid:13)(cid:13) D Z R T d Z R T F + R,T ( ω , ˆ ω , ω , ˆ ω ) (cid:13)(cid:13)(cid:13) Z R = O (cid:0) e − aT (cid:1)(cid:16)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , + (cid:13)(cid:13) ω (cid:13)(cid:13) Z , (cid:17) , (5.124)where a > is a universal constant. NALYTIC TORSION FORMS 48
By Corollary 1.4, (5.121), (5.123) and (5.124), we have (cid:13)(cid:13)(cid:13)(cid:0) Id − P [ − , R,T (cid:1) G + R,T ( ω , ˆ ω , ω , ˆ ω ) (cid:13)(cid:13)(cid:13) Z R + (cid:13)(cid:13)(cid:13) D Z R T (cid:0) Id − P [ − , R,T (cid:1) G + R,T ( ω , ˆ ω , ω , ˆ ω ) (cid:13)(cid:13)(cid:13) Z R = O (cid:0) R − κ (cid:1)(cid:16)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , + (cid:13)(cid:13) ω (cid:13)(cid:13) Z , (cid:17) . (5.125)From Proposition 5.1 and (5.125), we obtain (5.122). This completes the proof ofProposition 5.9. (cid:3) We define(5.126) I + R,T : H • ( Y, F ) → Ω • +1 ( Z R , F ) as follows: for ˆ ω ∈ H • ( Y, F ) , I + R,T (ˆ ω ) (cid:12)(cid:12) Z j, = 0 , for j = 1 , ,I + R,T (ˆ ω ) (cid:12)(cid:12) IY R = χ e T f T − T du ∧ ˆ ω . (5.127)We have(5.128) d Z R T I + R,T (ˆ ω ) = 0 . Proposition 5.10.
For T = R κ ≫ and ˆ ω ∈ H • ( Y, F ) , we have (5.129) (cid:13)(cid:13)(cid:13)(cid:0) Id − P [ − , R,T (cid:1) I + R,T (ˆ ω ) (cid:13)(cid:13)(cid:13) H ,Z R = O (cid:0) e − aT (cid:1)(cid:13)(cid:13) ˆ ω (cid:13)(cid:13) Y . Proof.
By (3.8), (3.11), (3.15) and the construction of I + R,T (see (5.126)), we have(5.130) (cid:13)(cid:13)(cid:13) D Z R T I + R,T (ˆ ω ) (cid:13)(cid:13)(cid:13) Z R = O (cid:0) e − aT (cid:1)(cid:13)(cid:13) ˆ ω (cid:13)(cid:13) Y , (cid:13)(cid:13)(cid:13) D Z R , T I + R,T (ˆ ω ) (cid:13)(cid:13)(cid:13) Z R = O (cid:0) e − aT (cid:1)(cid:13)(cid:13) ˆ ω (cid:13)(cid:13) Y . By Corollary 1.4 and (5.130), we have(5.131) (cid:13)(cid:13)(cid:13)(cid:0) Id − P [ − , R,T (cid:1) I + R,T (ˆ ω ) (cid:13)(cid:13)(cid:13) Z R + (cid:13)(cid:13)(cid:13) D Z R T (cid:0) Id − P [ − , R,T (cid:1) I + R,T (ˆ ω ) (cid:13)(cid:13)(cid:13) Z R = O (cid:0) e − aT (cid:1)(cid:13)(cid:13) ˆ ω (cid:13)(cid:13) Y , where a > is a universal constant. From Proposition 5.1 and (5.131), we obtain(5.129). This completes the proof of Proposition 5.10. (cid:3) We identify C , • = W • ⊕ W • = H • ( Z , F ) ⊕ H • ( Z , F ) with H • abs ( Z , ∞ ⊔ Z , ∞ , F ) viathe map (3.22). We identify C , • = V • = H • ( Y, F ) with H • ( Y, F ) via the isomorphism H • ( Y, F ) ≃ H • ( Y, F ) given by the Hodge theorem. We define a map S R,T : C • , • → E [ − , ,R,T , S R,T (cid:12)(cid:12)(cid:12) C , • = P [ − , R,T G + R,T , S R,T (cid:12)(cid:12)(cid:12) C , • = P [ − , R,T I + R,T . (5.132) Proposition 5.11.
The vector subspaces S R,T ( C , • ) , S R,T ( C , • ) ⊆ Ω • ( Z R , F ) are orthog-onal with respect to (cid:10) · , · (cid:11) Z R . NALYTIC TORSION FORMS 49
Proof.
We consider σ ∈ C , • and σ ∈ C , • . Since the supports of G + R,T ( σ ) and I + R,T ( σ ) are mutually disjoint, we have(5.133) (cid:10) G + R,T ( σ ) , I + R,T ( σ ) (cid:11) Z R = 0 . On the other hand, by (5.121) and (5.128), we have G + R,T ( σ ) ∈ Ker (cid:0) d Z R , ∗ T (cid:1) = Ker (cid:0) D Z R T (cid:1) ⊕ Im (cid:0) d Z R , ∗ T (cid:1) ,I + R,T ( σ ) ∈ Ker (cid:0) d Z R T (cid:1) = Ker (cid:0) D Z R T (cid:1) ⊕ Im (cid:0) d Z R T (cid:1) . (5.134)Since P R \ [ − , R,T := Id − P [ − , R,T commutes with d Z R T and d Z R , ∗ T , we have(5.135) P R \ [ − , R,T G + R,T ( σ ) ∈ Im (cid:0) d Z R , ∗ T (cid:1) , P R \ [ − , R,T I + R,T ( σ ) ∈ Im (cid:0) d Z R T (cid:1) , which implies(5.136) D P R \ [ − , R,T G + R,T ( σ ) , P R \ [ − , R,T I + R,T ( σ ) E Z R = 0 . From (5.133), (5.136) and the obvious identity D G + R,T ( σ ) , I + R,T ( σ ) E Z R = D S R,T ( σ ) , S R,T ( σ ) E Z R + D P R \ [ − , R,T G + R,T ( σ ) , P R \ [ − , R,T I + R,T ( σ ) E Z R , (5.137)we obtain D S R,T ( σ ) , S R,T ( σ ) E Z R = 0 . This completes the proof of Proposition 5.11. (cid:3) Theorem 5.12.
For T = R κ ≫ , the map S R,T is bijective.Proof.
We will use the identifications (5.50). We construct a vector subspace U • , • ⊆ H • ( C • , • , ∂ ) as follows, U , • = n ( ω , ω , ˆ ω ) ∈ H • abs ( Z , ∞ , F ) : ω j is a generalized eigensectionof D Z j, ∞ associated with , j = 1 , o ,U , • = H ( C • , • , ∂ ) = L • , ⊥ , abs ∩ L • , ⊥ , abs . (5.138) Step 1 . We show that for σ ∈ S HR,T (cid:0) U , • (cid:1) or σ ∈ S HR,T (cid:0) U , • (cid:1) , (cid:13)(cid:13) σ zm (cid:13)(cid:13) Y, max = O (cid:0) R − κ (cid:1)(cid:13)(cid:13) σ zm (cid:13)(cid:13) IY R , (cid:13)(cid:13) σ (cid:13)(cid:13) Z , ∪ Z , + (cid:13)(cid:13) σ nz (cid:13)(cid:13) IY R = O (cid:0) (cid:1)(cid:13)(cid:13) σ zm (cid:13)(cid:13) Y, max . (5.139)By the construction of S H (see (5.51)), for σ ∈ S HR,T (cid:0) U , • (cid:1) , there exists ( ω , ω , ˆ ω ) ∈ U , • such that σ = P R,T F R,T ( ω , ω , ˆ ω ) , where F R,T was defined in (5.20). We denote e σ = F R,T ( ω , ω , ˆ ω ) . By (5.20), we have(5.140) e σ zm = e − T f T ˆ ω . By (3.8), (3.11), (5.140) and the assumption T = R κ , we have(5.141) (cid:13)(cid:13)e σ zm (cid:13)(cid:13) Y, max = O (cid:0) R − κ (cid:1)(cid:13)(cid:13)e σ zm (cid:13)(cid:13) IY R . NALYTIC TORSION FORMS 50
By (5.15), (5.20), (5.31) and [50, (2.36)-(2.38)], we have (cid:13)(cid:13) ω (cid:13)(cid:13) Z , + (cid:13)(cid:13) ω (cid:13)(cid:13) Z , = (cid:13)(cid:13)e σ (cid:13)(cid:13) Z , ∪ Z , = O (cid:0) (cid:1)(cid:13)(cid:13)e σ zm (cid:13)(cid:13) Y, max , (cid:13)(cid:13)e σ nz (cid:13)(cid:13) IY R = O (cid:0) (cid:1)(cid:13)(cid:13)e σ (cid:13)(cid:13) Z , ∪ Z , . (5.142)By (5.142), we have(5.143) (cid:13)(cid:13)e σ (cid:13)(cid:13) Z , ∪ Z , + (cid:13)(cid:13)e σ nz (cid:13)(cid:13) IY R = O (cid:0) (cid:1)(cid:13)(cid:13)e σ zm (cid:13)(cid:13) Y, max . From the Trace theorem for Sobolev spaces, Proposition 5.2 and (5.141)-(5.143), weobtain (5.139) with σ ∈ S HR,T (cid:0) U , • (cid:1) .By the construction of S H (see (5.51)), for σ ∈ S HR,T (cid:0) U , • (cid:1) , there exists ˆ ω ∈ U , • such that σ = P R,T I R,T (ˆ ω ) , where I R,T was defined in (5.44). We denote e σ = I R,T (ˆ ω ) .By (5.44), we have (cid:16) O (cid:0) e − aT (cid:1)(cid:17)(cid:13)(cid:13) ˆ ω (cid:13)(cid:13) Y = (cid:13)(cid:13)e σ zm (cid:13)(cid:13) Y, max = O (cid:0) R − κ (cid:1)(cid:13)(cid:13)e σ zm (cid:13)(cid:13) IY R , (cid:13)(cid:13)e σ nz (cid:13)(cid:13) IY R = (cid:13)(cid:13)e σ (cid:13)(cid:13) Z , ∪ Z , = 0 , (5.144)where a > is a universal constant. From the Trace theorem for Sobolev spaces,Proposition 5.3 and (5.144), we obtain (5.139) with σ ∈ S HR,T (cid:0) U , • (cid:1) . Step 2 . We show that for σ ∈ E { λ } R,T with λ ∈ [ − , \{ } , (cid:13)(cid:13) σ zm (cid:13)(cid:13) Y, max = O (cid:0) R − κ (cid:1)(cid:13)(cid:13) σ zm (cid:13)(cid:13) IY R , (cid:13)(cid:13) σ (cid:13)(cid:13) Z , ∪ Z , + (cid:13)(cid:13) σ nz (cid:13)(cid:13) IY R = O (cid:0) (cid:1)(cid:13)(cid:13) σ zm (cid:13)(cid:13) Y, max . (5.145)By the Trace theorem for Sobolev spaces, Lemma 4.4 and Proposition 5.1, we have(5.146) (cid:13)(cid:13) σ nz (cid:13)(cid:13) IY R = O (cid:0) (cid:1)(cid:13)(cid:13) σ nz (cid:13)(cid:13) ∂Z , ∪ ∂Z , = O (cid:0) (cid:1)(cid:13)(cid:13) σ (cid:13)(cid:13) ∂Z , ∪ ∂Z , = O (cid:0) (cid:1)(cid:13)(cid:13) σ (cid:13)(cid:13) Z , ∪ Z , . By (4.42), σ zm is an eigensection of D H • ( Y,F ) T associated with λ , i.e.,(5.147) (cid:16) c ∂∂s + T f ′ T ˆ c (cid:17) σ zm = D H • ( Y,F ) T σ zm = λσ zm . The first inequality in (5.145) follows from the Sobolev inequality, (5.147) and theassumption λ ∈ [ − , . The second inequality in (5.145) follows from Lemma 5.7 and(5.146).Let E [ − , T ⊆ Ω • (cid:0) [ − , , H • ( Y, F ) (cid:1) be the eigenspace of D H • ( Y,F ) T associated witheigenvalues in [ − , . Let P [ − , T : Ω • (cid:0) [ − , , H • ( Y, F ) (cid:1) → E [ − , T be the orthogonalprojection. Step 3 . We show that the map π R,T : S HR,T (cid:0) U • , • (cid:1) ⊕ E [ − , \{ } ,R,T → E [ − , T σ P [ − , T σ zm (5.148)is injective.Let(5.149) σ , · · · , σ m ∈ S HR,T (cid:0) U • , • (cid:1) ⊕ E [ − , \{ } ,R,T NALYTIC TORSION FORMS 51 be a basis such that each σ i belongs to one of the following vector spaces(5.150) S HR,T (cid:0) U , • (cid:1) , S HR,T (cid:0) U , • (cid:1) , E { λ } R,T with λ ∈ [ − , \{ } . We suppose that for i = j with σ i , σ j belonging to the same vector space in (5.150),(5.151) (cid:10) σ i , σ j (cid:11) Z R = 0 . By the constructions of F R,T and I R,T , we have(5.152) F R,T (cid:0) U , • (cid:1) ⊥ I R,T (cid:0) U , • (cid:1) . By Propositions 5.2, 5.3 and (5.152), for σ i ∈ S HR,T (cid:0) U , • (cid:1) and σ j ∈ S HR,T (cid:0) U , • (cid:1) , wehave(5.153) (cid:10) σ i , σ j (cid:11) Z R = O (cid:0) R − κ/ (cid:1)(cid:13)(cid:13) σ i (cid:13)(cid:13) Z R (cid:13)(cid:13) σ j (cid:13)(cid:13) Z R . Since S HR,T (cid:0) U • , • (cid:1) ⊆ Ker (cid:0) D Z R T (cid:1) , for σ i ∈ S HR,T (cid:0) U • , • (cid:1) and σ j ∈ E [ − , \{ } R,T , we have(5.154) (cid:10) σ i , σ j (cid:11) Z R = 0 . By (5.151), (5.153) and (5.154), we have(5.155) (cid:10) σ i , σ j (cid:11) Z R = (cid:16) δ ij + O (cid:0) R − κ/ (cid:1)(cid:17)(cid:13)(cid:13) σ i (cid:13)(cid:13) Z R (cid:13)(cid:13) σ j (cid:13)(cid:13) Z R , where δ ij is the Kronecker delta.By Steps 1, 2 and the obvious identity(5.156) (cid:10) σ i , σ j (cid:11) Z R = (cid:10) σ zm i , σ zm j (cid:11) IY R + (cid:10) σ nz i , σ nz j (cid:11) IY R + (cid:10) σ i , σ j (cid:11) Z , ∪ Z , , we have(5.157) (cid:10) σ zm i , σ zm j (cid:11) IY R = (cid:10) σ i , σ j (cid:11) Z R + O (cid:0) R − κ (cid:1)(cid:13)(cid:13) σ i (cid:13)(cid:13) Z R (cid:13)(cid:13) σ j (cid:13)(cid:13) Z R . Recall that the maps P ⊥ j with j = 1 , were defined by (5.111)-(5.112). By (5.113),for σ ∈ E { λ } R,T with λ ∈ [ − , \{ } , we have(5.158) (cid:13)(cid:13)(cid:13) P ⊥ j σ zm (cid:13)(cid:13)(cid:13) ∂Z j, = O (cid:0) R − κ (cid:1)(cid:13)(cid:13) σ zm (cid:13)(cid:13) Y, max , for j = 1 , . By the constructions of F R,T and I R,T , for e σ ∈ F R,T (cid:0) U , • (cid:1) or e σ ∈ I R,T (cid:0) U , • (cid:1) , we have(5.159) P ⊥ j e σ zm (cid:12)(cid:12) ∂Z j, = 0 , for j = 1 , . By the Trace theorem for Sobolev spaces, Propositions 5.2, 5.3, (5.142), (5.144) and(5.159), for σ ∈ S HR,T (cid:0) U , • (cid:1) or σ ∈ S HR,T (cid:0) U , • (cid:1) , we have(5.160) (cid:13)(cid:13)(cid:13) P ⊥ j σ zm (cid:13)(cid:13)(cid:13) ∂Z j, = O (cid:0) R − κ (cid:1)(cid:13)(cid:13) σ zm (cid:13)(cid:13) Y, max , for j = 1 , . Applying Proposition 4.3 to (5.158) and (5.160) with ǫ = 1 , we get(5.161) (cid:13)(cid:13)(cid:13) σ zm i − P [ − , T σ zm i (cid:13)(cid:13)(cid:13) IY R = O (cid:0) R − κ (cid:1)(cid:13)(cid:13) σ zm i (cid:13)(cid:13) IY R . By Theorem 4.1, we have P [ − , T = P [ − , T . Then equation (5.161) yields(5.162) (cid:10) P [ − , T σ zm i , P [ − , T σ zm j (cid:11) IY R = (cid:10) σ zm i , σ zm j (cid:11) IY R + O (cid:0) R − κ (cid:1)(cid:13)(cid:13) σ zm i (cid:13)(cid:13) IY R (cid:13)(cid:13) σ zm j (cid:13)(cid:13) IY R . NALYTIC TORSION FORMS 52
By (5.155), (5.157) and (5.162), we have(5.163) (cid:10) P [ − , T σ zm i , P [ − , T σ zm j (cid:11) IY R = (cid:16) δ ij + O (cid:0) R − κ (cid:1)(cid:17)(cid:13)(cid:13) σ i (cid:13)(cid:13) Z R (cid:13)(cid:13) σ j (cid:13)(cid:13) Z R . By (5.148) and (5.163), the Gram matrix (cid:16)(cid:10) π R,T ( σ i ) , π R,T ( σ j ) (cid:11) IY R (cid:17) i,j m is positive-definite. Hence the map π R,T is injective.
Step 4 . We show that the map S R,T is bijective.By Theorems 4.2, 5.4 and Step 3, we have dim (cid:16) E [ − , \{ } ,R,T (cid:17) + dim U • , • = dim (cid:16) E [ − , \{ } ,R,T (cid:17) + dim S HR,T (cid:0) U • , • (cid:1) dim E [ − , T = dim C • , • r . (5.164)By Theorem 5.4, we have(5.165) dim Ker (cid:0) D Z R T (cid:1) = dim H • ( C • , • , ∂ ) . By (2.4), (2.5) and (3.20), we have(5.166) dim C • , • − dim C • , • r = dim K • + dim K • . By the construction of U • , • and (3.23)-(3.25), we have(5.167) dim H • ( C • , • , ∂ ) − dim U • , • = dim K • + dim K • . From (5.164)-(5.167), we obtain(5.168) dim E [ − , ,R,T = dim E [ − , \{ } ,R,T + dim Ker (cid:0) D Z R T (cid:1) dim C • , • . By Propositions 5.9-5.11 and (5.132), the map S R,T : dim C • , • → E [ − , R,T is injective.Then, by (5.168), it is bijective. This completes the proof of Theorem 5.12. (cid:3)
De Rham operator on E [ − , ,R,T .Proposition 5.13. For T = R κ ≫ , we have (5.169) d Z R T S R,T (cid:0) C , • (cid:1) = 0 , d Z R T S R,T (cid:0) C , • (cid:1) ⊆ S R,T (cid:0) C , • (cid:1) . Proof.
Since d Z R T commutes with P [ − , R,T , (5.128) and (5.132) yield(5.170) d Z R T S R,T (cid:0) C , • (cid:1) = d Z R T P [ − , R,T I + R,T (cid:0) C , • (cid:1) = P [ − , R,T d Z R T I + R,T (cid:0) C , • (cid:1) = 0 . Since d Z R , ∗ T commutes with P [ − , R,T , (5.121) and (5.132) yield(5.171) d Z R , ∗ T S R,T (cid:0) C , • (cid:1) = d Z R , ∗ T P [ − , R,T G + R,T (cid:0) C , • (cid:1) = P [ − , R,T d Z R , ∗ T G + R,T (cid:0) C , • (cid:1) = 0 . Thus S R,T (cid:0) C , • (cid:1) is perpendicular to the image of d Z R T . On the other hand, by Proposi-tion 5.11 and Theorem 5.12, we have an orthogonal decomposition(5.172) E [ − , ,R,T = S R,T (cid:0) C , • (cid:1) ⊕ S R,T (cid:0) C , • (cid:1) . Hence d Z R T S R,T (cid:0) C , • (cid:1) must lie in S R,T (cid:0) C , • (cid:1) . This completes the proof of Proposition5.13. (cid:3) NALYTIC TORSION FORMS 53
For ω ∈ Ω • ( Z R , F ) , we will view ω zm as an element in Ω • (cid:0) [ − R, R ] , H • ( Y, F ) (cid:1) . Set(5.173) τ R,T ( ω ) = Z R − R e T f T ω zm ∈ H • ( Y, F ) . Lemma 5.14.
There exists a > such that for T = R κ ≫ and ˆ ω ∈ H • ( Y, F ) , we have (5.174) (cid:13)(cid:13)(cid:13) e − T τ R,T (cid:16) S R,T (cid:0) ˆ ω (cid:1)(cid:17) − √ πR − κ/ ˆ ω (cid:13)(cid:13)(cid:13) Y = O (cid:0) e − aT (cid:1)(cid:13)(cid:13) ˆ ω (cid:13)(cid:13) Y . Proof.
By (3.8), (3.11), (5.42) and (5.127) and the assumption T = R κ , we have Z R − R e T f T (cid:16) I + R,T (cid:0) ˆ ω (cid:1)(cid:17) zm = Z R − R χ e T f T − T du e T ˆ ω = (cid:16) O (cid:0) e − aT (cid:1)(cid:17) Z + ∞−∞ e − T u /R du e T ˆ ω = (cid:16) O (cid:0) e − aT (cid:1)(cid:17) √ πR − κ/ e T ˆ ω . (5.175)From Proposition 5.10, (5.132) and (5.175), we obtain (5.174). This completes theproof of Lemma 5.14. (cid:3) We will use the notation in (3.37).
Theorem 5.15.
For T = R κ ≫ , we have (5.176) S − R,T ◦ d Z R T ◦ S R,T = π − / R − κ/ e − T (cid:16) ∂ + O End( C • , • ) (cid:0) R − κ/ (cid:1)(cid:17) . Proof.
For ω ∈ S R,T (cid:0) C , • (cid:1) , by (3.10), (3.15) and (5.173), we have(5.177) τ R,T (cid:0) d Z R T ω (cid:1) = Z R − R d (cid:0) e T f T ω zm (cid:1) = i ∂∂u du ∧ (cid:16) ω zm (cid:12)(cid:12) ∂Z , − ω zm (cid:12)(cid:12) ∂Z , (cid:17) . By the Trace theorem for Sobolev spaces, Proposition 5.9 and (5.132), for j = 1 , and ( ω , ˆ ω , ω , ˆ ω ) ∈ H • abs ( Z , ∞ ⊔ Z , ∞ , F ) , we have (cid:13)(cid:13)(cid:13)(cid:16) S R,T ( ω , ˆ ω , ω , ˆ ω ) − G + R,T ( ω , ˆ ω , ω , ˆ ω ) (cid:17) zm (cid:13)(cid:13)(cid:13) ∂Z j, = O (cid:0) R − κ/ (cid:1)(cid:16)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , + (cid:13)(cid:13) ω (cid:13)(cid:13) Z , (cid:17) . (5.178)By (5.119), we have(5.179) (cid:16) G + R,T ( ω , ˆ ω , ω , ˆ ω ) (cid:17) zm (cid:12)(cid:12)(cid:12) ∂Z j, = ˆ ω j ∈ H • ( Y, F ) . By (5.177)-(5.179), we have (cid:13)(cid:13)(cid:13) ˆ ω − ˆ ω − τ R,T (cid:16) d Z R T S R,T ( ω , ˆ ω , ω , ˆ ω ) (cid:17)(cid:13)(cid:13)(cid:13) Y = O (cid:0) R − κ/ (cid:1)(cid:16)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , + (cid:13)(cid:13) ω (cid:13)(cid:13) Z , (cid:17) . (5.180)From Proposition 5.13, Lemma 5.14, (2.1), (3.20) and (5.180), we obtain (5.176).This completes the proof of Theorem 5.15. (cid:3) NALYTIC TORSION FORMS 54 L -metric on E [ − , ,R,T . Recall that the Hermitian metric h C • , • R,T on C • , • was con-structed in (3.30). We denote by (cid:13)(cid:13) · (cid:13)(cid:13) R,T the norm on C • , • associated with h C • , • R,T . Proposition 5.16.
