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O. SB is a basis for a metrizable uniform structure --=-----:-=::-:-,b, m
b,mu(coo(E)). Let (COO (E)
U
_
,b,mu) be the completion. Then
we get properties absolutely parallel to the assertions 2.5 - 2.9,
jEJ
b,m comp (O define b,m,aU(COO(E)) , the completion
b,m,an(COO(E)) =
L
b,m,acomp(
+ big -
g2lg)
m-1
+
L Pr+1,r+1 r=O
=
Pm (b,il g1 - glg1> b,jlg - g2Ig)·
Therefore, we established (U~). Denote by b,mU(M) the corresponding uniform structure. It is metrizable since it is 0 trivially Hausdorff and {V1/ n }n::::no is a countable basis. We see, the proof of 2.16 is parallel to that of 1.19, replacing b,mll by IIp,r.
104
Relative Index Theory, Determinants and Torsion
Denote ~M = (M, b,mU(M)) and by b,m M the completion. It has been proven in [202] that b,m M still consists of positive definite elements, which are of class
em.
Remark 2.17 We endowed by our procedure M with a canonical intrinsic em-topology without choice of a cover or a special 9 to define the em-distance. According to the definition of the uniform topology, for 9 E b,m M
o
is a neighbourhood basis in this topology. Proposition 2.18 The space b,m M is locally contractible.
For a proof we refer to [27], [35] and to proposition 1.9 of chapter VII. 0 Corollary 2.19 In b,m M components and arc components coincide. 0
Set
b,mU(g) = {g'
E
b,m M I big - g'lgl
~ + 1. MP,r(I, B k ) has a representation as a topological sum
MP,r(I, B k ) =
Then
L up,r(gi). iEI
o We finish at this point the example of uniform structures of Riemannian metrics and turn to our last class of examples, uniform structures of metric spaces. We start with the GromovHausdorff uniform structure. Let Z = (Z, d z ) be a metric space, X, Y c Z subsets, E > 0, define Ue(X) := {z E Z I dist( z, X) < E}, analogously Ue (Y). Then the Hausdorff distance dH(X, Y) = dfI(X, Y) is defined as
drr(X, Y)
:= inf{E
I X c Ue(Y), Y c Ue(X)}.
If there is no such E then we set dfI(X, Y) := 00. Then dfI factorized by d(·,·) = 0 is an almost metric on all closed subsets, i.e. it has values in [O,ooJ but satisfies all other conditions of a metric. If Z is compact then dfI is a metric on the the set of all closed subsets. A metric space (X, d) is called proper if the closed balls Be(x) are compact for all x E X, E > O. This
111
Non-linear Sobolev Structures
implies that X is separable, complete and locally compact. In the sequel we restrict our attention to proper metric spaces. Let X = (X,d x ), Y = (Y,d y ) be metric spaces, Xu Y their disjoint union. A metric d on X U Y is called admissible if d restricts to dx and dy , respectively. The Gromov-Hausdorff distance dGH(X, Y) is defined as
dGH(X, Y)
:=
inf{dH(X, Y) I d is admissible on XU Y}.
Note that the Gromov-Hausdorff distance can be infinity. Originally Gromov defined dGH as
dGH(X, Y) := inf{d~(i(X),j(Y))i: X
-t
Z,j : Y
-t
Z
isometric embeddings into a metric space Z}. It is a well known fact and can simply be proved that both definitions coincide. Lemma 2.43 If X and Yare compact metric spaces and dGH(X, Y) = 0 then X and Yare isometric.
This follows from the definition and an Arzela-Ascoli argument.
o Remark 2.44 The class of all metric spaces, even only up to isometry, is not a set. But the class of isometry classes of proper metric spaces is, so we only consider proper metric spaces in this chapter. A proper metric space X can be covered by a countable number of compact metric balls of fixed radius. Each such ball is isometric to a subset of Loo([O, 1]) and we obtain X after identification of the overlappings, i.e. we can understand X as a subset of (IT:l Loo([O, 1]))2.
o Denote by 9J1 the set of all isometry classes [X] of proper metric spaces X and 9J1GH = 9J1/ rv where [X] rv [Y] if dGH([X], [Y]) =
O.
112
Relative Index Theory, Determinants and Torsion
Proposition 2.45 dCH defines an almost metric on 9J1, i. e. it is a metric with values in [0,00]. D
We write in the sequel X = [X]CH if there cannot arise any confusion. Now we define the uniform structure. Let 0 and set v" := {(X, Y) E 9J1~H I dcH(X, Y) < o is a basis for a metrizable uniform structure UCH (9J1). Proof. Q3 is defined by a local metric. Hence it satisfies all D desired conditions. Let 9J1CH be the completion of 9J1CH with respect to UCH and denote the metric in 9J1 by dCH .
Lemma 2.47 9J1CH = 9J1CH as sets and dCH and d CH are locally equivalent, i. e. for any X E 9J1CH = 9J1CH there exist equivalent neighbourhood bases by metric balls. Proof. Cauchy sequences with respect to dCH and UcH(9J1 CH ) coincide. But according to [56], proposition 10.1.7, p. 277, any Cauchy sequence in 9J1 with respect to dCH converges in 9J1CH . This proposition is formulated there for compact metric spaces, but the proof does not use compactness at any stage. The assertion concerning the local equivalence of dCH and dCH follows immediately from the formulae (2.4)-(2.6) in [68], 11.2.7, p. 117. We refer also to 2.3, 2.4. D Let X E 9J1CH = 9J1CH and denote by comp(X) and arccomp(X) the component and arc component of X in 9J1CH , respecti vely. A key role for all what follows is played by
Proposition 2.48 9J1CH = 9J1CH is locally arcwise connected.
Non-linear Sobolev Structures
113
We refer to [33] for the proof.
D
Corollary 2.49 In m GH components and arc components coincide. Moreover, each component is open and m GH = m GH is the topological sum of its components,
m= Lcomp(X
i ).
iEI
D
Proposition 2.50 Let X E m GH = m GH . Then comp(X) zs given by comp(X) = {Y E mGH I dGH(X, Y) < oo}.
Proof. Let Y E comp(X) = arccomp(X). Then there exists an arc between X and Y. For given c > 0 this can be covered by a finite number, say r, of c-balls. Hence dGH(X, Y) ~ 2rc < 00. If dGH(X, Y) = c < 00 then we can construct an arc from X to Y as given in the proof of proposition 2.48. D Hence we have a quite natural splitting of mGH = mGH into its components = arc components and a canonical topology and convergence inside each component. This is not - as usual until now - the pointed convergence of all metric balls but uniform convergence. We discuss this later. Any complete Riemannian manifold determines a unique component. First we give another characterization of components. We call a map : X --+ Y metrically semilinear if it satisfies the following two conditions.
1. It is uniformly metrically proper, i.e. for each R > 0 there is an S > 0 such that the inverse image under of a set of diameter R is a set of diameter at most S. 2. There exists a constant Gil> 2': 0 such that for all
Xl, X2
E X
114
Relative Index Theory, Determinants and Torsion
Two metric spaces X and Yare called metrically semilinearly equivalent if there exist metrically semilinear maps : X - t Y, W : Y - t X and constants Dx and D y , such that for all x E X and y E Y d(x, wx) :S D x ,
d(wy, y) :S D y
.
Proposition 2.51 Y E comp(X), i.e. dCH(X, Y) < 00 if and only if X and Yare metrically semilinearly equivalent.
o We refer to [33] for the proof. The other class of uniform structures of particular meaning are Lipschitz uniform structures. A map : X ----t X is called Lipschitz if there is a constant C> 0 such that
for all Xl, X2 EX. Restricting now to Lipschitz maps, we create a local metric which takes into account the measure of expansivity. Define for a Lipschitz map : X - t Y dil :=
Xl~~~X
d( x I , X2) d(XI' X2) .
xli' x 2
Set dL(X, Y)
.- inf{max{O,logdil } +max{O,logdil w} + sup d(Wx, x) + sup d(wy, y) xEX
I : X
yEY -t
Y, W : Y
-t
X Lipschitz maps},
if { ... } =1= 0 and inf{ ... } is < 00 and set ddX, Y) = 00 in the other case. Then d L ~ 0, symmetric and dL(X, Y) = 0 if X and Yare isometric. Set ML = Mj "', where X '" Y if ddX, Y) = O. That", is in fact an equivalence relation follows from 2.52 below. Let is > 0 and define V" = {(X, Y) E MildL(X, Y) < oo}.
Non-linear Sobolev Structures
115
Proposition 2.52 ~ = {V8}o>o is a basis for a metrizable uniform structure tiL(M L). We refer to [33] for the proof.
0
Denote by 9J1~L the completion of 9J1L with respect to tiL, We come back to the topological properties of 9J1L below. Before doing this we introduce still another important Lipschitz uniform structure. Define for X, Y E 9J1
dL,top(X, Y)
:=
inf{max{O, log dil }
I : X if { ... }
i- 0 and = 00
--t
+ max{O, log dil
-l} Y is a hi-Lipschitz homeomorphism}
in the other case.
Then dL,top(X, Y) ~ 0, symmetric and dL,top(X, Y) = 0 if X and Yare isometric. Set 9J1L ,top := 9J1/ ~ where X "" Y if dL,top(X, Y) = O. Then dL,top is an almost metric on 9J1L ,topo
Remark 2.53 Gromov defined in [39] dL,top(X, Y)
:=
inf{llogdil 1 + Ilogdil -11 I : X ---t Y is a bi-Lipschitz homeomorphism. }
But this does not work since this dL,top does not satisfy the triangle inequality. log(x· y) = logx + logy, log is monotone increasing, but Zl ~ Z2 + Z3 does not imply IZ11 ~ IZ21 + IZ31· There are explicit counterexamples to the triangle inequality of the local I log dil (.) I-metric. 0 Set for 8 > 0
V8 = {(X, Y) Proposition 2.54 ~ form structure tiL, top'
E
9J11,topldL,top(X, Y) < 8}.
= {Vo}o>o is a basis for a metrizable uni0
Denote by 9J1~~;;P the completion of 9J1 L,top with respect to tiL,top.
116
Relative Index Theory, Determinants and Torsion
Proposition 2.55 9Jh,top is complete with respect to llL,top, i. e. 9JtL,top = 9Jt L,top.
o We refer to [33] for the proof. Next we return to 9Jt L. Above we discussed 9Jt L,top since the pro?fs for 9JtL are modelled by the proofs for 9Jt L,top- The next result is valid only for noncom pact proper metric spaces (= classes of spaces, exactly spoken). Such spaces have infinite diameter. The restriction to noncom pact proper spaces is no restriction for us since we have in mind them only. Denote by 9Jt L(nc) the (classes of) noncompact proper metric spaces. We proved in [33] Proposition 2.56 9Jt L(nc) is complete with respect to llL(9Jt L) restricted to 9Jtdnc). 0 Remark 2.57 If we restrict to compact metric spaces then by Arzela-Ascoli arguments dL,top(X, Y) = 0 or ddX, Y) = 0 al-
ways imply X isometric to Y. Hence the factorization by or
rv
L
rv L,top would not be necessary. For noncom pact proper metric
spaces the corresponding question is open (at least for us). If a sequence (~+s
op,r (J*T N) 0, 8·D:::;
8N < rinj(N)/2, 1 :::; p < 00, 118 = {(J, g) E coo,m(M, N) x coo,m(M, N) I:3Y E O~(J*TN) such that 9 = gy = exp Y and IYlp,r < 8}. Proposition 3.1 ~
= {V,,}o 0 there exist j E coo,r(M, N) and a Sobolev vector field X along j, X E D,p,r(j*T N), IXlp,r < E, such that f(x) = (exp X)(x) = exp!(x) X/(x) = (exp! X
0
j)(x).
(3.3)
In particular coo,r(M, N) is dense in op,r(M, N). A special case is given if we restrict to diffeomorphisms. Let (Mn, g), (Nn, h)
123
Non-linear Sobolev Structures
as in the hypothesis of theorem 3.2. Define
1)p,r(M, N) = {f E np,r(M, N) I f is a diffeomorphism and there exists constants c, C > 0 such that c:::; inf Idflx :::; sup Idflx :::; C}. (3.4) xEM
x
(3.4) automatically implies the existence of constants such that C1 :::; inf Idf-1lx :::; sup Idf-1lx :::; C 1. x
C1,
C1 > 0
x
(3.5)
In fact, for diffeomorphisms (3.4) and (3.5) are equivalent. Moreover, (3.4) is an open condition in np,r(M, N). Hence we have Theorem 3.3 Suppose the hypothesis of 3.2. Then each component of1)p,r(M, N) is a cl+k-r -Banach manifold and for p = 2 it is a Hilbert manifold. 0
Corollary 3.4 Suppose for M = N the hypothesis of 3.2. Then each component of 1)p,r(M) is a cl+k-r -Banach manifold and for p = 2 it is a Hilbert manifold. 0
In the sequel, we need still a relative version of these manifolds of maps. Suppose (Mn,g), (Nn,h), k, r, p as in 3.2 and that there exist compact submanifolds KM C M, KN C N such that there exists f E np,r(M, N) with the following properties. a) fIM\KM maps M \ KM diffeomorphic onto N \ KN and b) there exist constants c, C > 0 such that
Then, automatically,
124
Relative Index Theory, Determinants and Torsion
We denote for fixed K M, KN the subset c D,p,r(M, N) of these I by V~~~(M, N). Clearly, V~~~(M, N) depends on the choice of KM,KN . Finally, we need this construction still for Riemannian vector bundles. Let (Ei' hi, \7 hi ) ----t (Mr,9i), i = 1,2 be Riemannian vector bundles satisfying (1), (Bk(Mi , 9i)), (Bk(Ei , hi, \7i)), k 2: r > ~ + 1. If we endow the total spaces Ei with the Kaluza-Klein metric 9E(X, Y) = h(XV, yV) + 9M(-rr*X, 71"*Y) , Xv, yv vertical components, then E 1, E2 are again manifolds with bounded geometry (cf. [30]) and D,p,r(E1, E 2) is well defined. If we restrict to bundle maps (fE, 1M = 71" 0 IE 071"-1 then we obtain a subset D,~b(E1' E 2 ) c D,p,r(E1, E2)' Quite analogously to above we define V~{ (E1' E 2 ) c D,~: (E1' E 2 ) if the bundles are isomorphic and V~brrel(E1' E 2 ) if they are isomorphic over M \ KM and N \ K N . Here we require (3.3) and (3.5) both for IE and 1M' We apply this notations in the next section.
Uniform structures of manifolds and Clifford bundles 4
We introduce in chapters IV - VI relative index theory, relative eta and zeta functions, relative determinants and relative analytic torsion. The whole approach relies on the following construction. We endow the set of isometry classes of Clifford bundles (of bounded geometry) with a metrizable uniform structure, define generalized components gen comp( E) (= set of Clifford bundles E' with finite Sobolev distance from a given E), associate the corresponding generalized Dirac operators D, D' and make all constructions for the pair D, D', where D' is running through gen comp( E). The first step for doing this is the introduction of the corresponding uniform structure(s). This is the content of this section. The applications will be performed in chapters IV - VI. Denote by fJItn(ml, I, B k ) the set of isometry classes of n-dimensional Riemannian manifolds (Mn, 9) satisfying the con-
Non-linear Sobolev Structures
125
ditions (1) and (Bk)' We defined for (Mf,gl), (M:;,g2) E mn(mf, I, B k ) and k > ~ + 1 the diffeomorphisms DP,r(M1, M 2) and for appropriate compact submanifolds Ki C M i , M1 \ K1 9:: M2 \ K2 the maps V:~~(M1' M2) == D~~~(M1' M2; Kl, K 2) c np,r(M1, M 2) as at the end of section 3. Recall bldfl = sup Idflx. x The elements f of D~:Z, np,r(M1, M 2) are not smooth. For k ~ r > ~ + 2 they are C 2 . Hence f* 9 is a C 1 metric. This would cause some troubles if we would consider in the sequel only classical derivatives which would disappear if we work with distributional derivatives. Another way to work with the nonsmoothness of f*g is to work with smooth approximations of f. We decide to go this way and define cr,p,r(M1, M 2) = U E np,r(M1, M 2)lf E cr+1(M1' M 2) and bl\7idfl < 00, i = 1, ... , r}. Completing the uniform structure below, we end up with fs E D~~~, np,r, i.e. the restriction at the beginning to Ck fs implies at the end no further restriction. We restrict in the sequel to k -> r > !!:p + 2. Further we remark that the conditions c :::; blf*1 :::; C and C1 :::; blf;11 :::; C1 are equivalent: blf;11 ~ = C1 follows from blf*f;11 = 1 and blf*1 :::; C and blf;11 :::; C 1(c, C) follows from elementary matrix calculus. If f* is the induced map between twofold covariant tensors then 1 2 2 C2 = c :::; blf*1 :::; C = C2, similarly C3 :::; blf*- 1 :::; C 3 . Under these conditions, I\7 i f* I :::; d, 1 :::; i :::; v implies I\7 i f* I :::; d1 , 1 :::; i :::; v, and l\7if*- 1 1 :::; d2, 1 :::; i :::; v, where d1, d2 are continuous functions in c, C, d. All this follows from f;1 f* = id*, f*-1 f* = id* and 0 = \7id* = \7id*. Consider now pairs (Mf,g1),(M:;,g2) E mn(mf,I,Bk ) with this property: There exist compact submanifolds Kf C Mf, K'2 C M:; and an f E D~~~(M1' M 2, Kl, K2)' For such pairs
b
Relative Index Theory, Determinants and Torsion
126
define
d1j{diff,rel(M1 , 91)' (M2,92)) := inf { max{O,logbldfl} + max{O, log bldhl} + sup dist(x, hfx) XEMl + sup dist(y, hfy) + sup IV'idfl + sup IV'idhl yEM2 "'EMl "' EM2 l~i~r l$i~r +IUIMl\Kl)*92 - 91IMl\KlI9l,p,r If E r,p,r(M1 , M 2), hE r,p,r(M1 , M 2)
c
c
and for some Kl C M 1 , K2 C M2 holds flMl\Kl E ,])p,r(M1 \ KI, M2 \ K 2) and hM2\K2 =
UI Ml\Kl)-I},
if { ... } f= 0 and inf{ ... } < 00. d~4iff,rel((Ml' 91)' (M2' 92)) = 00. Set
v"
(4.1) In the other case set
= {((MI, 91), (M2' 92)) E (mn(mf,I, Bk))21 ~:diff,rel(Ml' 91), (M2' 92)) < 5}.
Proposition 4.1 Suppose r > ~ + 2. Then Q3 = {V,,},,>o is a basis for a metrizable uniform structure on mn(mf, I, B k )/ rv, where (Ml' 91) rv (M2,92) if d~:diff,rel(Ml' 91), (M2' 92)) = 0. Proof. We have to verify (U~) and (U~), i.e. the symmetry and transitivity of the basis (not of dt~ifJ,rel) and start with (U~). For this it is sufficient that
(4.2) implies such that
(4.4)
127
Non-linear Sobolev Structures
We consider the single numbers in the set (4.1). The first number, sum of 4 terms is symmetric in f and h. The second number sup l\7 i dfl + sup l\7 i dhl is symmetric in f and h too. Suppose yE M 2 l$i$r
yEMl l$i$r
1
(fIMl\Kl)*g2 - glIMl\K1191,P,T
== {
J
1(fIMl\Kl)*g2 -
Ml\Kl T-l
+
L
(\7 91 )i(\7 91
11
-
gll~l'x
\7r92)1~1,xdvolx(gl)
i=O
r; 1
< 61 . (4.5)
Now we have to estimate
We omit in the notation 1Mi\Ki since in the remaining part of the proof we restrict to this. Then
Now
Hence, in the case r = 0,
Consider now the case r
= 1. Then
1\7(f*-l(gl - j*g2))ly =
1\7(j*-l)(gl - j*g2) +j*-l\7(gl - j*g2)ly ::; bl\7(f*-l)l· Igl - j*g21 +blj*- l l . 1\7(gl - j*g2)1· (4.8)
128
Relative Index Theory, Determinants and Torsion
We briefly discuss the case r = 2, to indicate the general rule.
1\7[\7(j*-I(gl - j*g2))lI92,Y = 1\7[\7(j*-I)(gl - j*g2) + j*-I\7(gl - j*g2)lI92,Y = 1\7 2 (j*-I)(gl - j*g2) + \7(j*-I)\7(gl - j*g2) +\7 j*-I\7(gl - j*g2) + j*-1\72(gl - j*g2)192,Y :S bl\7 2j*- 11. Igl - j* g21 + bl\7 j*- I II\7(gl - j* g2) 1 +bl\7 j*- I II\7(gl - j*g2) 1 + blj*- 111\72(gl - j*g2)1. (4.9) Continuing in this manner, we obtain on the right hand side linear polynomials in l\7 i (gl - j*g2) 1 without constant term and where the coefficients can be estimated by is. Summing up (4.7), (4.8), (4.9) and integrating over Ml \ K 1 , we obtain
In particular, (4.11) implies (4.12) This proves (U~). Completely similar is the proof of the transitivity of the basis.
= = h
We assume (M1, gl)
h
(M2, g2)
hI
(U~),
(M3, g3), fi : Mi \Ki ~
i.e.
MHI \
h2
K H1 , i = 1,2, with the desired properties. The triangle inequality for the sum of the first 4 terms in the set (4.1) is just proposition 2.52. Consider the next to numbers in the set (4.1). Applying the Leibniz rule, immediately yields sup l\7i(!2*fh) 1+ sup l\7i(hhh 2*)1 "'EMI
zEM3
l~i~r
l~i:::;r
:S c[ sup l\7 i fhl . sup l\7i!2*1
+
",EM
yEM2
l:::;i:::;r
l:::;i~r
i
sup l\7ih2 *1· sup l\7 hh IJ, zEM3
yE M 2
l~i:::;r
l~i:::;r
(4.13)
129
Non-linear Sobolev Structures
where C essentially is an expression in binomial coefficients. (4.13) expresses the desired transitivity of the basis. The desired transitivity of the last number in the set (4.1) would be established if (4.14) would imply (4.15) We estimate by the triangle inequality for Sobolev norms
l(hh)*93 - 91IMj\Kj,9j,p,r
== If{U;93 - f{-1 91 19j,p,r
+ If{(92 - f{-1 91 )19l,p,r = If{U;93 - 92)19j,p,r + 1f{92 - 91)1 9l,p,r i :::; sup IV fhl·lf;93 - 92192,p,r + If{92 - 9119l,p,r :::; If{U;93 - 92)19j,p,r
xEMl
O~i~r
(4.16) D
Denote the corresponding uniform structure with ll~~iff,rel and 9Jt2',~iff,rel for the completion of 9Jtn(mf, I, B k ) with respect to this uniform structure. It follows again from the definition that d~~dif f,rel (( M 1, 91)' (M2,92)) < 00 implies dL((M1, 91)' (M2, 92)) < 00, where dL is the Lipschitz distance of section 2. Hence (M2,92) E comPL (MI, 91), where comPL denotes the corresponding Lipschitz component, i.e.
{(M2,92) E 9Jt2',~;ff,rel ~ compL(M1, 91)'
I ~~diff,rel(MI,M2) < oo}
For this reason we denote the left hand side { ... } by gen compt~iff,rel(M1' 91) = {... } = {... } n comPL(M1,91)
130
Relative Index Theory, Determinants and Torsion
keeping in mind that this is not an arc component but a subset (of manifolds) of a Lipschitz arc component, endowed with the induced topology. We extend all this to Riemannian vector bundles (E, h, V'h) ------t (Mn, g) of bounded geometry. First we have to define vp,r (E --> M). For this we consider as at the end of section 3 the total space E as open Riemannian manifold of bounded geometry with respect to the Kaluza-Klein metric and restrict the uniform structure to bundle maps f = (fE, fM)' Quite similar we define for Ei = ((Ei' hi, V'hi) ------t (Mr,gi)), i = 1,2, v p,r(E1 ,E2 ) by corresponding bundle isomorphisms and v r,p,r(E1 , E 2 ) = v p,r(E1 , E 2 ) n c r,p,r(E1 , E 2) c v p,r(E1 , E 2) as the C r+1 elements with r-bounded differential, i.e. for f E (fE, fM) E v r,p,r(E1 , E 2) there holds sup lV'idfMI :S d, sup lV'idfEI :S d. xEMl
eEEl
l~i~r
l~i~r
Quite analogously to fJJrI(mf, I, B k ) we denote the bundle isometry classes of Riemannian vector bundles (E, h, V') ------t (Mn,g) with (I), (Bk(g)), (Bk(V')) of rkN over n-manifolds by BN,n(I, Bk)' Set for k ~ r > ~ + 2, Ei = ((Ei' hi, V'hi) ------t (Mr, gi)) E BN,n(I, B k), i = 1,2 d~:diff(El' E 2 )
inf { max{O, log bldfEI}
+ +
+ max{O, log bldfE/I}
max{O, log bldfMI} + max{O, log bldfi/I} sup lV'idfMI + sup IV'idfE I xEM l
l~i$r
eEEl l~i$r
if { ... } i- 0 and inf{ ... } < 00. In the other case set 1 d~~iff(El' E 2) = 00. Here we remark that bldfEI, bldfE 1, bldfMI, bldfiil < 00 automatically imply the quasi isometry of hI, fih2 or gl, fl.Jg2, respectively. A simple consideration
Non-linear Sobolev Structures shows that d(E 1 , E 2 ) = 0 is an equivalence relation BZ'~ff(I, B k ) := BN,n(I, B k )/ rv and for 5 > 0 Vj
= {(Eb E2) E (BZ~ff(I, Bk))2
1
131
rv.
Set
d!i,~iff(El' E 2) < 5}.
Proposition 4.2 ~ = {Vj}8>0 is a basis for a metrizable uniform structure ll~,~if f . The proof is quite analogous to that of proposition 4.1 with
Kl = K2 = 0.
0
The set (4.17) contains some more terms as the set (4.1). The new terms are Ihl - fE h 21 9I,hI '\1 hI ,p,r and l\7 hI - fE \7 h2 19I ,hI '\1 hI ,p,r' For the symmetry we consider
Ih2 - f;;-lh I192,h2'\1h2,p,r
= If;;-IU;;h 2 - h 1)1 92,h2,'\1h2,p,r ~ b,rlf;;-11 ·If;;h2 - h 119I ,hI'\1hI,p,r ~ k1 (5) . 5 - - - t 0 8-.0
and
f;; \7 h2 ) 192,h2,'\1h2 ,p,r < b,rlf;;-11·I\7h l - f;;\7 h2 19j ,hI'\1hI,p,r < k2 (8)· 5 - - - t 0 If;;-1 (\7 hI
-
8-.0
The proof of the transitivity is completely parallel to (4.14) (4.16). Denote BZ~;hr(I, B k) for the pair (BN,n(I, Bk),ll~~diff) and BZ~rf(I, B k) for the completion. The next task would be to prove the locally arcwise connectedness of BZ~rr If we restrict to (E, h) - - - t (Mn, g), i.e. we forget the metric connection \7 k , then the corresponding space is locally arcwise connected according to 5.19 of [33]. Taking into account the metric connection \7\ the situation becomes much worse. Given (g, h, \7 h), (g', h', \7h') sufficiently neighboured, we have to prove that they could be connected by a (sufficiently short) arc {(gt, ht, \7 ht )}. Here \7ht must be metric w. r. t. ht .