For T = R κ ≫ and σ ∈ C • , • , we have (5.181) (cid:13)(cid:13)(cid:13) S R,T ( σ ) (cid:13)(cid:13)(cid:13) Z R = (cid:13)(cid:13) σ (cid:13)(cid:13) R,T (cid:16) O (cid:0) R − / κ/ (cid:1)(cid:17) . Proof.
Let (cid:13)(cid:13) · (cid:13)(cid:13) ′ R,T be the norm on C • , • defined as follows: for σ ∈ C , • and σ ∈ C , • ,(5.182) (cid:13)(cid:13) σ + σ (cid:13)(cid:13) ′ R,T = (cid:13)(cid:13)(cid:13) G + R,T ( σ ) (cid:13)(cid:13)(cid:13) Z R + (cid:13)(cid:13)(cid:13) I + R,T ( σ ) (cid:13)(cid:13)(cid:13) Z R , where G + R,T and I + R,T were constructed in (5.119) and (5.127). By (5.119) and (5.127),we have(5.183) D G + R,T ( σ ) , I + R,T ( σ ) E Z R = 0 . By Propositions 5.9, 5.10, (5.132), (5.182) and (5.183), we have(5.184) (cid:13)(cid:13)(cid:13) S R,T ( · ) (cid:13)(cid:13)(cid:13) Z R = (cid:13)(cid:13) · (cid:13)(cid:13) ′ R,T (cid:16) O (cid:0) R − κ/ (cid:1)(cid:17) . By (5.119) and (5.182), the decomposition C • , • = W • ⊕ W • ⊕ V • is orthogonal withrespect to (cid:13)(cid:13) · (cid:13)(cid:13) ′ R,T . Thus it remains to show that(5.185) (cid:13)(cid:13) σ (cid:13)(cid:13) ′ R,T = (cid:13)(cid:13) σ (cid:13)(cid:13) R,T (cid:16) O (cid:0) R − / κ/ (cid:1)(cid:17) for σ belonging to W • or W • or V • .First we consider σ ∈ V • = H • ( Y, F ) . By (3.30), we have(5.186) (cid:13)(cid:13) σ (cid:13)(cid:13) R,T = (cid:13)(cid:13) σ (cid:13)(cid:13) Y R − κ/ √ π . On the other hand, by (5.127), we have(5.187) (cid:13)(cid:13) σ (cid:13)(cid:13) ′ R,T = (cid:13)(cid:13)(cid:13) I + R,T ( σ ) (cid:13)(cid:13)(cid:13) Z R = (cid:13)(cid:13) σ (cid:13)(cid:13) Y R − κ/ √ π (cid:16) O (cid:0) e − aT (cid:1)(cid:17) , where a > is a universal constant. From (5.186) and (5.187), we obtain (5.185) for σ ∈ V • .Now we consider σ ∈ W • . For ( ω, ˆ ω ) ∈ K • , ⊥ ⊆ W • = H • abs ( Z , ∞ , F ) , where K • , ⊥ was constructed in (3.25), by (5.119), we have (cid:13)(cid:13)(cid:13) G + R,T ( ω, ˆ ω, , (cid:13)(cid:13)(cid:13) Z R = (cid:13)(cid:13) ω (cid:13)(cid:13) Z , + (cid:13)(cid:13)(cid:13) χ e − T f T ˆ ω (cid:13)(cid:13)(cid:13) IY R + (cid:13)(cid:13)(cid:13) e T f T d Z R , ∗ (cid:16) χ R d F, ∗ ( ω, ˆ ω ) (cid:17)(cid:13)(cid:13)(cid:13) IY R + 2Re D χ e − T f T ˆ ω, e T f T d Z R , ∗ (cid:16) χ R d F, ∗ ( ω, ˆ ω ) (cid:17)E IY R . (5.188)Similarly to (5.175), we have(5.189) (cid:13)(cid:13)(cid:13) χ e − T f T ˆ ω (cid:13)(cid:13)(cid:13) IY R = (cid:13)(cid:13) ˆ ω (cid:13)(cid:13) Y R − κ/ √ π (cid:16) O (cid:0) e − aT (cid:1)(cid:17) , NALYTIC TORSION FORMS 55 where a > is a universal constant. By the Trace theorem for Sobolev spaces andProposition 5.1, we have(5.190) (cid:13)(cid:13) ˆ ω (cid:13)(cid:13) Y (cid:13)(cid:13) ω (cid:13)(cid:13) ∂Z , = O (cid:0) (cid:1)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , . By (5.15), (5.31), (5.32) and (5.190), we have (cid:13)(cid:13)(cid:13) e T f T d Z R , ∗ (cid:16) χ R d F, ∗ ( ω, ˆ ω ) (cid:17)(cid:13)(cid:13)(cid:13) IY R = O (cid:0) (cid:1)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , , D χ e − T f T ˆ ω, e T f T d Z R , ∗ (cid:16) χ R d F, ∗ ( ω, ˆ ω ) (cid:17)E IY R = O (cid:0) (cid:1)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , . (5.191)By (3.25), ω is a generalized eigensection of D Z , ∞ . Then, by [50, (2.38)], we have(5.192) (cid:13)(cid:13) ω (cid:13)(cid:13) Z , = O (cid:0) (cid:1)(cid:13)(cid:13) ˆ ω (cid:13)(cid:13) Y . By (5.188)-(5.192), we have(5.193) (cid:13)(cid:13)(cid:13) G + R,T ( ω, ˆ ω, , (cid:13)(cid:13)(cid:13) IY R = (cid:13)(cid:13) ˆ ω (cid:13)(cid:13) Y (cid:16) R − κ/ √ π/ O (cid:0) (cid:1)(cid:17) . For ( τ, ∈ K • ⊆ W • = H • abs ( Z , ∞ , F ) , where K • was constructed in (3.25), by(5.119), we have(5.194) (cid:13)(cid:13)(cid:13) G + R,T ( τ, , , (cid:13)(cid:13)(cid:13) Z R = (cid:13)(cid:13) τ (cid:13)(cid:13) Z , + (cid:13)(cid:13)(cid:13) e T f T d Z R , ∗ (cid:16) χ R d F, ∗ ( τ, (cid:17)(cid:13)(cid:13)(cid:13) IY R . We will use the canonical embedding Z ,R ⊆ Z , ∞ . Since e T f T d Z R , ∗ (cid:16) χ R d F, ∗ ( τ, (cid:17) vanishes near ∂Z ,R , it may be extended to a section on [0 , + ∞ ) × Y ⊆ Z , ∞ . We usethe identification IY R = [0 , R ] × Y ⊆ [0 , + ∞ ) × Y . By (5.15), (5.17), (5.18), (5.31)and (5.32), we have (cid:13)(cid:13) τ nz (cid:13)(cid:13) [0 , + ∞ ) × Y = (cid:13)(cid:13) τ nz (cid:13)(cid:13) IY R + O (cid:0) e − aR (cid:1)(cid:13)(cid:13) τ (cid:13)(cid:13) Z , = O (cid:0) (cid:1)(cid:13)(cid:13) τ (cid:13)(cid:13) Z , , (cid:13)(cid:13)(cid:13) e T f T d Z R , ∗ (cid:16) χ R d F, ∗ ( τ, (cid:17) − τ nz (cid:13)(cid:13)(cid:13) IY R = O (cid:0) R − κ (cid:1)(cid:13)(cid:13) τ (cid:13)(cid:13) Z , , (5.195)where a > is a universal constant. By (5.195), we have(5.196) (cid:13)(cid:13)(cid:13) e T f T d Z R , ∗ (cid:16) χ R d F, ∗ ( τ, (cid:17)(cid:13)(cid:13)(cid:13) IY R = (cid:13)(cid:13) τ nz (cid:13)(cid:13) , + ∞ ) × Y + O (cid:0) R − κ (cid:1)(cid:13)(cid:13) τ (cid:13)(cid:13) Z , . By (3.25), the zero-mode τ zm vanishes. As a consequence, we have(5.197) (cid:13)(cid:13) τ (cid:13)(cid:13) Z , ∞ = (cid:13)(cid:13) τ (cid:13)(cid:13) Z , + (cid:13)(cid:13) τ nz (cid:13)(cid:13) , + ∞ ) × Y . By (5.194)-(5.197), we have(5.198) (cid:13)(cid:13)(cid:13) G + R,T ( τ, , , (cid:13)(cid:13)(cid:13) Z R = (cid:13)(cid:13) τ (cid:13)(cid:13) Z , ∞ (cid:16) O (cid:0) R − κ (cid:1)(cid:17) . NALYTIC TORSION FORMS 56
For ( ω, ˆ ω ) ∈ K • , ⊥ and ( τ, ∈ K • , we have D G + R,T ( ω, ˆ ω, , , G + R,T ( τ, , , E Z R = (cid:10) ω, τ (cid:11) Z , + D e T f T d Z R , ∗ (cid:16) χ R d F, ∗ ( ω, ˆ ω ) (cid:17) , e T f T d Z R , ∗ (cid:16) χ R d F, ∗ ( τ, (cid:17)E IY R + D χ e − T f T ˆ ω, e T f T d Z R , ∗ (cid:16) χ R d F, ∗ ( τ, (cid:17)E IY R . (5.199)Similarly to (5.191), by (5.15), (5.31) and (5.32), we have D e T f T d Z R , ∗ (cid:16) χ R d F, ∗ ( ω, ˆ ω ) (cid:17) , e T f T d Z R , ∗ (cid:16) χ R d F, ∗ ( τ, (cid:17)E IY R = O (cid:0) (cid:1)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , (cid:13)(cid:13) τ (cid:13)(cid:13) Z , , D χ e − T f T ˆ ω, e T f T d Z R , ∗ (cid:16) χ R d F, ∗ ( τ, (cid:17)E IY R = O (cid:0) (cid:1)(cid:13)(cid:13) ω (cid:13)(cid:13) Z , (cid:13)(cid:13) τ (cid:13)(cid:13) Z , . (5.200)By (5.192), (5.199) and (5.200), we have(5.201) D G + R,T ( ω, ˆ ω, , , G + R,T ( τ, , , E Z R = O (cid:0) (cid:1)(cid:13)(cid:13) ˆ ω (cid:13)(cid:13) Y (cid:13)(cid:13) τ (cid:13)(cid:13) Z , . From (5.193), (5.198) and (5.201), we obtain (5.185) with σ ∈ W • . We can prove(5.185) with σ ∈ W • in the same way. This completes the proof of Proposition 5.16. (cid:3) We will use the following identifications, H ( C • , • , ∂ ) = Ker (cid:0) ∂ : C , • → C , • (cid:1) ⊆ C , • ,H ( C • , • , ∂ ) = (cid:16) Im (cid:0) ∂ : C , • → C , • (cid:1)(cid:17) ⊥ ⊆ C , • , (5.202)where the orthogonal is taken with respect to the metric h V • R,T in (3.29). Since all the h V • R,T are mutually proportional, this is independent of
R, T . Corollary 5.17.
For T = R κ ≫ and σ ∈ H • ( C • , • , ∂ ) , we have (5.203) (cid:13)(cid:13)(cid:13) S HR,T ( σ ) (cid:13)(cid:13)(cid:13) Z R = (cid:13)(cid:13) σ (cid:13)(cid:13) R,T (cid:16) O (cid:0) R − / κ/ (cid:1)(cid:17) . Proof.
By (5.20) and (5.119), for σ ∈ H ( C • , • , ∂ ) = H • abs ( Z , ∞ , F ) ⊆ H • abs ( Z , ∞ ⊔ Z , ∞ , F ) = C , • , we have(5.204) (cid:13)(cid:13)(cid:13) F R,T ( σ ) − F + R,T ( σ ) (cid:13)(cid:13)(cid:13) Z R = O (cid:0) e − aT (cid:1)(cid:13)(cid:13) σ (cid:13)(cid:13) R,T , where a > is a universal constant. By the first identity in (5.123) and (5.204), wehave(5.205) (cid:13)(cid:13)(cid:13) F R,T ( σ ) − G + R,T ( σ ) (cid:13)(cid:13)(cid:13) Z R = O (cid:0) R − κ/ (cid:1)(cid:13)(cid:13) σ (cid:13)(cid:13) R,T . By (5.44) and (5.127), for σ ∈ H ( C • , • , ∂ ) = L • , ⊥ , abs ∩ L • , ⊥ , abs ⊆ H • ( Y, F ) = C , • , wehave(5.206) I R,T ( σ ) = I + R,T ( σ ) . NALYTIC TORSION FORMS 57
By Propositions 5.2, 5.3, 5.9, 5.10, (5.51), (5.132), (5.205) and (5.206), we have(5.207) (cid:13)(cid:13)(cid:13) S HR,T ( σ ) − S R,T ( σ ) (cid:13)(cid:13)(cid:13) Z R = O (cid:0) R − / κ/ (cid:1)(cid:13)(cid:13) σ (cid:13)(cid:13) R,T . From Proposition 5.16 and (5.207), we obtain (5.203). This completes the proof ofCorollary 5.17. (cid:3)
Proof of Theorem 3.3.