132
Relative Index Theory, Determinants and Torsion
We were not able to construct the arc {V'ht h for given {hth. One could also try to set V't = (1 - t) V' + tV' and to construct h t from V't s. t. V't is metric w. r. t. ht. In local bases el, ... , en, 1 , ... , N this would lead to the system
= ri,iaht,"!(3 + ri,i(3ht,a"!, i = 1, ... ,n, a, (3 = 1, ... ,N, where h t,a(3 = ht(a, (3), V'~a = ri,ia"!. This is a system V'~iht,a(3
of n N(~+I) equations for the N(~H) components h a (3, i.e. it is overdetermined. With other words, we don't see a comparetively simple and natural proof for locally arc wise connectedness. B~~JF(I, B k) is a complete metric space. Hence locally and locally arcwise connectedness coincide. But to prove locally connectedness amounts very soon to similar questions just discussed. Consider for E = ((E, h, V'h) ---7 (M, g)) E BN,n(I, Bk) (4.18) The set is open and contains the arc component of E. If B~~JF(I, B k) would be locally arcwise connected = locally connected then we would have arccomp(E) = comp(E).
(4.19)
If we endow the total spaces E with the Kaluza-Klein metric gE(X, Y) = h(XV, yV) + gM(7r*X, 7r*Y) , Xv, yV vertical components,
then (E, gE) becomes a Riemannian manifold of bounded geometry, hence a proper metric space. It follows from the definition that {E' E B~~JF(I,Bk)
I d~~iff(E,E') < oo} ~ comPL(E).
(4.20) (4.18), (4.19) and the foregoing considerations are for us motivation enough to define the generalized component gen comp( E) by
gencomp~~diff(E)
:= {E' E
B~'~JF(I,Bk) I d~~diff(E,E') < oo}. (4.21)
Non-linear Sobolev Structures
133
In particular gencomp(E) is a subset of a Lipschitz component and is endowed with a well defined topology coming from ll~~diff' The next step in this section consists of the additional admission of compact topological perturbations, quite similar to the case above of manifolds. We consider pairs Ei = ((Ei' hi, \7 hi ) ----+ (Mi,9i)) E BN,n(I, B k ), i = 1,2, with the following property. There exist compact submanifolds Ki C Mi and f = (fE, fM) E cr,p,r(EI , E 2), flMl\Kl E vr,p,r(EIIMI\KI' E2IM2\K2)' For such pairs define d~~iff,rel(EI, E 2) = inf{ max{O, log bldfel}
+ max{O, log bldhEI} + max{O, log bldfMI} + max{O, log bldhMI} + sup d(hEfEel' el) el
+ sup d(fMhMX2, X2) + + eEEl sup
sup l'VidfMI xEMl
X2
l-:;i::;r
l'VidfEI
+
1
(fMI Ml\Kl)*92 - gIIMl\Kllgl,p,r
1-:; i::; r
+1(fEIEIMI\KI)*h2 - hIIElIMl\Kllgl,hl,V'hl,p,r +1 (fEIEIMI \KJ*'Vh2 - 'VhllEllMl \Kllgl,hl,V'hl ,p,r r 1 f = (fE,fM) E c ,p,r(EI ,E2), h = (hE, hM) E c r,p,r(E2, E I )}
bundle maps and for some KI C MI holds flEllMl\Kl E V;:z,r(EIIMI\KJl E2IfM(Ml\K l )) and
hI E2 If(Ml\Kl) = (fIEIMl\Kl)-I}
if { ... }
0 and inf { ... }
O, obtain the metrizable uniform structure U~~diJj,rel(CLBN,n(I, B k )) and finally the completion
N,n,p,r 1XT t . CLB L,diff,rel. vve se agam gen comp (E)
gen comp~~dif f,rel (E)
CLB~~fFrel(I, Bk)) dt~iJj,rel(E, E') < oo}
= {E'
E
which contains the arc component and inherits a Sobolev topol(IP,r ogy f rom J.AL,diJj,rel. As in the preceding considerations we obtain by requiring additionally hI = fEh2 or hllEllMl\Kl = fE(h2IE2IM2\K) local distances d~~iff,F(-'·) or d~~diff,F,rel,·) and corresponding uniform spaces CLBf'~fFF(I, B k ) or CLBf~f:/F,rel(I, B k ) respectively. We obtain generalized components gen comp~~if f,F (E)
(4.33)
141
Non-linear Sobolev Structures
and (4.34) gen COmp~:diJ J,F,rel (E) as before. One of our main technical results in chapter IV will be that E and E' being in the same generalized component tD,2 into the Hilbert space implies that after transforming etD,2 tD,2 tD2 tD2 L2((M, E), g, h), e- _eand e- D_eD' are of trace class and their trace norm is uniformly bounded on compact tIntervalls lao, ad, ao > O. For our later applications the components (4.33), (4.34) are most important, excluded one case, the case D2 = .6. (g) , D,2 = \1(g'). In this case variation of 9 automatically induces variation of the fibre metric and we have to consider (4.32) and gen comp~:diff,rel (E). Perhaps, for the reader the definitions for the gen comp(E) look very involved. We recall, roughly speaking, the main points are as follows. The distance which defines gen comp ... (E) measures step by step the distance between the main ingredients of a Clifford bundle: the smooth Lipschitz distance between the diffeomorphic parts of the manifolds and the bundles and the Sobolev distance between the manifold metrics, the fibre metrics, the fibre connections and the Clifford multiplications. Remark 4.11 The gen comPL',diJ J,rez(- )-definition extended canonically to structures with boundary.
can
be D
5 The classification problem, new (CO-) homologies and relative characteristic numbers As we already indicated, we understand this treatise as a contribution to the classification problem for open manifolds. We proved in chapter I that meaningful number valued invariants for all open manifold do not exist. The way out from this situation is to introduce relative number valued invariants or to give up the claim for number valued invariants and to admit group valued invariants as e.g. in classical algebraic topology. We go
142
Relative Index Theory, Determinants and Torsion
both ways. The heart of this treatise are new number valued relative invariants like relative determinants, relative analytic torsion, relative eta invariants, relative indices. This will be the content of chapters IV - VI. Our general approach consists in two steps, 1. to decompose the class of manifolds/bundles under consideration into generalized components and to try to "count", to "classify" them, 2. to "count", to "classify" the elements inside a generalized component. Chapters IV - VI are exclusively devoted to the second step. Concerning the first step, we developed in [34] some new (co-) homologies which are invariants of the corresponding generalized component and hence represent steps within the first task above. In this section, we give a brief review of these (co-) homologies. In the second part, we give an outline of bordism theory for open manifolds and corresponding relative characteristic numbers. Let X and Y be proper metric spaces. We call a map : X coarse if it is
-+
Y
1. metrically proper, i.e. for each bounded subset Bey the inverse image -l(B) is bounded in X, and 2. uniformly expansive, i.e. for R > 0 there is S > 0 such that d(Xl,X2) ~ R implies d(Xl,X2)::; S. A coarse map is called rough if it is additionally uniformly metrically proper. X and Yare called coarsely or roughly equivalent if there exist coarse or rough maps : X -+ Y, 'It : Y -+ X, respectively, such that there exist constants D x, Dy satisfying
d('ltx, x) ~ D x ,
d('lty, y) ~ D y
.
Proposition 5.1 X and Yare coarsely equivalent if and only if they are roughly equivalent. We refer to [33] for the proof.
o
Non-linear Sobolev Structures
143
The equivalence class of X under coarse equivalence is called the coarse type of X. Let X be a proper metric space. Then we have sequences of inclusions (5.1) coarse type (X) :J compCH(X), coarse type (X) :J comPCH(X) :J arccompL,h,rel(X) :J :J arccompL,h(X) :J compL,top(X),
(5.2)
coarse type (X) :J compL(X) :J arccompL,h,rel(X) :J :J arccompL,top,rel(X) :J compL,top(X),
(5.3)
The arising task is to define for any sequence of inclusions invariants depending only on the component and becoming sharper and sharper if we move from the left to the right. Start with the coarse type which has been extensively studied by J. Roe. Given X = (X, d), xq+1 becomes a proper metric space by d((xo,,,.,xq), (Yo,,,.,Yq)) = max{d(xo,Yo),,,.,d(xq,Yq)}' Let ~ = ~q C xq+1 be the multidiagonal and set Pen(~,R)
= {y
E xq+lld(~,y):s
R}.
Then J. Roe defines in [63] the coarse complex (CX*(X), 8) = (CXq(X),8)q by
cxq(X) := {J: xq+1
I
f is locally bounded Borel function and for each R > 0 is supp f n Pen(~, R) relatively compact in Xq+l}, ---t
IR
q+l
8f(xo, ... ,Xq+1) := 2)-1)if(xo, ... ,Xi,'" ,Xq+l)
(5.4)
i=O
The coarse cohomology HX*(X) of X is then defined as
HX*(X) := H*(CX*(X)). Theorem 5.2 H X*(X) is an invariant of the coarse type, i.e.
coarse equivalences phisms.
:
X
---t
Y, \II : Y
---t
X induce isomor-
144
Relative Index Theory, Determinants and Torsion
o
We refer to [63] for a proof.
Corollary 5.3 H X* (X) is an invariant for all components right from the coarse type. Remark 5.4 It is well known that without the support condition supp f n Pen(b., R) relatively compact (5.5) the complex GX*(X) would be contractible. After fixing a base point x E X the map D : Gq ~ Gq-1,
Df(xl, ... ,Xq):= f(X,X1, ... ,xq) would be a contracting homotopy.
(5.6)
o
It is now possible to define in a canonical way a cohomology theory which is an invariant of comp L (.). One only has to choose the" right category". Let
Gt(X) = {f : Xq+1 ~ IR I f is Lipschitz continuous and supp f n Pen(b., R) is relatively compact for all R}. (5.7) Then, with 0 from (3.4), GL(X) = (Gt(X),O)q is a complex and we define
H'L(X) := H*(G'L(X)). If : X ~ Y is (u.p.) Lipschitz then induces t : Gt(Y) ~ GUX) by (t(X)f)(xo, ... , Xq) := f(xo, ... , Xq), f E Gt(Y), and 'L : HL(Y) ~ H'L(X).
Using Roe's anti Cech systems and uniqueness of the cohomology of uniform resolutions by appropriate sheafs as in [63], one easily obtains
Theorem 5.5 If Y E comPL(X) then there exist : X ~ Y, W : Y ~ X wich induce inverse to each other isomorphisms
w* 'blH'L(X) ~ H'L(Y). *L
Non-linear Sobolev Structures
145
But this approach is very unsatisfactory since we did in fact not define a really new invariant but the categorial restriction of a coarse invariant. The situation rapidly changes if we factorize or impose decay conditions. Let C1(X) as above, bC1(X) the subspace of bounded functions in C1(X) and CL(X) = C1(X)j bC1(X). Then 8 maps bC1(X) into bC1+ 1 (X), i.~. bCL is a sub complex and we obtain a complex CL b(X) = (C1 b(X), 8)q. Define "
HL,b(X) := H*(CL,b(X)), Any (u.p.) Lipschitz map : X ---t Y induces # : bCL(Y) bCL(X), hence # : bCL,b(Y) ---t bCL,b(X) and * : HL,b(Y)
---t
---t
HL,b(X),
Theorem 5.6 HL,b(X) is an invariant of comPL(X), Let Y E comPL(X), ddX, Y) < E, : X ---t Y, W : X, d(wx,x) < E, d(wy,y) < E and let [J] E HL(X). Then (w 0 is (#{ singular simplexes (Jq of c I supp (Jq c BR(x)})/R:::; N}.
149
Non-linear Sobolev Structures
Roughly speaking for all singular chains E Cq,b,ulf(N) simultanously holds that every metric ball of radius R contains at most R . N singular simplexes. From the definition Cq,b,ulf;;2 lim Cq,b,ulf(N) ----t
=
UCq,b,ulf(N). N
N
Set Cq,p,ulf(N)
{c = L C~(J E Cq,b,ulf(N)
I
~q
(Cq,p,ulf(N) , lip) is a normed space (nonseparable) and we have ... ~ Cq,p,ulf(N) ~ Cq,p,ulf(N + 1) ~ .... Denote Cq,p,ulf( (0) = lim Cq,p,ulf (N) with the inductive limit topol---->
ogy. Then 8 : Cq,p,ulf( (0) ----t Cq-1,p,ulf( (0) is continuous since 8 : Cq,p,ulf(N) ----t Cq-1,p,ulf(N) is norm-continuous. We obtain Hq,p,ulf( 00 )(X),
H q,p,ulf( 00 )(X),
Hq,p,ulf(oo)(X),
Hq,p,ulf(oo)(X),
(5.8)
where H denotes the reduced (co ) homology.
Theorem 5.14 The (co-)homologies of (5.8) are invariants of arccompL,h(')'
D
Corollary 5.15 (a) H*,b,ulf,oo and H*,b,ulf,oo are invariants of arccomp L,top,rel (.). (b) H*,b,ulf, H*,b,u1f and the (co-)homologies of (5.8) are invariants of compL,top(-)' D
150
Relative Index Theory, Determinants and Torsion
The proof of 5.14, 5.15 follows from the fact that the admitted maps induce chain maps and chain homotopy equivalences between the corresponding complexes. There are many other classes of invariants which we did not consider explicit ely until now. These include the K -theory of C*-algebras, K*( C* X), and Kasparovs K-homology for locally compact spaces, K*X. We conclude this section with a brief review of bordism for open manifolds and relative characteristic numbers. We consider as before oriented open manifolds (Mn , g) satisfying
and
(1) (Bn+! , 9B) is a bordism between (Mr, gl) and (M2', g2) if it satisfies the following conditions.
1) (8B, gBI8B) ~ (Ml' gl) U (-M2' g2), 2) there exists 0 such that gBl u6(8B) ~ g8B + dt 2, 3) (B, gB) satisfies (B k ) and inf rinj(gB, x) > 0, XEB\U6(8B) 4) there exists R> 0 such that B c UR (M1 ), Be UR (M2 ). We denote (Ml' gl) (M2' g2). (Bn+!, gB) is called a bordism. f"V
b
Sometimes we denote additionally
f"V,
b,b g
bg stands for bounded
geometry, i.e. (1) and (Bk).
Lemma 5.16 a)
f"V
b
is an equivalence relation. Denote by [Mn, g]
the bordism class. b) [MUM',gUg'] = [M#M',g#g']. c) Set [M,g] + [M',g']:= [MUM',gUg'j = [M#M',g#g'j. Then + is well defined and the set of all [Mn, 9 j becomes an abelian semigroup. D Denote by n~c = n~c(1, B k ) the corresponding Grothendieck group. Similarly one defines n~C(X) generated by pairs ( (Mn, g), f : Mn ~ X), f bounded and uniformly proper.
151
Non-linear Sobolev Structures
Remarks 5.17 1) Condition 4) above looks like dCH(M, M') ::; R, where dCH is the Gromov-Hausdorff distance. But this is wrong. 2) There is no chance to calculate n~c. 3) One would like to have a geometric representative for and for -[M, g]. 0
°
The way out from this is to establish bordism theory for special classes of open manifolds or/and further restrictions to bordism. Our first example is bordism with compact support. Here condition 1) above remains but one replaces 2) - 4) by the condition There exists a compact submanifold C n +! C B n +1 such that (B \
c, gB\d
(B \
c, gBIB\C)
is a product bordism, i.e.
~ (M \ C x [0,1]' gM\C
+ dt 2 ).
(cs)
Then one gets a bordism group n~c (cs) (= b,cs Grothendieck group). At the first glance, the calculation of n~C( cs) or at least the characterization of the bordism classes seems to be very difficult. But we will see, that this is not the case. For this, we introduce still some uniform, structures. Denote by mn(mJ) := mn(mJ, nc) C mL the set of isometry classes of complete, open, oriented Riemannian manifolds. Consider pairs (Mf, gl), (M2,g2) E mn(mf) with the following property: We write
f'V.
There exist compact submanifolds Kr
c
Mr and K~
and an isometry Ml \ Kl ~ M2 \ K 2 .
c
M~
(5.9)
For such pairs, we define in analogy to sections 2 and 4
bdL,iso,rez((M1,gl), (M2,g2» := inf{max{O, logbldJI} + max{O, logbldhl} + sup dist(x, hJx) + sup dist(y, Jhy) I XEMI
yE M 2
J E COO(Ml' M 2), 9 E COO (M2' M1), and for some Kl C K,JIMl\Kl is an isometry and 9If(Ml\K = J-l}.
152
Relative Index Theory, Determinants and Torsion
If (MI, 91) and (M2,92) do not satisfy (5.9), then we define bd L,iso,rel((M1,91), (M2,92)) = 00. We have bdL,iso,rel((M1,91), (M2,92)) = 0 if (M1, 91) and (M2,92) are isometric. Remarks 5.18 1) The notions Riemannian isometry and distance isometry coincide for Riemannian manifolds. FUrthermore, if ! is an isometry!, then we have bld!1 = l. 2) Any! that occurs in the definition of dL,iso,rel is automatically an element of C'X),m(M1, M 2) for all m. The same holds true for 9.
0
We write 9J1Lisorel(m!) = mn(mJ)j ",where (M1,91) '" (M2,92) if bd L,iso,rel((M1: 91), (M2, 92)) = O. Set
Va = {((M1,91), (M2,92)) E (9J1)'i,iso,rel(m!))2 bd L,iso,rel((M1, 9d, (M2, 92)) < 6}.
I
Proposition 5.19 .c = {Va}o>o is a basis for a metrizable uni0 form structure bUL,iso,rel. Denote by b9J1L ,iso ,rei (m J) the corresponding uniform space.
Proposition 5.20 If rinj(Mi , 9i) = ri > 0, r = min{rI, r2} and bd L,iso,rel((M1,91), (M2,92)) < r then Ml and M2 are (uniformly 0 proper) bi-Lipschitz homotopy equivalent. Corollary 5.21 If we restrict ourselves to open manifolds with injectivity radius 2': r, then manifolds (M1,91) and (M2,92) with bdL,iso,rel-distance less than r are automatically (uniformly proper) bi-Lipschitz homotopy equivalent. 0 Remark 5.22 If (MI, 91) satisfies (1) or (1) and (B k ) and bdL,iso,rel(M1, 91)' (M2, 92)) < 00 then (M2,92) also satifies (1) or (I) and (Bk). 0
Non-linear Sobolev Structures
153
We cannot show that b9J1L,iso,rel is locally arcwise connected, that components are arc components and bcompLisorel(M,g) = {(M',g')lbdL,iso,rel((M,g), (M',g')) < oo} is wrong. 'The reason is that we cannot connect non-homotopy-equivalent manifolds by a continuous family of manifolds. A parametrization of nontrivial surgery always contains bifurcation levels where we leave the category of manifolds. A very simple case comes from corollary 5.21.
Corollary 5.23 If we restrict bUL,iso,rel to open manifolds with injectivity radius ~ r > 0, then the manifolds in each arc component of this subspace are bi-Lipschitz homotopy equivalent. Proof. This subspace is locally arcwise connected and components are arc components. Consider an (arc) component and two elements (M1,gl) and (M2,g2) of it, connect them by an arc, cover this arc by sufficiently small balls, and apply 5.21. 0 By definition, we have
bd L,iso,rel((M1,91), (M2,92)) < 00 => dL((M1,9d, (M2,92)) < 00, where dL is the Lipschitz distance of section 2. Hence, (M2, g2) E comPL(M1,gl), i. e. {(M2,g2) E mn (mfWd L,iso,rel((M1,gl), (M2,g2)) < oo} 0 s.t. for all x, y E U(c:) holds
Here c: stands for C:I, ... ,c: s , c:~, ... hood of c:, U(c:) n C = 0.
,c:~,
and U(c:) for a neighbour-
Lemma 5.38 (GH) and (GH I ) are equivalent.
162
Relative Index Theory, Determinants and Torsion
Proof. Assume (CHI)' Then (CH) holds since for x-Y' Y-y E 11'1 c U(E), U(E) n C = 0, du(c) (x-y, Y-y) = d-y (x-y , y-y). If conversely x, Y E U(E) then there exists x-Y' Y-y E 11'1 C U(E) s.t. du(c) (x, x-y) ::; R M , du(c)(Y, Y-y) ::; R M . Then the assertion follows from
du(c) (x, y) - du(c) (x-y, Y-y) ::; du(c) (x, x-y) + du(c)(Y, Y-y), du(c)(x, y) - du(c)(x, x-y) - du(c)(Y' Y-y) ::; d-y(x-y, Y-y) = d-y (x-y , Y-y) - c' + c' ::; dB\C(x-y, Y-y) + c', du(c) - 2RM - c' ::; dB\c(x-y, Y-y), du(c) - 4RM - c' ::; dB\C(x, y).
o Remark 5.39 (CHI) immediately implies that dCH(B \ C, U(UEa)) < 00, where dC,He·) is the Gromov-Hausdorff disa
tance between proper metric spaces. This follows from the following facts. dCH(B \ C,U(UE a)) < 00 if we endow U(UEa) a
a
with the induced lengths metric and use (B \ C C UR(U(UEa)). a
Then we use dCH(U(E)), its own lengths metric, U(E), induced lengths metric < 00, which follows from (CHI)' As a matter of fact, we introduced (CH) to assure dCH(B \ C, U(E)) < 00. 0 Proposition 5.40
rv
ne
is an equivalence relation.
o We refer to [26] for the proof. O~C(ne) == onc(je, ne, bg ) is again defined as Grothendieck group. Next we develop geometric realizations for 0 and -[M, g]ne in O~C(ne).
Let (Mn, g) be as above, i.e. oriented, with (I), (Boo), finitely many ends Eb' .. , Es , each of them nonexpanding. Let E be one of them, C C M compact and so large that E is defined by one
163
Non-linear Sobolev Structures
of the components of M \ C, Ue C M \ C a neighbourhood, "I a ray in U(c). "I admits a tubular neighbourhood of radius 53 > O. Consider (B, gB) = (M x I, gM + dr 2). Then c x I =
{Uj(c) X I}jEJ is an end of M x I, U(c x I) = U(c) x I a neighbourhood disjoint to C MxI = C x I, and for 0 < 51 < 1, the curve "101 = "I x {5tl = b,(1 ) is a ray in U(c x I). c x I is nonexpanding. "101 admits a tubular neighbourhood with a radius 52 > 0, To 2 bo1)' Theorem 5.41 8T02 ("101) has bounded geometry, one nonexpanding end and there holds
We refer to [26] for the proof. 0 n Next we shall see, (chc (5),gst) will play the role of our zero in n~C(ne).
Lemma 5.42
chc n (r2)'
a)
For r1 < r2 is chcn(rl)
b)
[i~lchCn(ri)Le = [chcn(r)lne for r > rl + ... + rk·
rv
ne
(5.20)
(5.21)
Proof. a) is immediately clear (or follows from b)). Set for b) r = r1 + ... + rk + 5, place chcn (r1) u· .. U chcn(rk) all with parallel [0, oo[ direction into int(chcn(r)), where int(chcn(r)) coresponds to b~ x ]0,00[. Then CL(int( chcn(r)) \int( chc n(r1) u· .. U
chcn(rk))) defines the desired ne-bordism.
0
Theorem 5.43 For any oriented manifold (Mn, g) of bounded geometry and a finite number of ends, each of them nonexpanding, there holds (5.22)
164
Relative Index Theory, Determinants and Torsion
Proof. We must construct a ne-bordism between (Mn, g) and -((Mn,g) U (chcn(r),gst)). Let (Bn+l,gB) = (M x [0, l],gM + dt 2 ), E be an end of M, 'Y a ray in E, form 'Y,h = b, (h) c M x [0,1], T"2bch), 62 < inf{~,rinj(M)/2} and set B-y = Bn+1 \ intT"2 ('Y"1) with the induced metric. From our assumption rinj > 0 follows easily that aT"2 b"J has a smooth collar U" (aT). Endow U§. with the product metric g§. and form on U" - U§. the 2 2 2 smooth bg-convex combination of g§. and gB getting gB-y' En2 dow aTy2 b"1) with the induced orientation. Then (R(lgB-y) is a bg, ne-bordism between (Mn,g) and (Mn,g) U (aT"2b"1),gaT)' Theorem 5.41 yields
o
Theorem 5.44 n~C(ne) == n~C(bg, ne) is an abelian group with -[Mn,g] = [(-Mn,g)] and 0 = [chcn(r),gst]. 0
Our next goal is to produce independent generators of n~C(ne). As we shall see in the sequel, infinite connected sums of complex projective spaces (or their cartesian products) supply such elements. We prepare this by several assertions Lemma 5.45 Let (M[t, gi), i = 1,2, be open, oriented of bounded geometry and with a finite number of ends, each of them non expanding. Let further (Bn+l, gB) be a ne-bordism between them and K c B compact such that the ends of B coincide with the components of B \ K. Let Ce C B \ K a component of B \ K and Xo E Ce. Then there exists a constant C 1 > 0 such that the diameter of any metric sphere
is ~ C1 • Here we understand the diameter with respect to the induced length metric dB of B.
Non-linear Sobolev Structures
We refer to [26] for the proof.
165
o
Now we recall once again the chopping theorem of Cheeger / Gromov (cf. [17]) which is a consequence of Abresch's habilitation (cf. [1]) and was our I 1.33.
Theorem 5.46 Suppose (Mn, g) open, complete with bounded sectional curvature IKI :::; C. Given a closed set X c Mn and o < r :::; 1, there is a submanifold, un, with smooth boundary,
8U n, such that for some constant c( n, C)
Xc U c Tr(X), vol(8U) :::; c(n, C)vol(Tr(X) \ X)r- 1 , III(8U)1 :::; c(n, C)r-l. Moreover, U can be chosen to be invariant under I(r, X) group of isometries of Tr(X) which fix x.
0
In our case, X = X(2 = B(2(xo) c Bn+l. To apply 5.46, we form (vn+l, gv) = (Bn+l U Bn+l, gB U gB) which is well defined and smooth since we assumed the Riemannian collar gBlcoll ar = gaB + dt 2 . Now we set Xv = X U X and apply 5.46. Fix 0< r :::; 1. Then we get Uv , H(2,v,r = 8Uv .
Xv c Uv C Tr(Xv)(= {x E Vldv(x, Xv) :::; r}),(5.23) vol(H(2,v,r) = vol(8Uv ) (5.24) :::; c(n + 1, C)vol(Tr(Xv) \ Xv)r- 1 1 (5.25) III(8Uv ) 1 :::; c(n + 1, C)rand Uv is invariant under I(r,Xv). The main idea of the proof consists in considering the distance function F = d(·,X v ) where for points E V \ Xv, d(·,X v ) = d(·, X(2) = d(·,8(2). Then one applies Yomdin's theorem to F in Abresch's smoothed out metric. All constructions are invariant under the metric involution and this involution remains an isometry also with respect to Abresch's smoothed out metric.