First we consider the case j = 0 . We will show that the map S R,T constructed in this section satisfies the desired properties. The first property followsfrom (5.119), (5.127) and (5.132). The second property follows from Theorem 5.12.The third property follows from Proposition 5.16. The fourth property follows fromTheorem 5.15 and Proposition 5.16.For the cases j = 1 , , , we will only give the constructions of S Hj,R,T and S j,R,T , theproof of these properties is then essentially the same as in the case j = 0 .Concerning the cases j = 1 , , we construct F j,R,T : H • abs ( Z j, ∞ , F ) → Ω • ( Z j,R , F ) asfollows: for ( ω, ˆ ω ) ∈ H • abs ( Z j, ∞ , F ) , F j,R,T ( ω, ˆ ω ) (cid:12)(cid:12) Z j, = ω ,F j,R,T ( ω, ˆ ω ) (cid:12)(cid:12) IY R = e − T f T ˆ ω + e − T f T d Z R (cid:16) χ j R d F ( ω, ˆ ω ) (cid:17) . (5.208)Let P j,R,T : Ω • ( Z j,R , F ) → Ker (cid:0) D Z j,R T (cid:1) be the orthogonal projection with respect to (cid:13)(cid:13) · (cid:13)(cid:13) Z j,R . We identify H • ( C • , • j , ∂ ) = H ( C • , • j , ∂ ) with H • abs ( Z j, ∞ , F ) . We define(5.209) S Hj,R,T = P j,R,T F j,R,T : H • ( C • , • j , ∂ ) → Ker (cid:0) D Z j,R T (cid:1) . We construct G + j,R,T : H • abs ( Z j, ∞ , F ) ⊕ H • ( Y, F ) → Ω • ( Z j,R , F ) as follows: for ( ω, ˆ ω ) ∈ H • abs ( Z j, ∞ , F ) and ˆ µ ∈ H • ( Y, F ) , G + j,R,T ( ω, ˆ ω, ˆ µ ) (cid:12)(cid:12) Z j, = ω ,G + j,R,T ( ω, ˆ ω, ˆ µ ) (cid:12)(cid:12) IY R = e − T f T (cid:16) χ j ˆ ω + χ − j ˆ µ (cid:17) + e T f T d Z R , ∗ (cid:16) χ j R d F, ∗ ( ω, ˆ ω ) (cid:17) . (5.210)We also construct I + j,R,T : H • ( Y, F ) → Ω • +1 ( Z j,R , F ) as follows: for ˆ ω ∈ H • ( Y, F ) ,(5.211) I + j,R,T (ˆ ω ) (cid:12)(cid:12) Z j, = 0 , I + j,R,T (ˆ ω ) (cid:12)(cid:12) IY R = χ e T f T − T du ∧ ˆ ω . Let P [ − , j,R,T : Ω • ( Z j,R , F ) → E [ − , j,R,T be the orthogonal projection with respect to (cid:13)(cid:13) · (cid:13)(cid:13) Z j,R .We identify C , • j with H • abs ( Z j, ∞ , F ) ⊕ H • ( Y, F ) . We identify C , • j with H • ( Y, F ) . Wedefine(5.212) S j,R,T (cid:12)(cid:12)(cid:12) C , • j = P [ − , j,R,T G + j,R,T , S j,R,T (cid:12)(cid:12)(cid:12) C , • j = P [ − , j,R,T I + j,R,T . Now, concerning the case j = 3 , we construct F ,R,T : H • ( Y, F ) → Ω • ( IY R , F ) asfollows: for ˆ ω ∈ H • ( Y, F ) ,(5.213) F ,R,T (ˆ ω ) (cid:12)(cid:12) IY R = e − T f T ˆ ω . Let P ,R,T : Ω • ( IY R , F ) → Ker (cid:0) D IY R T (cid:1) be the orthogonal projection with respect to (cid:13)(cid:13) · (cid:13)(cid:13) IY R . We identify H • ( C • , • , ∂ ) = H ( C • , • , ∂ ) with H • ( Y, F ) . We define(5.214) S H ,R,T = P ,R,T F ,R,T : H • ( C • , • , ∂ ) → Ker (cid:0) D IY R T (cid:1) . NALYTIC TORSION FORMS 58
We construct G +3 ,R,T : H • ( Y, F ) ⊕ H • ( Y, F ) → Ω • ( IY R , F ) as follows: for (ˆ µ , ˆ µ ) ∈ H • ( Y, F ) ⊕ H • ( Y, F ) ,(5.215) G +3 ,R,T (ˆ µ , ˆ µ ) = e − T f T (cid:16) χ ˆ µ + χ ˆ µ (cid:17) . We also construct I +3 ,R,T : H • ( Y, F ) → Ω • +1 ( IY R , F ) as follows: for ˆ ω ∈ H • ( Y, F ) ,(5.216) I +3 ,R,T (ˆ ω ) = χ e T f T − T du ∧ ˆ ω . Let P [ − , ,R,T : Ω • ( IY R , F ) → E [ − , ,R,T be the orthogonal projection with respect to (cid:13)(cid:13) · (cid:13)(cid:13) IY R .We identify C , • with H • ( Y, F ) ⊕ H • ( Y, F ) , and identify C , • with H • ( Y, F ) . Wedefine(5.217) S ,R,T (cid:12)(cid:12)(cid:12) C , • = P [ − , ,R,T F +3 ,R,T , S ,R,T (cid:12)(cid:12)(cid:12) C , • = P [ − , ,R,T I +3 ,R,T . This completes the proof of Theorem 3.3. (cid:3)
Remark . The proof of Theorem 3.3 may be summarized as follows: all the resultshold with S HR,T replaced by S Hj,R,T and S R,T replaced by S j,R,T .6. A NALYTIC TORSION FORMS ASSOCIATED WITH A FIBRATION
The purpose of this section is to prove Theorem 3.4. Many arguments in this sectionfollow [10]. This section is organized as follows. In § § § T = R κ , where κ ∈ ]0 , / is a fixed constant. For easeof notations, we will systematically omit a parameter ( R or T ) as long as there is noconfusion.6.1. Decomposition of analytic torsion forms.
For j = 0 , , , and R > , wedenote F j,R = Ω • ( Z j,R , F ) , which is a complex vector bundle of infinite dimensionover S . Let ∇ F j,R be the connection on F j,R defined in (1.73). Let h F j,R be the L -metric on F j,R with respect to g T Z j,R and h F . Let ω F j,R ∈ Ω (cid:0) S, End( F j,R ) (cid:1) be asin (1.79) with ( ∇ F , h F ) replaced by ( ∇ F j,R , h F j,R ) . We may extend the constructionabove to R = + ∞ as follows,(6.1) F j, ∞ = (cid:8) ω ∈ Ω • ( Z j, ∞ , F ) : ω is L -integrable (cid:9) . By [12, Prop. 4.15], ω F j, ∞ ∈ Ω ( S, End( F j, ∞ )) is well-defined. Let D j,R,t be the opera-tor in (1.81) with ( ∇ F , h F ) replaced by ( ∇ F j,R , h F j,R ) . We have(6.2) D j,R,t = √ t (cid:0) d Z j,R , ∗ T − d Z j,R T (cid:1) + ω F j,R − √ t ˆ c ( T ) ∈ Ω • (cid:0) S, End (cid:0) F j,R (cid:1)(cid:1) . We remark that(6.3) (cid:16) d Z j,R , ∗ T − d Z j,R T (cid:17) = − (cid:16) d Z j,R , ∗ T + d Z j,R T (cid:17) = − D Z j,R , T . NALYTIC TORSION FORMS 59
We denote T j,R = T j,R,T (cid:12)(cid:12) T = R κ , where T j,R,T was defined in (3.13). By (1.84), wehave T j,R = − Z + ∞ (cid:26) ϕ Tr h ( − N TZ N T Z f ′ (cid:0) D j,R,t (cid:1)i − χ ′ ( Z j , F )2 − (cid:16) dim Z rk( F ) χ ( Z j )4 − χ ′ ( Z j , F )2 (cid:17) f ′ (cid:16) i √ t (cid:17)(cid:27) dtt . (6.4)Let T S j,R (resp. T L j,R ) be as in (6.4) with R + ∞ replaced by R R − κ/ (resp. R + ∞ R − κ/ ). Thefollowing identity is obvious,(6.5) T j,R = T S j,R + T L j,R . Small time contributions.
For r > and an operator A on a Hilbert space, theSchauder r -norm of A is defined as follows,(6.6) (cid:13)(cid:13) A (cid:13)(cid:13) r = (cid:16) Tr (cid:2) ( A ∗ A ) r/ (cid:3)(cid:17) /r . If A is orthogonally diagonalizable, we have(6.7) (cid:13)(cid:13) A (cid:13)(cid:13) r = (cid:18) X λ ∈ Sp( A ) | λ | r (cid:19) /r . Let (cid:13)(cid:13) A (cid:13)(cid:13) ∞ be the operator norm of A . These norms satisfy the H¨older’s inequality: for r , r , r ∈ [1 , + ∞ ] with /r + 1 /r = 1 /r , we have(6.8) (cid:13)(cid:13) AB (cid:13)(cid:13) r (cid:13)(cid:13) A (cid:13)(cid:13) r (cid:13)(cid:13) B (cid:13)(cid:13) r . Moreover, if A is of finite rank, we have(6.9) (cid:13)(cid:13) A (cid:13)(cid:13) r (cid:0) rk( A ) (cid:1) /r (cid:13)(cid:13) A (cid:13)(cid:13) ∞ . The proofs in this subsection involve sophisticated estimate of Schauder norms, whichfollows [10, § Sp (cid:0) D j,R,t (cid:1) ⊆ i R . Lemma 6.1.
There exist α, β > such that for r > dim Z + 1 , R ≫ , < t R − κ/ and λ ∈ C with Re( λ ) = ± , we have (6.10) (cid:13)(cid:13)(cid:13)(cid:0) λ − D j,R,t (cid:1) − (cid:13)(cid:13)(cid:13) r = O (cid:0) R β (cid:1) | λ | t − α , for j = 0 , , , . Proof.
We only consider the case j = 0 .We denote(6.11) a = (cid:16) λ − √ t (cid:0) d Z R , ∗ T − d Z R T (cid:1)(cid:17) − , b = ω F R − √ t ˆ c ( T ) . Since b ∈ Ω > (cid:0) S, End( F R ) (cid:1) , we have(6.12) (cid:0) λ − D R,t (cid:1) − = a + aba + · · · + a ( ba ) dim S . The same technique as above was used in [11, (2.45)]. Note that Sp (cid:0) d Z R , ∗ T − d Z R T (cid:1) ⊆ i R and Re( λ ) = ± , by (6.11), we have(6.13) (cid:13)(cid:13) a (cid:13)(cid:13) ∞ , (cid:13)(cid:13) b (cid:13)(cid:13) ∞ = O (cid:0) (cid:1) (1 + t − / ) . NALYTIC TORSION FORMS 60
We will temporarily treat R and T as independent parameters. Note that Sp (cid:0) d Z R , ∗ T − d Z R T (cid:1) depends continuously on R, T , there exist (cid:0) µ k,R,T ∈ i R (cid:1) k ∈ N such that- for R > and T > , we have (cid:8) µ k,R,T : k ∈ N (cid:9) = Sp (cid:0) d Z R , ∗ T − d Z R T (cid:1) ;- the function ( R, T ) µ R,T is continuous.We will estimate µ k,R,T with k ∈ N fixed. For ease of notations, we will omit the index k . By [50, (3.95)] and (6.3), we have(6.14) (cid:12)(cid:12) µ R, (cid:12)(cid:12) > R − (cid:12)(cid:12) µ , (cid:12)(cid:12) . Similarly to the first identity in (4.37), we have(6.15) (cid:0) d Z R , ∗ T − d Z R T (cid:1)(cid:12)(cid:12) IY R = ˆ cc (cid:0) d Y, ∗ − d Y (cid:1) − ˆ c ∂∂u − R − T f ′ T c . By (3.11), (6.3), (6.15) and the identity f T (cid:12)(cid:12) Z , ∪ Z , = 0 , there exists δ > independentof R, T, µ
R,T such that(6.16) (cid:12)(cid:12) µ R,T − µ R, (cid:12)(cid:12) δR − T .