166
Relative Index Theory, Determinants and Torsion
Restricting the obtained Uv , 8Uv to B, we obtain the desired result for X = Bg(xo) c B. Restricting for (J large to Co and using the construction of U as pre image under the smoothed F, we obtain in Co a hypersurface H = Hg which decomposes Co into a compact and noncompact part Co,e and Co,ne, respectively. Under our assumptions (8 B is totally geodesic) it is possible to arrange that Hn intersects 8B transversally under an angle> 5 and that there exists a constant C 1 independent of (J such that (5.26)
We infer from (5.23), bounded curvature and lemma 5.45 that for fixed 0 < r :::; 1 there is a constant C 2 > 0 such that (5.27)
for all (J. Moreover, H; has bounded geometry (at least of order 0) according to (5.24) and to the bounded geometry of B. Now we are able to present independent generators of n4~(ne). Let p2k(C) be the complex projective space with its standard orientation and with its Fubini-Study metric, fix two points ZI, Z2 and form by means of fixed spheres about ZI, Z2 the infinite connected sum 00 M4k = (M4k, g) = #P2k(C), (5.28) 1
always with same glueing metric. Then (M4k, g) is oriented, has bounded geometry, one end which is nonexpanding. Theorem 5.47 M4k
00
#p 2k (C) defines a non zero bordism 1
class in n4~(ne). Proof. Suppose [M4k] = O. Then there exists a bordism (Bn+\gB), 8B = M4k U -chc4k (r), gBlu,,(8B) = g8B + dt 2, UR (M4k) "2 B, UR (chc 4k (r)) "2 B and dB ~ dM-c, dB ~ dchc-c. We choose Zo E Plk(C), K = 0 and obtain for any (J > 0 a compact hypersurface H: k c B = B \ 0 = Co which decomposes B into a compact and noncompact part Be and B ne , respectively, and which satisfies (5.26), (5.27) and has bounded
167
Non-linear Sobolev Structures
geometry at least of order 0 with constants independent of (2. Then 8B4k+1 = (8B4k+l n M4k) U He U (8B4k+l n chc4k ) . Here c c c a(8Bdk+1 n chc 4k ) = O. a(8Bdk+1) must be zero since it is 0bordant (if one wants, after smoothing out). Hence (5.29)
But
a(H;k) =
J+ L
"7(8H;k)
+
H~k
J
expression(II(8H;k)). (5.30)
aH~k
The first expression on the r.h.s. of (5.30) is bounded by a bound independent of (2 according to (5.27) and (Bo) for H;k. The same holds for the second expression according to
1"7(8H;k) I ::; C3 vol(8H;k) and for the third expression according to (5.26), (5.27). On the other hand, choosing (2 sufficiently large, a(8B4k+l n M4k) can be made arbitrarily large. This contradicts (5.29). D Looking at the proof of theorem 5.47, we immediately infer Theorem 5.48 Let (M4k, g) be open, oriented, of bounded geometry and with a finite number of ends, each of them nonexpanding. If for any exhaustion Ml C M2 C ... by compact submanifolds, U Mi = M, there holds
lim a(Mr)
= 00
t---+OO
then [M4k, g]
i- 0 in n~Z(ne).
D
00
Corollary 5.49 #P2k(C) , or, more general, P 2h(C) x ... 1
p2i r1 #p 2jl (C) X ... X p 2j r2 # ... , i 1 + ... iT! k, . .. are not torsion elements in n~k( ne).
= k,
jl
+ ... + jr2
X
= D
168
Relative Index Theory, Determinants and Torsion
A special case for theorem 5.48 is given by manifolds M4k of the type
vol(Mi4k ) :::; G1 , !K(9i)! :::; G2 , rinj(9i) 2: G3 > 0, C7(Mfk) 2: o for i 2: io and > 0 for infinitely many i 2: io. Then, in particular, 7t 2k,2(M 4k ) is infinitedimensional and [M4k, 9] i=- 0 in n~k(ne), i.e. adding a finite number of closed manifolds with negative signature and an infinite number of closed manifolds with zero signature (such that the bg, ne-end struture remains preserved) does not transform a nonzero element into zero in n~k(ne). A finer characterization of nonzero elements in n~k(ne) will be presented at another place. Moreover there are very interesting specializations of the theory developed until now and generalizations, e.g. the restriction to manifolds with warped product structure at infinity or with prescribed volume growth of the ends etc .. This will be the topic of another investigation.
III The heat kernel of generalized Dirac operators Substantial estimates for the operator e- tD2 are more or less equivalent to estimates for the corresponding heat kernel. We present in the first section those estimates which are needed in the sequel and establish some invariance properties of the spectrum which we apply in chapters IV, V and VI.
1 Invariance properties of the spectrum and the heat kernel We start with an absolutely fundamental theorem.
Theorem 1.1 Let (E, h, \7 i) ----t (Mn, g) be a Clifford bundle, (Mn, g) complete and D the generalized Dirac operator. Then all powers Dn, n ~ 0, are essential self-adjoint.
o
We refer to [20] for the proof.
Corollary 1.2 Let (E, h, \7) ----t (Mn, g) be a Riemannian vector bundle, (Mn, g) complete and b. q the Laplace operator acting on q-forms with values in E. Then (b.q)n, n = 1,2, ... are essentially self-adjoint. In particular this holds for the Laplace operator acting on ordinary q-forms. Proof. b. q
= D2 for the Clifford bundle A*T* M
® E.
0
In what follows, we always consider the self-adjoint closure Dn and write Dn == Dn.
Corollary 1.3 There is a spectral decomposition
169
170
Relative Index Theory, Determinants and Torsion
where (Je denotes the essential and (Jpd the purely discrete point spectrum. In particular,
o A E (Je if and only if there exists a Weyl sequence for A. Properties of Weyl sequences imply very important invariance properties for the spectrum. Proposition 1.4 Let (E,h,'\lh,.) --> (Mn,g) be a Clifford bundle, Mn open and complete, K c M a compact subset, DF(EIM\K) Friedrichs' extension of Dlc,?O(EIM\K)' Then there hold (Je(D) = (Je(D F) = (Je(DF(EIM\K)) (1.3)
and
Proof. We start with (1.3) and (Je(D) ~ (Je(DF(EIM\K)). Let A E (Je(D), ('l/Jv)v be an orthonormal Weyl sequence for A, D'l/JvA'l/Jv --> O. Then (wv)v, Wv = 'l/J2v+l -'l/J2v is still a Weyl sequence for A. Let E CC: (M), 0 ::; ::; 1, = 1 on a neighbour hood U = U(K) of K. According to the Rellich chain property of Sobolev spaces (with real index) on compact manifolds, ('l/Jv) v contains an L 2 -convergent subsequence which we denote again by ('l/Jv) v' This yields w v --> 0 and grad . Wv --> 0 in L 2 . ((1 - 0, grad . Wv --> O. Hence (Je(D) ~ (Je(DF(EIM\K)). VDp(EIM\K) ~ V Dp and every Weyl sequence for A E (Je(DF(EIM\K)) is also a Weyl sequence for A E (Je(D). This finishes the proof of (1.3). (1.4)
Heat Kernel of Generalized Dirac Operators
171
is an immediate consequence of (1.3) by means of the spectral 0 theorem but it can also similarly be proven.
Corollary 1.5 The essential spectrum of D and D2 remains invariant under compact perturbations of the topology and the metric. In particular this holds for the Laplace operators acting 0 on forms with values in a vector bundle. As for compact manifolds, we can define the Riemannian connected sum for open Riemannian manifolds, even for Riemannian vector bundles (Ei' hi, '\1 hi ) -+ (Mt, gi), where at the compact glueing domain the metric and connection are not uniquely determined. Another corollary is then given by Proposition 1.6 Suppose (Ei' hi, '\1 hi ) - + (Mt,gi), i 1, ... ,r Riemannian vector bundles of the same rank, (Mt, gi) complete, and let .6. = .6.q be the Laplace operator acting on q-forms with values in Ei (resp. E). Then
O"e.6. q (
i~l
r
(Ei -+ Mi )) =
UO"e(.6. (Ei -+ Mi)). q
(1.5)
i=l
o 1.4 can be reformulated as the statement that the essential spectrum for an isolated end E is well defined. We denote it by
O"e(DF(E)), O"e(D~(E)). Proposition 1. 7 If (Mn, g) is complete and has finitely many ends El,' .. , Er then r
r
i=l
i=l
172
Relative Index Theory, Determinants and Torsion
Proposition 1.8 Assume the hypothesis of 1.4. Suppose A E
CTe(D). Then there exists a Weyl sequence ('Pv)v for A such that for any compact subset K
c
M
(1.7) For every A E CTe(D2) there exists a Weyl sequence ('Pv)v satisfying (1. 7) and (1.8)
Proof. Start with (1.7). Let ('l/Jv)v be a Weyl sequence for A E CTe (D), Kl C K2 C ... C Ki C KH 1 C "', UKi = M, an exhaustion by compact submanifolds. By a Rellich compactness argument there exists a subsequence ('l/J~l»)v of ('l/Jv)v ('l/J~O»)v converging on K 1. Inductively, there exists a subsequence ('l/J~Hl»)v of ('l/J~i»)v converging on K H1 . Set ('Pv)v = (('l/J~~~~l) - 'l/J~~v»)/V2)v. Then ('Pv)v is a Weyl sequence for A E CTe(D) satisfying (1.7). For A E CT e(D2) with Weyl sequence ('l/Jv) v, we choose the subsequence ('l/J~Hl»)v of ('l/J~i»)v such that ('l/J~Hl»)v and (D'l/J~Hl»)v converge on KHI (in L 2, as always). 0 1.8 means that w.l.o.g. Weyl sequences should "leave" (in the sense of the L 2-norm) any compact subset, i.e. there must be "place enough at infinity" .
Proposition 1.9 Let (E, h, \7, .) - - t (Mn, g) be a Clifford bundle with (1), (B r- 3 (M,g)), (B r- 3 (E, \7)), r > ~ + 1 and \7' a second Clifford connection satisfying 1\7' - \71V',2,r-l < 00. Then for D = D(\7) and D' = D(\7') there holds (1.9) and
(1.10)
(Mn,g) is complete, D and D' are self-adjoint. 1JD = 2 n ,I(E, D) = n2,1(E, \7) = n2,1(E, \7') = n2,1(E, D') = 1J D , Proof.
173
Heat Kernel of Generalized Dirac Operators
according to II 1.25 and II 1.32. We write \1' = \1 + ry. Then D' = Lei . \1~i = Lei' (\1 ei + ryei (.)) = D + ryOP, where the i
i
operator ryOP acts as ryOP ( 0, there exists a compact set K = K(E') eM such that
:s:
sup XEM\K
E'
Irylx < -C ' 1
i. e.
sup IryOPlx < xEM\K
(1.11)
E'.
Assume now ,\ E (Je(D), ( ~ + 1, g' another metric satisfying the same conditions and suppose g, g' quasi isometric and Ig' - glg,2,r = r-l (J(lg' - gl~,x + L 1(\7 g)i - \7gl~,x) dvolx(g))! < 00. Then i=O
v ~q(g) = V ~q(g')
as equivalent Hilbert spaces
(1.14)
and (1.15)
Proof. We write ~ = ~q(g), ~' = ~q(g'), \7 = \7g, \7' = \7g'. Then, according to II 1, V~ = nq,2,2(~, g) = nq,2,2(~Bochner' g) = n q,2,2(\7 , g) ~ n q,2,2(\7' , g') = nq,2,2(~'Bochner' g') = n q,2,2 (~', g') = V~,. Denote by R the Weitzenboeck endomorphism, ~
= \7*\7 + R.
(1.16)
175
Heat Kernel of Generalized Dirac Operators
We write t::..' - t::.. = \1'*'\1' - \1*\1 + R' - R. Let A E oAt::..) and (wv)v be a Weyl sequence for A as in proposition 1.8, for any K (1.17) and (1.18) We infer as above from (wv)v bounded, t::..wv - AWv (wv)v, (t::..wv)v are bounded, hence
(Wv)v is a bounded sequence in But under our assumption r >
~
----+
nq ,2,2(t::.., g).
+ 1, according to II
0 that
(1.19) 1.27,
n ,2,2(t::.., g) = n ,2,2(\I, g) ~ n ,2,2(\I', g') = n ,2,2(t::..', g') Q
Q
Q
Q
(1.20) as equivalent Sobolev spaces. Hence
(\lwv)v, (\l 2wv)v" (\I'Wv)v' , (\I,2wv )v' , (t::..'wv)v are bounded (1.21) and (1.22) To be very explicit, we choose a u. l. f. cover of (Mn,g) by normal charts with respect to g, U = {(Ua, ua)}a and an associated partition {'Pa}a of unity with bounded derivatives as in I 1. iq Then for a q-form wlu" = '" W·H ... 2q. dU i1a 1\ ... 1\ du a 6 hoo
(Ll'). Exchanging the role of 9 and g' yields the other inclusion. 0 E
(J"e
We state without proof the generalization to forms with values in a vector bundle. Proposition 1.13 Let (E,h, \7 h) ----t (Mn,g) be a Riemannian vector bundle satisfying (1), (Bk(Mn,g)), (Bk(E, \7)), k ~ r > ~ + 1, and let g' be a second metric, h' a second fibre metric with metric connection \7h', g, g' and h, h' quasi isometric, respectively,
Ig - g'lg,2,r < 00, Ih - h'lh,y>h,g,2,r < h h' g2r-1h , " ,
00,
(E,h', \7h') ----t (Mn,g') also satisfying (1) and (Bk). Then there holds for the Laplace operators Ll = Llq(g, h, \7 h), Ll' = Llq(g', h', \7h') acting on forms with values in E Vt, = Vt" as equivalent Hilbert spaces
(1.36)
and (1.37) The proof is quite similar to that of 1.12, taking the difference of the Weitzenboeck formulas and proceed as in the proof of 0 proposition 1.12. Now we collect some standard facts concerning the heat kernel 2 of e-tD • The best references for this are [9], [27]. We consider the self-adjoint closure of D in L2(E) = HO(E), +00
D =
J >.E>..
-00
178
Relative Index Theory, Determinants and Torsion
Lemma 1.14 {eitDhER defines a unitary group on the spaces
HT(E), for 0 :S h:S r holds (1.38)
o We can extend this action to H-T(E) by means of duality. Lemma 1.15 e- tD2 maps L 2 (E) r > 0 and
== HO(E)
------+
HT(E) for any (1.39)
Insert into e- tD2
Proof.
= J e- t ).,2 dE>.. the equation
J +00
t
e- >..2 =
V~7rt
ei>"se-ft. ds
-00
and use
o Corollary 1.16 Let r, s E Z be arbitrary.
HT(E)
------+
Then e- tD2
HS(E) continuously.
Proof. This follows from 1.15, duality and the semi group property of {e- tD2 h;:::o. 0
e- tD2 has a Schwartz kernel W E f(R x M x M, E tD2 W(t, m,p) = (6(m), e- 6(p)) ,
[8]
E),
where 6(m) E H-T(E) 0 Em is the map W E HT(E) ------+ (6(m), w) = w(m), r > The main result of this section is the fact that for t > 0, W(t, m,p) is a smooth integral kernel in L2 with good decay properties if we assume bounded geometry. Denote by C(m) the best local Sobolev constant of the map W ------+ W(m), r > i, and by a-(D2) the spectrum.
i.
Heat Kernel of Generalized Dirac Operators
179
Lemma 1.17 a) W(t, m,p) is fort> 0 smooth in all Variables. b) For any T > 0 and sufficiently small E > 0 there exists C > 0
such that IW(t, m,p)1 :::; e-(t-e)inf(T(D2) . C· C(m) . C(p) for all t E]T, 00[. (1.40)
c) Similar estimates hold for
(D~D~W)(t, m,p).
Proof. a) First one shows W is continuous, which follows from (8(m), -) continuous in m and e- tD2 8(p) continuous in t and p. Then one applies elliptic regularity. b) Write 2 1(8(m), e- tD2 8(p)) 1 = 1((1 + D2)-~8(m), (1 + D2Ye-tD (1 + D2)~8(p))1
c) Follows similarly as b.
D
Lemma 1.18 For any E > 0, T > 0,8 > 0 there exists C > 0 such that for l' > 0, m E M, T > t > 0 holds
J
(r_e)2
IW(t, m,p)1 2 dp:::; C· C(m) . e- (4+O)t.
(1.41)
M\Br(m)
A similar estimate holds for D~D~W(t, m,p). D
We refer to [9] for the proof.
Lemma 1.19 For any E > 0, T > 0,8 > 0 there exists C > 0 such that for all m, p E M with dist( m, p) > 2E, T > t > 0 holds
IW(t,m,p)1 2
(diBt(m,p)-e)2
:::;
C· C(m)· C(p)· e-
(4+o)t
(1.42)
A similar estimate holds for D~D~W(t, m,p). We refer to [9] for the proof.
D
180
Relative Index Theory, Determinants and Torsion
Proposition 1.20 Assume (M n , g) with (/) and (B K ), (E, \7) with (B K ), k 2: r > ~ + 1. Then all estimates in (1·40) - (1.42) hold with uniform constants. Proof. From the assumptions Hr (E) ~ wr (E) and sUPm C(m) = C = global Sobolev constant for wr(E), according to II 1.4, II 1.6. D
Let U c M be precompact, open, (M+, g+) closed with U c M+ isometrically and E+ --t M+ a Clifford bundle with E+lu ~ Elu 2 isometrically. Denote by W+(t, m,p) the heat kernel of e-tD+ . Lemma 1.21 Assume E > 0, T > 0, J > O. Then there exists C > 0 such that for all T > t > O,m,p E U with B2c(m), B 2c (p) C U holds .2
IW(t,m,p) - W+(t,m,p)1 ::; C· e-(4+8)t
We refer to [8] for the simple proof. Corollary 1.22 trW(t, m, m) has for t totic expansion as fortrW+(t,m,m).
(1.43) D
--t
0+ the same asympD
2 Duhamel's principle, scattering theory and trace class conditions 2 -,2 We want to prove the trace class property of e- tD - e- tD , where fy is a perturbation of D. The key to get convenient - 2 expressions for e- tD2 - etD' is Duhamel's principle. For closed manifolds, this is a very well known fact. We establish it here for open complete manifolds. In principle, it follows from Stokes' theorem, or, what is the same, from partial integration. Having established Duhamel's principle, the proof of the trace class property amounts to the estimate of a certain number of operator valued integrals. Their estimate occupies the whole 30 pages of chapter IV.
Heat Kernel of Generalized Dirac Operators
181
The trace class property is the key for the application of scattering theory. We give an account on those facts of scattering theory which are of great importance in chapters V and VI. First we establish Duhamel's principle and make the following assumptions: D and D' are generalized Dirac operators acting in the same Hilbert space,
where
'fl
= 'flop
is an operator acting in the same Hilbert space.
Lemma 2.1 Assume t > O. Then
J t
e-tD2 - e _tD,2 --
2 e -SD (D,2 - D2) e -(t-s)D,2 ds.
(2.1)
o
Proof.
(2.1) means at heat kernel level
W(t,m,p) - W'(t,m,p)
JJ t
=-
(W(s, m, q), (D2 - D,2)W'(t - s, q,p))q dq ds,
o M (2.2) where (,)q means the fibrewise scalar product at q and dq = dvolq(g). Hence for (2.1) we have to prove (2.2). (2.2) is an immediate consequence of Duhamel's principle. Only for completeness, we present the proof of (2.1), which is the last of the following 7 facts and implications. 1. For
t > 0 is W(t, m,p)
2. If , W E formula).
Vb
E L 2(M, E,
dp)
n Vb
then J(D 2, w) - (, D2W) dvol = 0 (Greens
3. ((D2+ %7)(T, g)W(t-T, q))q-((T, g), (D2+~)W(t-T, q))q = = (D2( ( T, q), W(t-T, q) )q- (( T, q), D 2w(t-T, q) )q+ (( T, g),
tr
W(t-T,q))q.
Relative Index Theory, Determinants and Torsion
182
f3
4. J J((D 2 + tr)')
-00
(2.3)
Heat Kernel oj Generalized Dirac Operators
187
is defined as the set of all vector-valued functions J, J(>..) E X(>..) which are measurable and square-integrable with respect to the measure m. The scalar product in (2.3) is defined as +00
(1, g):= j (J(>..), g(>..))x(>,)dm(>"), -00
where (J(>..),g(>..))x(>.) is the scalar product in the Hilbert space
X(>..). We say, a Hilbert space X has a decomposition as a direct integral if there is a unitary mapping +00
F: X
7
j EBX(>..)du(>..). -00
We denote this by +00
X
~
j EBX(>")du(>"). -00
A special case of such a representation is given by the spectral resolution for a self-adjoint operator A,
-00
which induces a decomposition of type (2.3) in which the operator F AF* acts as multiplication by>... Let 8 c IR be a Borel set. Then FEA(8)F* reduces to XB, and we obtain
(EA(B)J,g) = j((FJ) (>..), (Fg) (>..))dm(>..). B
If we apply these considerations to the self-adjoint operator B (of the pair A, B above) and to Xac = Xac(B) == (Pac(B))(X),
Relative Index Theory, Determinants and Torsion
188
then we get
Xac(B)
J
~
EBX>.(B)d)..:= X(B)(ac).
(2.4)
&(B)
Here o-(B) is a so-called core of (J"(B), i.a. a Borel set of minimal measure such that EB(lR \ o-(B)) = O. The right hand side of (2.4) diagonalizes the operator Bac = BlxaJB). The scattering operator S = W~ W - commutes with B ac , hence under the correspondence (2.4) its action goes over into multiplication by an operator valued function S()") : X>.(B) ----t X>.(B). S()") = S()..; A, B) is called the scattering matrix. If we consider the left hand side of (2.2) then we obtain the version S
=
J
(2.5)
S()..)dEB ()..)
&
instead of FSF* =
J
(2.6)
S()")d)"
Both representation are equivalent. In chapter V and VI, the spectral shift function (S S F)~ will playa central role. We introduce it now. The main goal is to introduce a function ~()..) such that
tr( n + 2, n ~ 2, '\7' E comp('\7) n CE(Bk) c C~r(Bk)' D = D(g, '\7), D' = D(g, '\7') generalized Dirac operators. Then
are trace class operators for t > 0 and their trace norm is uni0 formly bounded on compact t-intervalls lao, ad, ao > O.
Here '\7' E compl,r('\7) means in particular 1'\7 - '\7'IV',l,r < and both connections satisfy (Bk(E)). Denote '\7 - '\7' = 'rJ.
192
00
Trace Class Properties
193
As we indicated in III 2, we have, writing D2 - D,2 D') + (D - D')D', to estimate
J = - J -J J +J
=
D(D -
t
e- tD2 _ e- tD'2
= _
e- sD2 (D2 - D,2)e-(t-S)D
I2
e- sD2 D(D - D')e-(t-S)D
I2
ds
o
t
ds
o
t
e- SD2 (D - D')D'e-(t-s)D
I2
ds
o
t
e- sD2 DT/e-(t-S)D
I2
ds
o
t
e- sD2 T/D' e-(t-s)D
,2
ds,
o n
where T/ = T/op is defined by T/oP(w)lx =
2: eiT/ei(w) and IT/OPlop,x
::;
i=l
C . IT/Ix, C independent of x. We split
~
t
t
J = J + J,
o
0
t
:I
t
J :I
e- tD2 _ e- tD'2
=
e- sD2 DT/e-(t-s)D
I2
ds
(h)
o t
J +J :I
+
e-sD2T/D'e-(t-s)DI2 ds
(h)
o
t
e- sD2 DT/e-(t-s)D
t
:I
I2
ds
(Is)
194
Relative Index Theory, Determinants and Torsion
J t
+
e- SD\,D'e-(t-S)D'2 ds.
t
'2
We want to show that each integral (II) - (14) is a product of Hilbert-Schmidt operators and to estimate their HilbertSchmidt norm. Consider the integrand of (1 4 ),
There holds
Write
Here f shall be a scalar function which acts by multiplikation. The main point is the right choice of f. e-~D2 f has the integral kernel s (1.1) W('2' m,p)f(p) and f-le-~D27] has the kernel (1.2) We have to make a choice such that (1.1), (1.2) are square integrable over M x M and that their L 2-norm is uniformly bounded on compact t-intervals.
Trace Class Properties
195
We decompose the L 2-norm of (1.1) as
JJIW(~,m,pWlf(mW = J J IW(~, + J J IW(~,m,p)12If(mW
(1.3)
dm dp
MM
m,pWlf(mW dp dm
(1.4)
dp dm
(1.5)
M dist(m,p)'2:c
M dist(m,p) there exists C > such that
°
IW(t,m,p)1 ~
and sufficiently small
e-(t-c)infa(D
for all t E]T, oo[ and obtain for
(1.5)
°
S
).
C1If(mWvolBc(m) dm
M
>
°
C· C(m). C(p)
EH, t[
J
~
2
E
~ C2
° °
J
If(m)12 dm
M
Moreover, for any E > 0, T > 0, 0 > there exists C > Osuch that for r > 0, m E M, T> t > holds
J
M\Br(m)
which yields
J J ~J
IW(~,m,pWlf(mW dp dm
M dist(m,p)'2:c
C3 e- -~+8)2 ~ If(mW dm
M
~ C3 . e- _(~+;)2 ~
J
If(m)12 dm,
M
c>
E.
(1.6)
196
Relative Index Theory, Determinants and Torsion
Hence the estimate of
J J IW(~, m,p)1 2If(m)1 2dpdm
for s E
MM
[~, t] is done if
J
If(mW dm
s as
.
= 'l,DCf>s,
Cf>o
=
Cf>,
Cf> C
1
with compact support.
(1.12)
197
Trace Class Properties
It is well known that (1.12) has a unique solution CPs which is given by (1.13) and supp CPs C Uisl (supp cp)
Uisl =
lsi -
(1.14)
neighbourhood. Moreover,
We fix a uniformly locally finite cover U = {Uv},J = {Bd(Xv)}v by normal charts of radius d < rinj (M, g) and associated decomposition of unity {'Pv}v satisfying
l\7i 'Pvl
:S C for all v, O:Si:Sk+2
(1.16)
Write
N(CP)
I(8(m), e- tD2 ryOPcp) I
J -l +00
1 J41ft
1(8(m),
e
eiSD(ryOPIP)
ds)1
L2(dp)
-00
J -4~2 +00
1 J41ft
I
e
(eisDryoPIP)(m)
dsl
.
(1.17)
L2(dp)
-00
We decompose (1.18) v
(1.18) is a locally finite sum, (1.12) linear. Hence (1.19) v
Denote as above in particular
198
Relative Index Theory, Determinants and Torsion
Then we obtain from (1.15), (1.16) and an Sobolev embedding theorem
(1.21) !: - '!! r - 1 > !: 2 > i for r > n + 2 and since r - 1 - '!!t > 2 2' 2' IIH~ ::; 0 5 . This yields together with the Sobolev embedding the estimate
l/
mE U.(U.,)
< 0 11
, l/
mE U.(U.,)
0 12 , vol(B2d+lsl(m)) . (
vo
IB 1 ( ) ·17J11,r-1,B2d+ I•I(m») . 2d+lsl m (1.22)
There exist constants A and B, independent of m s. t.
vol(B2d+lsl(m)) ::; A. eBI • I . Write .2
e- 4t ·
9 .2 vol(B2d+lsl(m)) ::; 0 13 , e-W4t,
0 13 = A. elOB
2 t,
(1.23)
thus obtaining
0 14
=
0 12 .013
=
0 12 . A . e lOB2t .
Now we apply Buser/Hebey's inequality in chapter II, proposition 1.9,
Jlu - ucl M
dvolx(g) ::; O· c
Jl'\lul M
dvolx(g)
Trace Class Properties
199
for U E W1,1(M)nC1(M), c EjO, R[, Ric (g) ~ k, C and
uc(x)
:=
VOI~c(x)
J
= C(n, k, R)
u(y) dvoly
Bc(x)
with R
= 3d + s and infer
J
vo
M
IB 1
( ). 11J!t,r-l,B2d +181(m) dm
2d+lsl m
:::; 11J!t,r-l + C(3d + s) . (2d + s)I\71Jh,r-l :::; 11Jll,r-l + C(3d + s) . (2d + s)I1Jll,r-l.