We consider the triangle spanned by λ, √ tµ R,T ∈ C . Let A be its area. As Re( λ ) = ± ,we have(6.17) | λ | (cid:12)(cid:12) λ − √ tµ R,T (cid:12)(cid:12) > A = (cid:12)(cid:12) √ tµ R,T (cid:12)(cid:12) . Equivalently, we have(6.18) (cid:12)(cid:12) λ − √ tµ R,T (cid:12)(cid:12) − | λ | (cid:12)(cid:12) √ tµ R,T (cid:12)(cid:12) − . If | µ R,T | > /R , by (6.14)-(6.18), we have (cid:12)(cid:12) λ − √ tµ R,T (cid:12)(cid:12) − | λ | t − / (cid:12)(cid:12) µ R, (cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) µ R, − µ R,T µ R,T (cid:12)(cid:12)(cid:12)(cid:12) | λ | t − / (cid:12)(cid:12) µ , (cid:12)(cid:12) − R (1 + δT ) = O (cid:0) R κ (cid:1) | λ | t − / (cid:12)(cid:12) µ , (cid:12)(cid:12) − , (6.19)where O (cid:0) R κ (cid:1) is uniform, i.e., it is bounded by CR κ with C > independent of R, T, µ
R,T . On the other hand, as
Re( λ ) = ± and µ R,T ∈ i R , we have the obviousestimate(6.20) (cid:12)(cid:12) λ − √ tµ R,T (cid:12)(cid:12) − . By Theorem 3.1 and (6.3), there exists α > such that(6.21) Sp (cid:16) i (cid:0) d Z R , ∗ T − d Z R T (cid:1)(cid:17) ⊆ (cid:3) − ∞ , − α √ T /R (cid:3) ∪ (cid:2) − /R, /R (cid:3) ∪ (cid:2) α √ T /R, + ∞ (cid:2) . Moreover, by Theorem 3.3, the number of eigenvalues lying in [ − /R, /R ] is constantfor R κ = T ≫ . Let P R \{ } R,T : F R → (cid:16) Ker (cid:0) D Z R T (cid:1)(cid:17) ⊥ be the orthogonal projection. By(6.7), the first identity in (6.11) and (6.19)-(6.21), we have(6.22) (cid:13)(cid:13) a (cid:13)(cid:13) r = O (cid:0) (cid:1) + O (cid:0) R κ (cid:1) | λ | t − / (cid:13)(cid:13)(cid:13)(cid:0) D Z R T (cid:1) − P R \{ } R,T (cid:12)(cid:12) R =1 ,T =0 (cid:13)(cid:13)(cid:13) r . Since r > dim Z + 1 , by Weyl’s law, we have (cid:13)(cid:13)(cid:13)(cid:0) D Z R T (cid:1) − P R \{ } R,T (cid:12)(cid:12) R =1 ,T =0 (cid:13)(cid:13)(cid:13) r < + ∞ . Then(6.22) becomes(6.23) (cid:13)(cid:13) a (cid:13)(cid:13) r = O (cid:0) (cid:1) + O (cid:0) R κ (cid:1) | λ | t − / . NALYTIC TORSION FORMS 61
By (6.8) and (6.12), we have(6.24) (cid:13)(cid:13)(cid:13)(cid:0) λ − D R,t (cid:1) − (cid:13)(cid:13)(cid:13) r (cid:13)(cid:13) a (cid:13)(cid:13) r dim S X k =0 (cid:13)(cid:13) b (cid:13)(cid:13) k ∞ (cid:13)(cid:13) a (cid:13)(cid:13) k ∞ . From (6.13), (6.23), (6.24) and the assumption < t R − κ/ , we obtain (6.10).This completes the proof of Lemma 6.1. (cid:3) Let ρ : R → [0 , be a smooth even function such that(6.25) ρ ( x ) = 1 for | x | / , ρ ( x ) = 0 for | x | > . For ς > and z ∈ C , set F ς ( z ) = (1 + 2 z ) Z + ∞−∞ exp (cid:16) √ xz (cid:17) exp (cid:18) − x (cid:19) ρ ( √ ςx ) dx √ π ,G ς ( z ) = (1 + 2 z ) Z + ∞−∞ exp (cid:16) √ xz (cid:17) exp (cid:18) − x (cid:19) (cid:0) − ρ ( √ ςx ) (cid:1) dx √ π . (6.26)The construction above follows [10, Def. 13.3]. We have(6.27) F ς ( z ) + G ς ( z ) = f ′ ( z ) . Moreover, F ς (cid:12)(cid:12) i R and G ς (cid:12)(cid:12) i R take real values, and lie in the Schwartz space S ( i R ) . Proposition 6.2.
There exists α > such that for R ≫ and < t R − κ/ , we have (6.28) (cid:13)(cid:13)(cid:13) G tR − κ/ (cid:0) D j,R,t (cid:1)(cid:13)(cid:13)(cid:13) exp (cid:0) − αR − κ/ /t (cid:1) , j = 0 , , , . Proof.
We only consider the case j = 0 .Due to the relation ∂ m ∂x m exp (cid:0) √ xz (cid:1) = 2 m/ z m exp (cid:0) √ xz (cid:1) , we can integrate by partsin the expression of z m G ς ( z ) and obtain that for m ∈ N , there exists C m > such thatfor z ∈ C with | Re( z ) | , we have(6.29) (cid:12)(cid:12) z (cid:12)(cid:12) m (cid:12)(cid:12) G ς ( z ) (cid:12)(cid:12) C m exp (cid:18) − ς (cid:19) . The function G ς ( z ) is an even holomorphic function. Therefore there exists a holo-morphic function e G ς ( z ) such that(6.30) G ς ( z ) = e G ς ( z ) . Set(6.31) U = n z ∈ C : 4Re( z ) + | Im( z ) | < o . We have(6.32) √ U := n z ∈ C : z ∈ U o = n z ∈ C : | Re( z ) | < o . By (6.29), (6.30) and (6.32), for z ∈ U , we have(6.33) (cid:12)(cid:12) z (cid:12)(cid:12) m/ (cid:12)(cid:12) e G ς ( z ) (cid:12)(cid:12) C m exp (cid:18) − ς (cid:19) . The technique above follows [10, §
13 c)].
NALYTIC TORSION FORMS 62
For r ∈ N , let e G r,ς ( z ) be the unique holomorphic function satisfying(6.34) r ! d r dz r e G r,ς ( z ) = e G ς ( z ) , lim z →−∞ e G r,ς ( z ) = 0 . By (6.33) and (6.34), for m > r , there exists C m,r > such that for z ∈ U ,(6.35) (cid:12)(cid:12) e G r,ς ( z ) (cid:12)(cid:12) C m,r (cid:12)(cid:12) z (cid:12)(cid:12) r − m/ exp (cid:18) − ς (cid:19) . In the rest of the proof, we fix ς = tR − κ/ and N ∋ r > Z/ . We have(6.36) G ς (cid:0) D R,t (cid:1) = e G ς (cid:0) D R,t (cid:1) = 12 πi Z ∂U e G r,ς ( λ ) (cid:0) λ − D R,t (cid:1) − r − dλ . By (6.8), we have(6.37) (cid:13)(cid:13)(cid:13)(cid:0) λ − D R,t (cid:1) − r − (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)(cid:0) √ λ − D R,t (cid:1) − (cid:13)(cid:13)(cid:13) r +12 r +2 (cid:13)(cid:13)(cid:13)(cid:0) − √ λ − D R,t (cid:1) − (cid:13)(cid:13)(cid:13) r +12 r +2 . From Lemma 6.1 and (6.35)-(6.37), we obtain (6.28). This completes the proof ofProposition 6.2. (cid:3)
Let χ ′ ( Z j , F ) be as in (1.83) with Z replaced by Z j . Set(6.38) χ ′ = X j =0 ( − j ( j − / χ ′ ( Z j , F ) . Proposition 6.3.
There exists α > such that for R ≫ , we have X j =0 ( − j ( j − / T S j,R = − χ ′ Z R − κ/ (cid:26) f ′ (cid:16) i √ t (cid:17) − (cid:27) dtt + O (cid:16) exp (cid:0) − αR κ/ (cid:1)(cid:17) . (6.39) Proof.
Let F tR − κ/ (cid:0) D j,R,t (cid:1) ( x, y ) ∈ (cid:16) Λ • (cid:0) T ∗ Z j,R (cid:1) ⊗ F (cid:17) x ⊗ (cid:16) Λ • (cid:0) T ∗ Z j,R (cid:1) ⊗ F (cid:17) ∗ y ⊗ π ∗ j,R (cid:0) Λ • ( T ∗ S ) (cid:1) (6.40)be the integration kernel of the operator F tR − κ/ (cid:0) D j,R,t (cid:1) with respect to the Riemann-ian volume form associated with g T Z j,R . Let d ( · , · ) be the distance function on Z j,R . Bythe finite propagation speed of the wave equation for D j,R,t (cf. [55, § F ς (cid:0) D j,R,t (cid:1) ( x, y ) = 0 for d (cid:0) x, y (cid:1) > p t/ς . In particular, we have(6.42) F tR − κ/ (cid:0) D j,R,t (cid:1) ( x, y ) = 0 for d (cid:0) x, y (cid:1) > R − κ/ , t R − κ/ . Using (6.42) in the same way as in [50, Thm. 4.5], we get(6.43) X j =0 ( − j ( j − / Tr h ( − N TZ N T Z F tR − κ/ (cid:0) D j,R,t (cid:1)i = 0 , for t R − κ/ . NALYTIC TORSION FORMS 63
By Proposition 6.2, we have(6.44) Z R − κ/ Tr h ( − N TZ N T Z G tR − κ/ (cid:0) D j,R,t (cid:1)i dtt = O (cid:16) exp (cid:0) − αR κ/ (cid:1)(cid:17) . From (6.5), (6.27), (6.43), (6.44) and the identity(6.45) χ ( Z ) − χ ( Z ) − χ ( Z ) + χ ( IY ) = 0 , we get (6.39). This completes the proof of Proposition 6.3. (cid:3) Large time contributions.
Set U j,R,t = n λ ∈ C : (cid:12)(cid:12) Re( λ ) (cid:12)(cid:12) < , (cid:12)(cid:12) Im( λ ) (cid:12)(cid:12) > t / R − κ/ o ∪ [ µ ∈ i [ − R − ,R − ] ∩ Sp (cid:0) d Zj,R, ∗ T − d Zj,RT (cid:1) n λ ∈ C : (cid:12)(cid:12) Re( λ ) (cid:12)(cid:12) < , (cid:12)(cid:12) Im( λ − √ tµ ) (cid:12)(cid:12) < o . (6.46)By Theorem 3.1 and (6.3), for R ≫ , we have(6.47) Sp (cid:16) √ t (cid:0) d Z j,R , ∗ T − d Z j,R T (cid:1)(cid:17) ⊆ U j,R,t . Set(6.48) e D j,R,t = √ tP [ − , j,R,T (cid:0) d Z j,R , ∗ T − d Z j,R T (cid:1) P [ − , j,R,T + P [ − , j,R,T ω F j,R P [ − , j,R,T . We fix p, q ∈ N such that(6.49) q > dim Z , − κ κq − κp . Lemma 6.4.
There exists α > such that for R ≫ , t > R − κ/ and λ ∈ ∂U j,R,t , wehave (6.50) (cid:13)(cid:13)(cid:13)(cid:0) λ − D j,R,t (cid:1) − p − P [ − , j,R,T (cid:0) λ − e D j,R,t (cid:1) − p P [ − , j,R,T (cid:13)(cid:13)(cid:13) = O (cid:0) R − κ/ (cid:1) | λ | α t − / . Proof.