(1.24)
C (3d + s) depends on 3d + s at most linearly exponentially, i. e.
C(3d+s)· (2d+s):::; A1e B1S • This implies
J J :::;= J : :; J 00
2
e- 10984t"
o
vo
M
IB 1 ( ) ·11Jll,r-l,B2d +181(m) dm ds (1.25) 2d+lsl m
00
e-!oft (11Jh,r-l + C(3d + s) . (2d + s)I1Jll,r-l) ds
o
00
e- ilift ds(I1Jll,r-l + AlelOBrtl1Jh,r
o
= Vi· ~J57i=(I1Jll,r-l + AlelOBrtl1Jll,r) < The function JR+
X
00.
M ---. JR, 2
(s, m) ---. e- 109 4t" 8
(
VO
IB 1 ( ) ·11Jll,r-l,B2d+181(m) ) 2d+lsl m
is measurable, nonnegative, the integrals (1.24), (1.25) exist, hence according to the principle of Tonelli, this function is 1summable, the Fubini theorem is applicable and
200
Relative Index Theory, Determinants and Torsion
1= 0)
is (for TJ
everywhere
=1=
0 and 1-summable. We proved
(1.26) Now we set
(1.27)
I(m) = (fj(m))! and infer I(m)
=1=
1/-1e-~D2
0 everywhere, 0
1 E L2
and
TJIL
JJl(m)-21((W(~, : ; Jfj(~) J
=
m,p), TJOP)pI2 dp dm
MM
fj(m)2 dm =
M
fj(m) dm
M
2 In 1 ~(I 1 ::; C12 · A· elOB sV8' "2v57f TJ 1,r-1
2 + Ale lOB s1 TJ 11,r) 1
(1.28)
::; C15Vse10B2sITJI1,r, i. e.
1/- 1e- 2D 8
!
2
0
!
TJI2 ::; C 125 '8 4 . e5B s . ITJlr,r' 1
2
(1.29)
Here according to the term A1e10Brs, C 15 still depends on We obtain
IIL2 . 1/-10 e-~D2 C4 1/1L2 . C1!25 • 8 41 . e5B2 s.
le-~D2
::;
0
TJI ! ITJIf,r 0
::; C4 . C15VselOB2SITJiI.r = C 16 . Vs' e This yields e- sD2 0 TJ is of trace class,
le- sD2 TJl1 ::; e- sD2 le- SD2
0
0
TJ TJ
0
0
le-~D2
0
8.
lOB2s
ITJiI,r' (1.30)
112 '1/-1e-~D2TJI2 ::; C16VselOB2SITJ11,r, (1.31)
,2
D' 0 e-(t-s)D is of trace class, D' 0 e-(t-S)D
1 \
< le- sD2 TJl1
. ID' e-(t-s) DI2 Iop
< C16VIn8e lOB2s1 TJ 1,r' C . 1
'
1
~'
t-8 (1.32)
Trace Class Properties
I
201
t
(e-
SD2
O'T) 0
D'
0
e-(t-s)D
I2
ds
t
2"
I t
:S
le-
sD2
'T} 0
D' e-(t-S) DI2 11 ds
t
2"
:S C 16 • C , . elOB
2
tI I It (t _ s )! S
'T} 1,r .
(1.33)
ds,
t
2"
[VS(t - s)
t
2s - t
t
t7r
tn
2
22
2 2
-- + -- = -( - -
I
t
+"2 arcsin -t-l~ 1)
'
t
(e-
SD2
o'T) 0
D'
0
e-(t-s)D
,2
ds
t
2"
1
:S C 16 . C , . e lOB t . ( 2"n - 1) '"2t I'T} Il,r 2
=
C17 e lOB2t . t·
I'T} Il,r·
(1.34)
Here C17 = C 17 (t) and C17(t) can grow exponentially in t if the volume grows exponentially. (1.34) expresses the fact that (14) is of trace class and its trace norm is uniformly bounded on any t-intervall lao, all, ao > O. The treatment of (II) - (13) is quite parallel to that of (14). Write the integrand of (1 3 ), (1 2 ) or (II) as (1.35) or ( 1.36)
Relative Index Theory, Determinants and Torsion
202
or (1.37) respectively. Then in the considered intervals the expressions [ ... J are of trace class which can literally be proved as for (14). The main point in (14) was the estimate of j- I e- TD2 'T/. In (1.36), TDI2 (1.37) we have to estimate expressions 'T/ej-l. Here we use the fact that 'T/ = 'T/OP is symmetric with respect to the fibre metric h: the endomorphism 'T/ei (.) is skew symmetric as the Clifford multiplication ei' which yields together that 'T/0P is symmetric. Then the Lrestimate of ('T/OP . W' ( T, m, p), .) is the same as that of W'(T,m,p),'T/°P(p)') and we can perform the same procedure as that starting with (1.6). The only distinction are other constants. Here essentially enters the equivalence of the D- and D'-Sobolev spaces i.e. the symmetry of our uniform structure. The factors outside [... J produce on [~, tl, and on [0, ~J (up to constants). Hence (Id - (h) are of trace class with uniformly bounded trace norm on any t-intervall [aD, all, aD > O. This finishes the proof of the first part of theorem 1.1. We must still prove the trace class property of
Js
e-tD2D -e -
vbs
tD,2 D'
.
Js
(1.38)
Consider the decomposition
e-tD2D -e -
.
tDI2 D'
(1.39)
tD2
tD,2 .
Accordmg to the first part, e-"2 - e-"2 IS for t > 0 of trace class. Moreover, e-~D2 D = De-~D2 is for t > 0 bounded, its operator norm is ::; ~. Hence their product is for t > 0 of trace class and has bounded trace norm for t E [aD, all, aD > O. (1.39)
Trace Class Properties
203
is done. We can write (1.40) as
t
J '2
+
e- sD\,D'e-(&-s)D
/2
/ ds](D'e-&D \
(1.41)
o
Now (1.42) (1.42) is of trace class and its trace norm is uniformly bounded on any lao, all. ao > 0, according the proof of the first part. If t
'2
we decompose
i
4
i
2
J = J + J then we obtain back from the integrals o
0
t
4
in (1.41) the integrals (II) - (I4), replacing t ---t ~. These are /2 done. D'e-&D generates C / yt in the estimate of the trace norm. Hence we are done. 0
2
Variation of the Clifford structure
Our procedure is to admit much more general perturbations than those of 'V = 'V h only. Nevertheless, the discussion of more general perturbations is modelled by the case of 'V -perturbation. In this next step, we admit perturbations of g, 'Vh,., fixing h, the topology and vector bundle structure of E ----+ M. The next main result shall be formulated as follows.
Theorem 2.1 Let E = (E, h, 'V = 'V h, .) ----+ (Mn, g) be a Clifford bundle with (1), (Bk(M,g)), (Bk(E, 'V)), k ~ r+1 > n+3, E' = (E, h, 'V' = 'V,h, .') ----+ (Mn, g') E gencomp~~d~Jf,F(E) n
204
Relative Index Theory, Determinants and Torsion
CLBN,n(I, B k ), D = D(g, h, \7 = \7 h, .), D' = D(g', h, \7' = \7,h, .') the associated generalized Dirac operators. Then for
t>O (2.1)
is of trace class and the trace norm is uniformly bounded on compact t-intervalls lao, al], ao > O. Here D'L is the unitary transformation of D,2 to L2 = L2 ((M, E), g, h). 2.1 needs some explanations. D acts in L2 = L 2((M, E), g, h), D' in L; = L 2((M, E), g', h). L2 and L; are quasi isometric Hilbert spaces. As vector spaces they coincide, their scalar products can be quite different but must be mutually bounded at the diagonal after multiplication by constants. D is self adjoint on V D in L 2 , D' is self adjoint on V D , in L; . L 2. H ence e- tD,2 an d e- tD2 - e- tD,2 but not necessan'1y m are not defined in L 2. One has to graft D2 or D,2. Write dvolq(g) == dq(g) = a(q) . dq(g') == dvolq(g'). Then
o < Cl
:S a(q) :S C2, a, a-I are (g, \79) and (g', \79') - boundedup to order 3,
la -
119,1,r+1,
la -
119',1,r+1
(t,q) = W(t,m,q), w(t,q) = WL(t,m,q) and obtain f3
- j j hq(W(T, m, q), (D2 Q
+ :t)WL(t -
M
= j[hq(W({3, m, q), W£2(t - (3, q,p) M
-hq(W(a, m, q), W£2 (t - a, q,p)] dq(g).
T, q,p)) dq(g) dT
206
Relative Index Theory, Determinants and Torsion
Performing a yields
----t
0+, (3
----t
t and using dq (g)
= a (q) dq (g')
t
- j j hq(W(s, m, q), (D2
+ !)W'(t -
s, q,p))dq(g)ds
°M
t
= - j j[hq(W(S, m, q), (D2 -
D'L)W~2(t -
°M
= W(t, m,p)a(p) - WL(t, m,p).
s, q,p)dq(g)ds (2.4)
(2.4) expresses the operator equation t
- j e- sD2 (D2 - D'L)e-(t-S)D'i2 ds. e- tD2 a _ e- tD'i 2 e- tD2 _ e- tD'i 2
°2 e-tD (a -1) + e- tD2 - e- tD'i2, hence _e-tD 2(a - 1) t
- j e- SD2 (D2 _ D'i 2)e-(t-S)D'i 2 ds.
°
(2.5)
Jt:tt;, -
As we mentioned in (2.2), (a - 1) = ::(~}) - 1 = 2 1 E 0°,1,1'+1 since 9 E COmp1,1'+1(g). We write e-tD (a - 1) = (e-~D2 f)(f-1e-~D2 (a -1)), determine f as in the proof of theorem 1.1 from T/c. = a-I and obtain e- tD2 (a - 1) is of trace class with trace norm uniformly bounded on any t-interval lao, a1], ao > O. Decompose D2 - D'L = D(D - D~J + (D - DL)DL. We need explicit analytic expressions for this. D(D - DL) = D(D - a-!D'a!) = D(D - D') - D gr~'aJ, (D - DL)D~2 = ((D - D') - gra;a'aJ)a-!D'a!. If we set again D - D' = -TJ J then we have to consider as before with gr;~ 'a = gra;a'a where
Trace Class Properties
grad'
==
grad
207
g'
grad 'a)D' e-(t-s)D'L 2 d s 2a L2 t
J +J
e -SD2D( 'TJ -
+
grad 'a) e -(t-s)D,2L2 ds 2a
t
"2
t
e- SD2 ('I1- grad'a)D' e-(t-s)D'L d '/ 2a L2 2 s,
t
"2
It follows immediately from g' E COmpl,r+l (g) that the vector field gra:' a E nO,l,r (T M). If we write 'TJr! = a.' then 'TJgP is a zero order operator, l'TJolr < 00 and we literally repeat the ad' , procedure for (11) - (14) as before, inserting 'TJo = - gr a a· for 'TJ there. Hence there remains to discuss the integrals
gr:'
J t
J t
e- sD2 D'TJ e -(t-S)D'L2 ds
°
+
e- sD2 'TJD~2e-(t-s)D'L2 ds.
(2.6)
°
The next main step is to insert explicit expressions for D - D'. Let mo E M, U = U(mo) a manifold and bundle coordinate neighbourhood with coordinates Xl, ... , xn and local bundle basis
"("'7
9 -. [( 9
ik
- 9
+glik
if-.
V i'¥ -
8xk
lik)
9
8
lik
8
- k.
1 "("'71 if-.
V·'¥
8x
"("'7
8x k ' v i
t
+ 9 lik 8x8 k
'
("("'7
8~k (. - ./)\7~lCf>,
"("'7/)
v i-V i
(2.8)
i. e. we can write -(D - D')Cf> = ('r/fP
+ 'r/i + 'r/~P)Cf>,
(2.9)
where locally (9
9
ik
lik
glik
Here
(glik)
- 9
8
lik)
8x k '
8
("("'7
8~k (. -
"("'7
if-.
8x k ' v i '¥, "("'7/) if-.
(2.10)
v i - V i '¥,
(2.11)
\7~Cf>.