The technique that we will apply is similar to [4, Theorems 9.30]. We onlyconsider the case j = 0 .Set(6.51) D ⊕ R,t = P [ − , R,T D R,t P [ − , R,T + P R \ [ − , R,T D R,t P R \ [ − , R,T . Step 1 . We show that(6.52) (cid:13)(cid:13)(cid:13) P R \ [ − , R,T (cid:0) λ − D ⊕ R,t (cid:1) − p P R \ [ − , R,T (cid:13)(cid:13)(cid:13) = O (cid:0) (cid:1) | λ | α t − / . We denote a = P R \ [ − , R,T (cid:16) λ − √ t (cid:0) d Z R , ∗ T − d Z R T (cid:1)(cid:17) − P R \ [ − , R,T ,b = P R \ [ − , R,T (cid:16) ω F R − √ t ˆ c ( T ) (cid:17) P R \ [ − , R,T . (6.53)By (6.2) and (6.53), we have(6.54) P R \ [ − , R,T (cid:0) λ − D ⊕ R,t (cid:1) − P R \ [ − , R,T = a + aba + · · · + a ( ba ) dim S . NALYTIC TORSION FORMS 64
By Theorem 3.1, (6.18) and the first identity in (6.53), we have(6.55) (cid:13)(cid:13) a (cid:13)(cid:13) ∞ = O (cid:0) R − κ/ (cid:1) | λ | t − / . By the second identity in (6.53), we have(6.56) (cid:13)(cid:13) b (cid:13)(cid:13) ∞ = O (cid:0) (cid:1) . By (6.54)-(6.56) and the assumption t > R − κ/ , we have (cid:13)(cid:13)(cid:13) P R \ [ − , R,T (cid:0) λ − D ⊕ R,t (cid:1) − P R \ [ − , R,T (cid:13)(cid:13)(cid:13) ∞ = O (cid:0) R − κ/ (cid:1) | λ | dim S +1 t − / = O (cid:0) R − κ/ (cid:1) | λ | dim S +1 . (6.57)Similarly to (6.22), by (6.7), (6.19) and the first identity in (6.53), we have(6.58) (cid:13)(cid:13) a (cid:13)(cid:13) q = O (cid:0) R κ (cid:1) | λ | t − / . By (6.8), (6.54)-(6.56) and (6.58), we have(6.59) (cid:13)(cid:13)(cid:13) P R \ [ − , R,T (cid:0) λ − D ⊕ R,t (cid:1) − P R \ [ − , R,T (cid:13)(cid:13)(cid:13) q = O (cid:0) R κ (cid:1) | λ | dim S +1 t − / . By (6.8), we have (cid:13)(cid:13)(cid:13) P R \ [ − , R,T (cid:0) λ − D ⊕ R,t (cid:1) − p P R \ [ − , R,T (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) P R \ [ − , R,T (cid:0) λ − D ⊕ R,t (cid:1) − P R \ [ − , R,T (cid:13)(cid:13)(cid:13) qq (cid:13)(cid:13)(cid:13) P R \ [ − , R,T (cid:0) λ − D ⊕ R,t (cid:1) − P R \ [ − , R,T (cid:13)(cid:13)(cid:13) p − q ∞ . (6.60)From (6.49), (6.57), (6.59) and (6.60), we obtain (6.52). Step 2 . We show that(6.61) (cid:13)(cid:13)(cid:13)(cid:0) λ − D R,t (cid:1) − p − (cid:0) λ − D ⊕ R,t (cid:1) − p (cid:13)(cid:13)(cid:13) = O (cid:0) R − κ/ (cid:1) | λ | α t − / . Since d Z R , ∗ T − d Z R T commutes with P [ − , R,T , we have P [ − , R,T D R,t P R \ [ − , R,T = P [ − , R,T ω F R P R \ [ − , R,T − √ t P [ − , R,T ˆ c ( T ) P R \ [ − , R,T ,P R \ [ − , R,T D R,t P [ − , R,T = P R \ [ − , R,T ω F R P [ − , R,T − √ t P R \ [ − , R,T ˆ c ( T ) P [ − , R,T . (6.62)As a consequence, we have(6.63) (cid:13)(cid:13)(cid:13) P [ − , R,T D R,t P R \ [ − , R,T (cid:13)(cid:13)(cid:13) ∞ = O (cid:0) (cid:1) , (cid:13)(cid:13)(cid:13) P R \ [ − , R,T D R,t P [ − , R,T (cid:13)(cid:13)(cid:13) ∞ = O (cid:0) (cid:1) . The same argument as in (6.54)-(6.57) yields(6.64) (cid:13)(cid:13)(cid:13) P [ − , R,T (cid:0) λ − D ⊕ R,t (cid:1) − P [ − , R,T (cid:13)(cid:13)(cid:13) ∞ = O (cid:0) (cid:1) . We denote A = n P [ − , R,T (cid:0) λ − D ⊕ R,t (cid:1) − P [ − , R,T , P R \ [ − , R,T (cid:0) λ − D ⊕ R,t (cid:1) − P R \ [ − , R,T o , B = n P [ − , R,T D R,t P R \ [ − , R,T , P R \ [ − , R,T D R,t P [ − , R,T o . (6.65) NALYTIC TORSION FORMS 65
We have (cid:0) λ − D R,t (cid:1) − − (cid:0) λ − D ⊕ R,t (cid:1) − = dim S X k =1 X a i ∈A ,b i ∈B a b a b a · · · b k a k = dim S X k =1 X a i ∈A ,b i ∈B ,a = a a b a b a · · · b k a k . (6.66)By (6.57) and (6.63)-(6.66), we have(6.67) (cid:13)(cid:13)(cid:13)(cid:0) λ − D R,t (cid:1) − − (cid:0) λ − D ⊕ R,t (cid:1) − (cid:13)(cid:13)(cid:13) ∞ = O (cid:0) R − κ/ (cid:1) | λ | (dim S +1) t − / . By (6.57), (6.64) and (6.67), we have(6.68) (cid:13)(cid:13)(cid:13)(cid:0) λ − D R,t (cid:1) − p − (cid:0) λ − D ⊕ R,t (cid:1) − p (cid:13)(cid:13)(cid:13) ∞ = O (cid:0) R − κ/ (cid:1) | λ | p (dim S +1) t − / . By (6.66), we have(6.69) Im (cid:16)(cid:0) λ − D R,t (cid:1) − p − (cid:0) λ − D ⊕ R,t (cid:1) − p (cid:17) ⊆ p X k =1 X a ∈A ,b ∈B Im (cid:0) a k b (cid:1) , whose dimension is bounded by p dim (cid:0) E [ − , ,R,T (cid:1) dim (cid:0) Λ • ( T ∗ S ) (cid:1) . Hence(6.70) rk (cid:16)(cid:0) λ − D R,t (cid:1) − p − (cid:0) λ − D ⊕ R,t (cid:1) − p (cid:17) p dim (cid:0) E [ − , ,R,T (cid:1) dim (cid:0) Λ • ( T ∗ S ) (cid:1) . From (6.9), (6.68) and (6.70), we obtain (6.61).
Step 3 . We show that(6.71) (cid:13)(cid:13)(cid:13) P [ − , R,T (cid:0) λ − D ⊕ R,t (cid:1) − p P [ − , R,T − P [ − , R,T (cid:0) λ − e D R,t (cid:1) − p P [ − , R,T (cid:13)(cid:13)(cid:13) = O (cid:0) (cid:1) t − / . Using the identity(6.72) P [ − , R,T (cid:16) D ⊕ R,t − e D R,t (cid:17) P [ − , R,T = − √ t P [ − , R,T ˆ c ( T ) P [ − , R,T and proceeding in the same way as (6.54)-(6.57), we can show that(6.73) (cid:13)(cid:13)(cid:13) P [ − , R,T (cid:0) λ − D ⊕ R,t (cid:1) − p P [ − , R,T − P [ − , R,T (cid:0) λ − e D R,t (cid:1) − p P [ − , R,T (cid:13)(cid:13)(cid:13) ∞ = O (cid:0) (cid:1) t − / . Since the rank of the operator in (6.73) is bounded by dim (cid:0) E [ − , ,R,T (cid:1) dim (cid:0) Λ • ( T ∗ S ) (cid:1) ,from (6.9) and (6.73), we obtain (6.71).By Steps 1-3, we have (6.50). This completes the proof of Lemma 6.4. (cid:3) Recall that the complexes ( C • , • j , ∂ ) with j = 0 , , , were defined by (2.1), (2.6)and (3.20). We denote(6.74) χ ′ ( C • , • j ) = X p =0 dim Z X q =0 ( − p + q ( p + q ) dim C p,qj . NALYTIC TORSION FORMS 66
Set f T j,R = − Z + ∞ (cid:26) ϕ Tr h ( − N TZ N T Z f ′ (cid:0) e D j,R,t (cid:1)i − χ ′ ( Z j , F ) − (cid:0) χ ′ ( C • , • j ) − χ ′ ( Z j , F ) (cid:1) f ′ (cid:16) i √ t (cid:17)(cid:27) dtt . (6.75)By [11, Remark 2.21], Theorem 3.3 and (6.48), f T j,R is well-defined. Proposition 6.5.
For R ≫ , we have (6.76) X j =0 ( − j ( j − / T j,R = X j =0 ( − j ( j − / f T j,R + O (cid:0) R − κ/ (cid:1) . Proof.
Let f T S j,R (resp. f T L j,R ) be as in (6.75) with R + ∞ replaced by R R − κ/ (resp. R + ∞ R − κ/ ).The following identity is obvious,(6.77) f T j,R = f T S j,R + f T L j,R . We fix N ∋ r > dim Z . Let f r : U j,R,t → C be a holomorphic function satisfying(6.78) r ! d r dz r f r ( z ) = f ′ ( z ) , lim z →± i ∞ f r ( z ) = 0 . We further assume that for each bounded connected component V ⊆ U j,R,t , thereexists z ∈ V such that f r ( z ) = 0 . Note that the bounded (resp. unbounded) connectedcomponents of U j,R,t cover the small (resp. large) eigenvalues of √ t (cid:0) d Z j,R , ∗ T − d Z j,R T (cid:1) ,by Theorem 3.3 and (6.47), the total area of the bounded connected components of U j,R,t is bounded by a universal constant. Then, there exists C > such that for any z ∈ U j,R,t , we have(6.79) (cid:12)(cid:12) f r ( z ) (cid:12)(cid:12) Ce −| z | . By (6.47) and (6.78), for R ≫ , we have f ′ (cid:0) D j,R,t (cid:1) = 12 πi Z ∂U j,R,t f r ( λ ) (cid:0) λ − D j,R,t (cid:1) − r − dλ ,f ′ (cid:0) e D j,R,t (cid:1) = 12 πi Z ∂U j,R,t f r ( λ ) (cid:0) λ − e D j,R,t (cid:1) − r − dλ . (6.80)By Lemma 6.4, (6.79) and (6.80), for R ≫ and t > R − κ/ , we have(6.81) (cid:13)(cid:13)(cid:13) f ′ (cid:0) D j,R,t (cid:1) − f ′ (cid:0) e D j,R,t (cid:1)(cid:13)(cid:13)(cid:13) = O (cid:0) R − κ/ (cid:1) t − / . On the other hand, by (0.22), (3.20), (6.4), (6.5), (6.75) and (6.77), we have X j =0 ( − j ( j − / (cid:16) T L j,R − f T L j,R (cid:17) = − X j =0 ( − j ( j − / Z + ∞ R − κ/ ϕ Tr h ( − N TZ N T Z (cid:16) f ′ (cid:0) D j,R,t (cid:1) − f ′ (cid:0) e D j,R,t (cid:1)(cid:17)i dtt . (6.82) NALYTIC TORSION FORMS 67
By (6.81) and (6.82), we have(6.83) X j =0 ( − j ( j − / T L j,R = X j =0 ( − j ( j − / f T L j,R + O (cid:0) R − κ/ (cid:1) . By Theorem 3.3 and (6.3), for R ≫ , we have(6.84) Sp (cid:16)(cid:0) d Z j,R , ∗ T − d Z j,R T (cid:1)(cid:12)(cid:12) E [ − , j,R,T (cid:17) ⊆ i [ − exp( − R κ ) , exp( − R κ )] , where the exponential term comes from e − T = e − R κ in (3.34). As a consequence, wehave(6.85) (cid:13)(cid:13)(cid:13) √ tP [ − , j,R,T (cid:0) d Z j,R , ∗ T − d Z j,R T (cid:1) P [ − , j,R,T (cid:13)(cid:13)(cid:13) ∞ exp( − R κ ) t / . By (6.48) and (6.85), we have Tr h ( − N TZ N T Z f ′ (cid:0) e D j,R,t (cid:1)i − Tr h ( − N TZ N T Z f ′ (cid:0) P [ − , j,R,T ω F j,R P [ − , j,R,T (cid:1)i = O (cid:0) exp( − R κ ) (cid:1) t / . (6.86)Note that f ′ is an even function, by the proof of [11, Prop. 1.3], the function(6.87) R ∋ s Tr h ( − N TZ N T Z f ′ (cid:0) sP [ − , j,R,T ω F j,R P [ − , j,R,T (cid:1)i ∈ Q S is constant. Taking s = 0 , in (6.87) and using the identity f ′ (0) = 1 , we get Tr h ( − N TZ N T Z f ′ (cid:0) P [ − , j,R,T ω F j,R P [ − , j,R,T (cid:1)i = Tr h ( − N TZ N T Z P [ − , j,R,T i . (6.88)By Theorem 3.3, (6.74) and (6.88), we have(6.89) Tr h ( − N TZ N T Z f ′ (cid:0) P [ − , j,R,T ω F j,R P [ − , j,R,T (cid:1)i = 12 χ ′ ( C • , • j ) . By (6.38), (6.75), (6.77), (6.86) and (6.89), we have(6.90) X j =0 ( − j ( j − / f T S j,R = − χ ′ Z R − κ/ (cid:26) f ′ (cid:16) i √ t (cid:17) − (cid:27) dtt + O (cid:0) R − κ/ (cid:1) . From Proposition 6.3, (6.5), (6.77), (6.83) and (6.90), we obtain (6.76). This com-pletes the proof of Proposition 6.5. (cid:3)
We denote G = Ω • ( Y, F ) , which is a complex vector bundle of infinite dimensionover S . We define the connection ∇ G on G in the same way as in (1.73). Let h G be the L -metric on G with respect to h F and g T Y . Let ω G ∈ Ω (cid:0) S, End( G ) (cid:1) be as in (1.79)with ( ∇ F , h F ) replaced by ( ∇ G , h G ) .Let ∇ V • be the canonical flat connection on V • = H • ( Y, F ) (see [11, Def. 2.4]).We identify V • with H • ( Y, F ) ⊆ G via the Hodge theorem. Recall that h V • is the L -metric on V • defined after (3.28). Let ω V • ∈ Ω (cid:0) S, End( V • ) (cid:1) be as in (0.1) with ( ∇ F , h F ) replaced by ( ∇ V , h V ) . Let P V • : G → V • be the orthogonal projection withrespect to h G . By [11, Prop. 3.14], we have(6.91) ω V • = P V • ω G P V • . NALYTIC TORSION FORMS 68
Recall that the flat sub vector bundles V • j = Im (cid:0) τ j : W • j → V • (cid:1) ⊆ V • with j = 1 , were defined by (2.2) and (3.20). The identity (6.91) also holds with V • replaced by V • j . Let(6.92) τ ⊥ j : K • , ⊥ j → V • j be the restriction of τ j : W • j → V • j ⊆ V • defined in (3.20) to K • , ⊥ j ⊆ W • j defined in(3.25), which is bijective. Set(6.93) ω K • , ⊥ j = (cid:0) τ ⊥ j (cid:1) − ◦ ω V • j ◦ τ ⊥ j ∈ Ω (cid:0) S, End( K • , ⊥ j ) (cid:1) . For j = 1 , , let ∇ W • j be the canonical flat connection on W • j = H • ( Z j , F ) (see [11,Def. 2.4]). The sub vector bundle K • j ⊆ W • j is preserved by ∇ W • j . Then ∇ K • j := ∇ W • j (cid:12)(cid:12) K • j is a flat connection on K • j . The Hermitian metric h K • j was constructed in(3.26). Let ω K • j ∈ Ω (cid:0) S, End( K • j ) (cid:1) be as in (0.1) with ( ∇ F , h F ) replaced by ( ∇ K • j , h K • j ) .We identify K • j with F j, ∞ ∩ Ker (cid:0) D Z j, ∞ (cid:1) via the map (3.22). Let P K • j : F j, ∞ → K • j bethe orthogonal projection. By [11, Prop. 3.14], we have(6.94) ω K • j = P K • j ω F j, ∞ P K • j . Recall that C • , • = W • ⊕ W • ⊕ V • = K • ⊕ K • , ⊥ ⊕ K • ⊕ K • , ⊥ ⊕ V • . Set(6.95) ω C • , • = ω K • ⊕ ω K • , ⊥ ⊕ ω K • ⊕ ω K • , ⊥ ⊕ ω V • ∈ Ω (cid:0) S, End( C • , • ) (cid:1) . For j = 1 , , , we may construct ω C • , • j in the same way.Recall that the bijection S j,R,T : C • , • j → E [ − , j,R,T ⊆ Ω • ( Z j,R , F ) was defined in (5.132),(5.212) and (5.217). Lemma 6.6.
For j = 0 , , , and R ≫ , we have (6.96) S − j,R,T ◦ (cid:16) P [ − , j,R,T ω F j,R P [ − , j,R,T (cid:17) ◦ S j,R,T = ω C • , • j + O R,T (cid:0) R − / κ/ (cid:1) , where O R,T (cid:0) · (cid:1) was defined in the paragraph above Theorem 3.3.Proof. We only prove the case j = 0 .We consider σ ∈ C • , • . All the estimates in the proof of Proposition 5.16 holdwith (cid:13)(cid:13) σ (cid:13)(cid:13) Z R replaced by (cid:10) σ, ω F R σ (cid:11) Z R . Hence (5.181) holds with (cid:13)(cid:13) σ (cid:13)(cid:13) Z R replaced by (cid:10) σ, ω F R σ (cid:11) Z R , i.e.,(6.97) D S R,T ( σ ) , ω F R S R,T ( σ ) E Z R = (cid:10) σ, ω C • , • σ (cid:11) R,T + O (cid:0) R − / κ/ (cid:1)(cid:13)(cid:13) σ (cid:13)(cid:13) R,T . From the polarization identity, Proposition 5.16 and (6.97), we obtain (6.96) with j = 0 . This completes the proof of Lemma 6.6. (cid:3) For j = 0 , , , , let ∇ C • , • j be the flat connection on C • , • j induced by ∇ W • , ∇ W • and ∇ V • . Let ω C • , • j R,T ∈ Ω • (cid:0) S, End( C • , • j ) (cid:1) be as in (0.1) with ( ∇ F , h F ) replaced by ( ∇ C • , • j , h C • , • j R,T ) , where the metric h C • , • j R,T was defined in (3.30).
Lemma 6.7.
For j = 0 , , , and R ≫ , we have (6.98) ω C • , • j R,T = ω C • , • j + O R,T (cid:0) R − / κ/ (cid:1) . NALYTIC TORSION FORMS 69
Proof.
We only prove the case j = 0 .Let ω W • j R,T ∈ Ω (cid:0) S, End( W • j ) (cid:1) be as in (0.1) with ( ∇ F , h F ) replaced by ( ∇ W • j , h W • j R,T ) ,where the metric h W • j R,T was defined in (3.28). More precisely, we have(6.99) ω W • j R,T = (cid:0) h W • j R,T (cid:1) − ∇ W • j h W • j R,T . By (6.99) and the paragraph above Lemma 6.7, we have(6.100) ω C • , • R,T = ω W • R,T ⊕ ω W • R,T ⊕ ω V • . Comparing (6.95) with (6.100), it remains to show that(6.101) ω W • j R,T = ω K • j ⊕ ω K • , ⊥ j + O R,T (cid:0) R − / κ/ (cid:1) , for j = 1 , . Let P j : W • j → K • j and P ⊥ j : W • j → K • , ⊥ j be the projections with respect to thedecomposition W • j = K • j ⊕ K • , ⊥ j . Set(6.102) ∇ W • j , ⊕ = P j ∇ W • j P j + P ⊥ j ∇ W • j P ⊥ j . Since K • j ⊆ W • j is a flat sub vector bundle, we have(6.103) ∇ W • j − ∇ W • j , ⊕ = P j ∇ W • j P ⊥ j ∈ Ω (cid:16) S, Hom (cid:0) K • , ⊥ j , K • j (cid:1)(cid:17) . We denote by (cid:13)(cid:13) · (cid:13)(cid:13) R,T the operator norm on
Hom (cid:0) K • , ⊥ j , K • j (cid:1) with respect to h W • j R,T . By(3.28), we have(6.104) (cid:13)(cid:13) · (cid:13)(cid:13) R,T = R − / T / (cid:13)(cid:13) · (cid:13)(cid:13) , = R − / κ/ (cid:13)(cid:13) · (cid:13)(cid:13) , . By (6.103) and (6.104), we have(6.105) ∇ W • j − ∇ W • j , ⊕ = O R,T (cid:0) R − / κ/ (cid:1) . By (3.28) and (6.102), we have(6.106) (cid:0) h W • j R,T (cid:1) − ∇ W • j , ⊕ h W • j R,T = ω K • j ⊕ ω K • , ⊥ j . From (6.99), (6.105) and (6.106), we obtain (6.101). This completes the proof ofLemma 6.7. (cid:3)
Proof of Theorem 3.4.
Applying Remark 1.7 to the map S j,R,T : C • , • j → E [ − , j,R,T andusing Theorem 3.3, Lemmas 6.6, 6.7, (3.36) and (6.75), we get(6.107) f T j,R − c T j,R = O (cid:0) R − / κ/ (cid:1) . From Proposition 6.5 and (6.107), we obtain (3.38). This completes the proof ofTheorem 3.4. (cid:3)
7. T
ORSION FORMS ASSOCIATED WITH THE M AYER -V IETORIS EXACT SEQUENCE
The purpose of this section is to prove Theorem 3.5. This section is organized asfollows. In § § T = R κ , where κ ∈ ]0 , / is a fixed constant. For ease of notations, we willsystematically omit a parameter ( R or T ) as long as there is no confusion. NALYTIC TORSION FORMS 70
A filtration of the Mayer-Vietoris exact sequence.
Recall that W • , W • , V • , V • and V • were defined by (2.2) and (3.20). Recall that W • ⊆ W • ⊕ W • was defined by(2.12). For convenience, we denote V • quot = V • / ( V • + V • ) .For k ∈ N , we construct a truncation of the exact sequence (0.22) as follows,(7.1) · · · / / H k ( Z, F ) / / W k ⊕ W k / / V k / / V k quot / / . The truncations of (0.22) at degree k − and k fit into the following commutativediagram with exact rows and columns, · · · / / V k − / / (cid:15) (cid:15) · · · / / H k ( Z, F ) / / (cid:15) (cid:15) W k ⊕ W k / / (cid:15) (cid:15) V k / / (cid:15) (cid:15) V k quot / / (cid:15) (cid:15) / / W k / / W k ⊕ W k / / V k / / V k quot / / . (7.2)We equip H • ( Z, F ) , W • , W • and V • in (7.2) with Hermitian metrics induced by (cid:13)(cid:13) · (cid:13)(cid:13) Z j,R ( j = 0 , , , ) via the identification (3.16). We equip W • in (7.2) with theHermitian metric induced by h W • R,T ⊕ h W • R,T , which were defined in (3.28), via the em-bedding W • ֒ → W • ⊕ W • . Set(7.3) a R,T = R Z − χ ( s ) e T f T ( s ) − T ds , where χ : R → R was defined in (5.42). We equip V • quot in (7.2) with the quotientmetric of a − R,T h V • R,T , where h V • R,T was defined in (3.29). Let T k hor ,R,T be the torsion formassociated with the third row in (7.2). Let T k vert ,R,T be the torsion form associated withthe (unique) non trivial column in (7.2). Proposition 7.1.
The following identity holds in Q S /Q S, , (7.4) T H ,R,T = n X k =0 ( − k T k hor ,R,T − n X k =1 ( − k T k vert ,R,T . Proof.
Let T H ,R,T ( k ) be the torsion form associated with (7.1). In particular, T H ,R,T ( −
1) =0 and T H ,R,T (dim Z ) = T H ,R,T . Applying [11, Thm. A1.4] to (7.2), we get(7.5) T H ,R,T ( k − − T H ,R,T ( k ) + ( − k T k hor ,R,T − ( − k T k vert ,R,T ∈ Q S, . Taking the sum of (7.5) for k = 0 , , · · · , dim Z , we obtain (7.4). This completes theproof of Proposition 7.1. (cid:3) Estimating T k vert ,R,T and T k hor ,R,T . For j = 0 , , , , under the identification (5.202),we view H • ( C • , • j , ∂ ) as a vector subspace of C • , • j . Let(7.6) P Hj : C • , • j → H • ( C • , • j , ∂ ) be the orthogonal projection with respect to h C • , • j R,T (see (3.30)). Note that K • j ⊆ H ( C • , • j , ∂ ) , by (3.28) and (3.30), P Hj is independent of R, T . Let ω H • ( C • , • j ,∂ ) (resp. NALYTIC TORSION FORMS 71 ω H • ( C • , • j ,∂ ) R,T ) be the -form on S with values in End( C • , • j ) induced by ω C • , • j in (6.95)(resp. ω C • , • j R,T in (6.98)) via the projection P Hj . More precisely, we have ω H • ( C • , • j ,∂ ) = P Hj ω C • , • j P Hj ∈ Ω (cid:0) S, End (cid:0) H • ( C • , • j , ∂ ) (cid:1)(cid:1) ,ω H • ( C • , • j ,∂ ) R,T = P Hj ω C • , • j R,T P Hj ∈ Ω (cid:0) S, End (cid:0) H • ( C • , • j , ∂ ) (cid:1)(cid:1) . (7.7)By (2.13), we have H • ( C • , • j , ∂ ) = H ( C • , • j , ∂ ) = W • j for j = 1 , . Then we have(7.8) ω H • ( C • , • j ,∂ ) R,T = ω W • j R,T for j = 1 , , where ω W • j R,T was defined in (6.99).For j = 0 , , , , let h H • ( Z j ,F ) R,T be the Hermitian metric on H • ( Z j , F ) = H • ( Z j,R , F ) induced by (cid:13)(cid:13) · (cid:13)(cid:13) Z j,R via the identification (3.16). Let ∇ H • ( Z j ,F ) be the canonical flatconnection on H • ( Z j , F ) (see [11, Def. 2.4]). Let ω H • ( Z j ,F ) R,T ∈ Ω (cid:0) S, End (cid:0) H • ( Z j , F ) (cid:1)(cid:1) be as in (0.1) with ( ∇ F , h F ) replaced by ( ∇ H • ( Z j ,F ) , h H • ( Z j ,F ) R,T ) .Let (cid:2) S Hj,R,T (cid:3) T : H • ( C • , • , ∂ ) → H • ( Z j , F ) be the map defined by (5.51), (5.55),(5.212) and (5.217). Recall that the notation O R,T (cid:0) · (cid:1) was defined in the paragraphabove Theorem 3.3. Lemma 7.2.
For R ≫ , we have (cid:16)(cid:2) S Hj,R,T (cid:3) T (cid:17) − ◦ ω H • ( Z j ,F ) R,T ◦ (cid:2) S Hj,R,T (cid:3) T = ω H • ( C • , • j ,∂ ) + O R,T (cid:0) R − / κ/ (cid:1) ,ω H • ( C • , • j ,∂ ) R,T = ω H • ( C • , • j ,∂ ) + O R,T (cid:0) R − / κ/ (cid:1) . (7.9) Proof.
Recall that P j,R,T : F j,R → Ker (cid:0) D Z j,R T (cid:1) was defined above Proposition 5.2, above(5.212) and above (5.217). By [11, Prop. 3.14], the following identity holds underthe identification (3.16),(7.10) ω H • ( Z j ,F ) R,T = P j,R,T ω F j,R P j,R,T , where ω F j,R was defined in § (cid:16) S Hj,R,T (cid:17) − ◦ (cid:16) P j,R,T ω F j,R P j,R,T (cid:17) ◦ S Hj,R,T = P Hj ω C • , • j P Hj + O R,T (cid:0) R − / κ/ (cid:1) . From (7.7), (7.10) and (7.11), we obtain the first identity in (7.9). The second identityin (7.9) is a direct consequence of Lemma 6.7 and (7.7). This completes the proof ofLemma 7.2. (cid:3)
Proposition 7.3.
For R ≫ , we have (7.12) T k vert ,R,T = O (cid:0) R − / κ/ (cid:1) . NALYTIC TORSION FORMS 72
Proof.
Recall that H ( C • ,k , ∂ ) = W k and H ( C • ,k − , ∂ ) = V k − . First we show that thefollowing diagram commutes,(7.13) V k − a R,T Id (cid:15) (cid:15) / / V k − ⊕ W k (cid:2) S HR,T (cid:3) T (cid:15) (cid:15) / / W k (cid:15) (cid:15) V k − δ / / H k ( Z, F ) α / / W k , where a R,T was defined by (7.3), the first row consists of canonical injection andprojection, the second row is the (unique) non trivial column in (7.2). We remark that(7.13) is not a commutative diagram of flat complex vector bundles over S .Let η : [ − R, R ] → R be a smooth function such that(7.14) η (cid:12)(cid:12) [ − R, − R/ = 0 , η (cid:12)(cid:12) [ R/ ,R ] = 1 . We will view η as a function on IY R . Let σ ∈ H k − ( Y, F ) = V k − . Let σ ∈ V k − be theimage of σ . Let ω ∈ Ω • ( Z R , F ) such that(7.15) ω (cid:12)(cid:12) Z , = 0 , ω (cid:12)(cid:12) Z , = 0 , ω (cid:12)(cid:12) IY R = dη ∧ σ . Then we have(7.16) δ ( σ ) = [ ω ] ∈ H k ( Z R , F ) = H k ( Z, F ) . Let σ ′ ∈ (cid:0) V k − + V k − (cid:1) ⊥ ⊆ V k − . By (5.43)-(5.45), we have(7.17) I R,T ( σ ′ ) (cid:12)(cid:12) Z , = 0 , I R,T ( σ ′ ) (cid:12)(cid:12) Z , = 0 , I R,T ( σ ′ ) (cid:12)(cid:12) IY R = χ e T f T − T du ∧ σ ′ . Let σ ′ ∈ V k − be the image of σ ′ . By (5.51), (5.55), (5.212) and (5.217), we have(7.18) (cid:2) S HR,T (cid:3) T ( σ ′ ) = (cid:2) e T f T I R,T ( σ ′ ) (cid:3) ∈ H k ( Z R , F ) = H k ( Z, F ) . By (7.3) and (7.14)-(7.18), we have(7.19) (cid:2) S HR,T (cid:3) T ( σ ) = (cid:16) Z R − R χ ( u ) e T f T ( u ) − T du (cid:17) δ ( σ ) = a R,T δ ( σ ) . Hence the left square in (7.13) commutes.Let ( ω , ω , ˆ ω ) ∈ H k abs ( Z , ∞ , F ) . Its image in W k via the identification (5.47) isgiven by (cid:16)(cid:2) ω (cid:12)(cid:12) Z , (cid:3) , (cid:2) ω (cid:12)(cid:12) Z , (cid:3)(cid:17) . By (5.51), (5.55), (5.212) and (5.217), we have (cid:2) S HR,T (cid:3) T (cid:16)(cid:2) ω (cid:12)(cid:12) Z , (cid:3) , (cid:2) ω (cid:12)(cid:12) Z , (cid:3)(cid:17) = (cid:2) e T f T F R,T ( ω , ω , ˆ ω ) (cid:3) ∈ H k ( Z R , F ) = H k ( Z, F ) . (7.20)By (5.19)-(5.21), we have(7.21) F R,T ( ω , ω , ˆ ω ) (cid:12)(cid:12) Z , = ω (cid:12)(cid:12) Z , , F R,T ( ω , ω , ˆ ω ) (cid:12)(cid:12) Z , = ω (cid:12)(cid:12) Z , . On the other hand, for [ ω ] ∈ H k ( Z R , F ) = H k ( Z, F ) , we have(7.22) α ([ ω ]) = (cid:16)(cid:2) ω (cid:12)(cid:12) Z , (cid:3) , (cid:2) ω (cid:12)(cid:12) Z , (cid:3)(cid:17) ∈ W k ⊆ W k ⊕ W k . NALYTIC TORSION FORMS 73
By (7.20)-(7.22), we have(7.23) α ◦ (cid:2) S HR,T (cid:3) T (cid:16)(cid:2) ω (cid:12)(cid:12) Z , (cid:3) , (cid:2) ω (cid:12)(cid:12) Z , (cid:3)(cid:17) = (cid:16)(cid:2) ω (cid:12)(cid:12) Z , (cid:3) , (cid:2) ω (cid:12)(cid:12) Z , (cid:3)(cid:17) . Hence the right square in (7.13) commutes.We equip H k ( Z, F ) = H k ( Z R , F ) in (7.13) with the Hermitian metric induced by (cid:13)(cid:13) · (cid:13)(cid:13) Z R via the identification (3.16). We equip W k in (7.13) with the Hermitian metricinduced by h W • R,T ⊕ h W • R,T via the embedding W k ֒ → W k ⊕ W k . We equip V • quot in thefirst row of (7.13) with the quotient metric of h V • R,T . We equip V • quot in the second rowof (7.13) with the quotient metric of a − R,T h V • R,T . Then the torsion form of the first rowin (7.13) vanishes, and the torsion form of the second row in (7.13) equals T k vert ,R,T .Applying Proposition 1.6 to (7.13) and using Corollary 5.17 and Lemma 7.2, we obtain(7.12). This completes the proof of Proposition 7.3. (cid:3) Proposition 7.4.
For R ≫ , the following identity holds in Q S /Q S, , (7.24) T k hor ,R,T = c T k H ,R,T + O (cid:0) R − / κ/ (cid:1) with T k hor ,R,T being as in (7.4) and c T k H ,R,T being as in (3.40) .Proof. We denote b R,T = π / RT − / e T . By (7.3), there exists a > such that(7.25) a R,T = b R,T (cid:16) O (cid:0) e − aT (cid:1)(cid:17) . Let p : V k → V k quot be the canonical projection. The following commutative diagramis obvious,(7.26) W k / / Id (cid:15) (cid:15) W k ⊕ W k / / Id (cid:15) (cid:15) V k b − R,T p / / Id (cid:15) (cid:15) V k quot b R,T Id (cid:15) (cid:15) W k / / W k ⊕ W k / / V k p / / V k quot . We equip W k in (7.26) with the Hermitian metric induced by h W k R,T ⊕ W W k R,T via theinclusion W k ⊆ W k ⊕ W k . We equip W k ⊕ W k in the first row of (7.26) with theHermitian metric h W k R,T ⊕ h W k R,T . We equip W k ⊕ W k = H k ( Z ,R , F ) ⊕ H k ( Z ,R , F ) in thesecond row of (7.26) with the Hermitian metric induced by (cid:13)(cid:13) · (cid:13)(cid:13) Z j,R ( j = 1 , ) via theidentification (3.16). We equip V k in the first row of (7.26) with the Hermitian metric h V k R,T . We equip V k in the second row of (7.26) with the Hermitian metric induced by (cid:13)(cid:13) · (cid:13)(cid:13) IY R via the identification (3.16). We equip V • quot in the first (resp. second) row of(7.26) with the quotient metric of h V • R,T (resp. a − R,T h V • R,T ). The torsion form of the firstrow of (7.26) is given by c T k H ,R,T , and the torsion form of the second row of (7.26) isgiven by T k hor ,R,T . For j = 1 , , , , let T j ∈ Q S be the torsion form of the j -th columnin (7.26). Applying [11, Thm. A1.4] to (7.26), we get(7.27) c T k H ,R,T − T k hor ,R,T − T + T − T + T ∈ Q S, . NALYTIC TORSION FORMS 74
Since the first vertical map is isometric, we have(7.28) T = 0 . By Corollary 1.8, Corollary 5.17, Remark 5.18 and Lemma 7.2, we have(7.29) T = O (cid:0) R − / κ/ (cid:1) , T = O (cid:0) R − / κ/ (cid:1) . By Corollary 1.8 and (7.25), we have(7.30) T = O (cid:0) e − aT/ (cid:1) . From (7.27)-(7.30), we obtain (7.24). This completes the proof of Proposition 7.4. (cid:3)
Proof of Theorem 3.5.
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