(2.12)
./)
= (g}Z)-l. We simply write 'r/v instead 'r/~, hence
J + J t
(2.6)
=
e-
sD2
D('r/1
+ 'r/2 + 'r/3)e-(t-S)D'i 2 ds + (2.13)
o
t
e-
sD2
('r/1
+ 'r/2 + 'r/3)D~2e-(t-S)D'i2
ds. (2.14)
o We have to estimate
J t
e- sD2 D'r/v e -(t-s)D''i 2 ds
(2.15)
D' e-(t-s)D''i 2 ds e -sD2'n 'IV L2 •
(2.16)
o and
t
J o
209
Trace Class Properties
t
Decompose
~
t
J = J + J which yields ~Ot
"2
t
J "2
e- sD2 D'fJv e -(t-S)D''i 2
ds,
o t
"2
J J
e- sD2 'Yl D' e-(t-s)D''i 2
·,v
L2
ds ,
o
t
e- sD2 D'fJv e -(t-S)D''i 2
ds,
t
"2
J t
e-
sD2
'fJvD~2e-(t-S)D''i2
ds.
t
"2
(Iv,l) - (Iv,4) look as (II) - (14) as before. But in distinction to that, not all 'fJv = 'fJ':! are operators of order zero. Only 'fJ2 is a zero order operator, generated by an EndE valued I-form 'fJ2. 'fJl and 'fJ3 are first order operators. We start with 1/ = 2, 'fJ2' /'fJ2/1,r < 00 is a consequence of E' E comp~~dtff(E) and we are from an analytical point of view exactly in the situation as before. (h,d-(I2,4) can be estimated quite parallel to (I1)-(I4) and we are done. There remains to estimate (Iv,j), 1/ i- 2, j = 1, ... ,4. We start with 1/ = 1, j = 3 and write
(2.17) De-~D2 and
are bounded in [~, tJ and we perform their estimate as in section 1. e-~D2 . f is Hilbert-Schmidt if f E L 2 . There remains to show that for appropriate f e-(t-s)D,2
f-le-~D2 'fJ1
210
Relative Index Theory, Determinants and Torsion
is Hilbert-Schmidt. Recall r + 1 > n + 3, n 2: 2, which implies r.2 > !!2 + 1' r - 1 - n -> r.2 - ! !2' r - 1 > r. 2 -> i. If we write in - 2' the sequel pointwise or Sobolev norms we should always write IWlgl,h,m l , IwIW(E,DI), Iw lgl,h,'V",2,¥, Ig - g'lgl,m, Ig - g'lgl,l,r etc. or the same with respect to g, h, V, D, depending on the situation. But we often omit the reference to g', h, V', D, m, g, h . .. in the notation. The justification for doing this in the Sobolev case is the symmetry of our uniform structure. Now
To estimate
n
2
L I~ I k=l x g,m
more concretely we assume that
xl, ... ,xn are normal coordinates with respect to g, i.e. we assume a (uniformly locally finite) cover of M by normal charts of fixed radius::; rinj(M, g). Then 1~1;,m = 9 (~,~) = gkk (m), and there is a constant C 2 = C2 (R, rinj (M, g)) s. t. n
(
2) ~ ::; C
~ IVilh,m
2.
Using finally
IVx I ::;
IXI .
IV I,
we
obtain (2.19) (2.19) extends by the Leibniz rule to higher derivatives IVk1h lm' where the polynomials on the right hand side are integrable by the module structure theorem (this is just the content of this theorem). (2.18), (2.19) also hold (with other constants) if we perform some of the replacements 9 ~ g', V ~ V': We remark that the expressions D(g, h, V h,., D(g', h, vh, .) are invariantly defined, hence
[D(g, h, vh,·)
- D(g, h, V h, ·)](lu) = ((gik - g'ik)8k)· Vi(lu). (2.20)
Trace Class Properties
211
We have to estimate the kernel of
hp(W(t, m, p), r/(·)
(2.21)
in L 2 ((M, E), g, h) and to show that this represents the product of two Hilbert-Schmidt operators in L2 = L2((M, E), g, h). We cannot immediately apply the procedure as before since r/fP is not of zero order but we would be done if we could write (2.21) as (2.22)
710J
71~~ of first order,
tV
of zeroth order. Then we would replace by 710 (p)W(t, m,'p), apply k ~ r + 1 > n + 3, and obtain
710 W(t, m,')
E H~ (E),
IW(t, m, ')I H 2 ::; C(t)
(2.23)
and would then literally proceed as before. Let E C':' (U). Then
j (W(t, m, p), 71((p) (p))p dvolp(g) = j(((gik _ glik)8k). V'iW,IL2(M,E,dp) +IW(t, m,p, 1]~,oif>IL2(dp) C· (N1(if» + N2(if»).
(2.27)
Hence we have to estimate sup
N1(if» =
E Cg"(E) 1IL2 = 1
sup E Cg"(E) 1IL2 = 1
(2.28) and sup
N 2(if» =
E Cg"(E) 1IL2 = 1
sup E Cg"(E) 1IL2 = 1
(2.29) According to k > r
+ 1 > n + 3,
D(W(t, m, .), W(t, m,·) E H~ (E), I(D(W(t,m, ')I H2 , IW(t,m, ')I H2 and we can restrict in (2.28), (2.29) to sup E Cg"(E) 1IL2 = 1 IIH~ : O. Proof. The proof is a simple combination of the proofs of 1.1 and 2.1. 0 Now we additionally admit perturbation of the fibre metric h. Before the formulation of the theorem we must give some explanations. Consider the Hilbert spaces L 2 (g, h) = L 2 ((M, E), g, h), L 2 (g', h) = L 2 ((M, E), g', h), L 2 (g', h') = L 2 ((M, E), g', h') L~ and the maps
i(g',h},(g',h'} : L 2 (g', h) U(g,h},(gl,h} : L 2 (g, h) where dp(g)
D~2(9,h)
---t ---t
L 2 (g', h'), i(gl,h),(gl,hl} = L 2 (g', h), U(g,h},(g',h} = a~
= a(p)dp(g'). Then we set D~2 := U(g,h), (gl ,h) i(gl ,h}, (g',h') D' i(gl ,h),(g' ,hi} U(g,h},(gl ,h} == U*i* D'iU. (2.54)
i;
Here i* is even locally defined (since g' is fixed) and = dual"hlo i' odualh', where dualh((p)) = hp (" (p)). In a local basis field 1 , ... ,N, (p) = e(p)i(p), (2.55)
218
Relative Index Theory, Determinants and Torsion
It follows from (2.55) that for h' E COmp1,r+1 (h) i*, i*-l are bounded up to order k,
i* - 1, i*-l - 1 E n,0,I,r+I(Hom((E, h', 'Vh') ----+
----+
(M,g'), (E, h, 'V h) ----+ (M,g')))
(2.56)
and i* -1,i*-1 -1 E n,0,2,!:f-(Hom((E,h', 'Vh')----+ ----+
(M, g'), (E, h, 'V h )
----+
(M, g'))).
(2.57)
D' == D' is self adjoint on DD' = ego (E) I lv, , where JJ~, = JJi,2 + JD'Ji,·2 i: L 2 (g', h) ----+ L 2 (g', h') == L~ and i* : L 2 (g', h') ----+ L 2 (g', h) are for h' E compl,r+l(h) quasi isometries with bounded derivatives, they map er;:(E) 1-1 onto er;:(E) and i* D'i is self adjoint on ego(E)lli*D'i = Di*D'i C L 2((M, E), g', h) == L 2 (g', h). We obtain as a consequence that e- t(i*D'i)2 is defined and selfadjoint in L 2 ((M, E), g', h) = L 2 (g', h), maps for t> 0 and i,j E Z Hi(E,i*D'i) continuously into Hj(E,i*D'i) and has the heat kernel W~"h(t, m,p) = (o(m), e- t(i*D'i)2 o(p)), W'(t, m,p) satisfies the same general estimates as W(t, m,p). By exactly the same arguments we obtain that e-W*(i* D'i)2U = e-t(U*i* D'iU)2 = U*e-t(i* D'i)2U is defined in L2 = L 2((M, E), g, h), self adjoint and has the heat kernel W{ 2 (t, m, p) = W g' , h (t, m, p) = a-! (m)W~"h(t, m,p)a(p)!. Here we assume g' E compl,r+l(g). Now we are able to formulate our main theorem.
Theorem 2.9 Let E = ((E, h, 'V = 'Vh,.) ----+ (Mn, g)) be a Clifford bundle with (1), (Bk(M, g)), (Bk(E, 'V)), k ~ r + 1 > n+3, E' = ((E,h,'V' = 'V h',.') ----+ (Mn,g)) E gencompZ~d;;f (E)nCLB N ,n(1, B k), D = D(g, h, 'V = 'V h, .), D' = D(g', h, 'V' = 'V h' ,.') the associated generalized Dirac operators, dp(g) = a(p)dp(g'), U = a!. Then for t > 0 e -tD2 - U* e -t(i* D'i)2U
(2.58)
is of trace class and the trace norm is uniformly bounded on compact t-intervalls lao, all, ao > O.
219
Trace Class Properties
Proof. We are done if we could prove the assertions for e- t (UD'u*)2 _ e- t (i*D'i)2 = Ue- tD2 U* _ e- t(i*D'i)2 (2.59) since U*(2.59)U = (2.58). To get a better explicit expression for (2.59), we apply again Duhamel's principle. This holds since Greens formula for U D 2U* holds,
J
hq(UD 2U*if?, w) - h(if?, UD 2U*w) dq(g') = O.
We obtain t
-JJhq(a~(m)W(s,m,q)a-~(q), o M
(UD 2U*
+ %t) W~"h(t -
JJhq(a~
s,q,p)) dq(g') ds
t
=-
o
(m)W(s, m, q)a-~ (q),
M
(U D 2U* - (i* D'i)2)W~"h(t - s, q, p) dq(g')) ds
= a~ (m)W(t, m, q)a-~ (q) - W~"h,(t, m,p) = Wg',h(t, m,p) - W~"h(t, m,p).
(2.60)
(2.60) expresses the operator equation e-t(U DU*)2 -e -t(i* D'i)2
-J -J t
e- s(U*DU)2((UDU*)2 - (i*D'i)2)e-(t-S)(i*D'i)2 ds
o
t
e- s(UDu*)2 UDU *(UDU* - i* D'i)e-(t-S)(i*D'i)2 ds
o
(2.61)
J t
e-s(UDU*)\U DU* - i* D'i)(i* D'i)e-(t-s)(i* D'if ds.
o
(2.62)
220
Relative Index Theory, Determinants and Torsion
We write (2.62) as
-J t
a~e-sD2 Da-~(a~Da-~ - i* D'i)e-(t-S)(i*D'i)2 ds
o t
J = - Ja~e-SD2Da-~i*((i*-1 a 12 e- sD2 D a _12 (D
=-
-
'*D" ~
~
-
o
grad a.) e -(t-s)(i* D'i)2 ds 2a
t
- l)D
+ (D -
D')
o
_i*-l grad a· )e-(t-s)(i* D'i)2 ds 2a
Ja~e-sD2 t
=
D('r/o
+ 'r/l + 'r/2 + 'r/3 + 'r/4)e-(t-s)(i*D'i)2
ds,
o
'r/o
=
= -a-~i*'r/i(2), i = 1,2,3, 'r/l(2) = (2.10), = (2.11), 'r/3(2) = (2.12), 'r/4 = a-~i*-l(i* - l)D. Here grad3 a',
'r/i
2a2"
'r/2(2) 'r/o and 'r/2 are of zeroth order. 'r/l and 'r/3 can be discussed as in (2.18)-(2.53). 'r/4 can be discussed analogous to 'r/b 'r/3 as before, i.e. 'r/4 will be shifted via partial integration to the left (up to zero order terms) and a-~i*(i* -1) thereafter again to the right. In the estimates one has to replace W by DW and nothing essentially changes as we exhibited in (2.35). We perform in (2.62) the same decomposition and have to estimate 20 integrals, t
J 2
a~ e- sD2 D'r/ve-(t-S)(i* D'i)2 ds,
o t
2
J J
-(t-s)(i* D'i)2 ds, a 12 e -sD2 'r/v ('*D") ~ ~ e
o
t
a~e-sD2 D'r/v e -(t-s)(i*D'i)2 ds,
t
2
Trace Class Properties
J
221
t
a! e- sD2 'TJv(i*D'i)e-(t-s)(i*D'i)2 ds,
(Iv,4)
t
'2 1/ = 0, ... ,4 and to show that these are products of HilbertSchmidt operators and have uniformly bounded trace norm on compact t-intervals. This has been completely modelled in the proof of 2.1. 0
Finally we obtain Theorem 2.10 Assume the hypotheses of 2.9. Then for t > 0
is of trace class and its trace norm is uniformly bounded on compact t-intervalls lao, al]' ao > O.
o The operators i* D,2 i and (i* D'i)2 are different in general. We should still compare e- ti*D'2i and e-t(i* D'i)2 . Theorem 2.11 Assume the hypotheses of 2.9. Then for t > 0
is of trace class and the trace norm is uniformly bounded on compact t-intervalls lao, al]' ao > O.
Relative Index Theory, Determinants and Torsion
222
Proof. We obtain again immediately from Duhamel's principle ,2 e -ti* D i - e -t(i* D' i)2 =
J =- J J t
e- S(i*D '2 i)(i* D,2 i - (i* D'i)2)e-(t-s)(i*D i)2 ds
=-
1
=
o
t
e- S(i*D '2 i)i* D'(l- ii*)D'ie-(t-S)(i*D' i)2 ds
=
o
t
I2 e- S(i*D i)(i*D'i)C 1(1_ ii*)i*-1(i*D'i)e-(t-s)(i*D'i)2 ds.
=-
o
(2.63)
W:_
In [~, t] we shift i* D'i again to the left of the kernel
S
(i*D 2 i)
via partial integration and estimate I2 I2 (i* D' ie-~(i* D i») [( e-1W D i») f) U-1e-1(i*DI2i)i-1(1_ ii*)i*-l)] (( i* D' i)e-(t-s)(i* Dli)2)
as before. In [0, ~] we write the integrand of (2.63) as (e-S(i* D'2 i)i* D' i)[( (i*it 1e- t 4s (i* D'i)2 f-1) U e- t 4
8
(i* Dli)2) 1
(e- t;s (i* D ' i)2 (i* D'i))
and proceed as in the corresponding cases.
o
Theorem 2.12 Assume the hypotheses of 2.9. Then for t > 0
is of trace class and the trace norm is uniformly bounded on any t-intervall lao, a1], ao > o.
Proof. This immediately follows from 2.9 and 2.11.
0
223
Trace Class Properties
3 Additional topological perturbations Finally the last class of admitted perturbations are compact topological perturbations which will be studied now. Let E = ((E, h, \7 h) ~ (Mn, g)) E CLBN,n(I, B k) be a Clifford bundle, k 2:: r + 1 > n + 3, E' = ((E, h', \7h') ~ (Mm, g')) E compi:~dthrel(E) n CLBN,n(I, Bk)' Then there exist K c M, K' C M' and a vector bundle isomorphism (not necessarily an isometry) f = (JE,jM) E 15 1,r+2(EIM\K,E /IM'\K') s. t.
gIM\K and f'Mg'IM\K are quasi isometric,
(3.1)
hIEIM\K and f~h/IEIM\K are quasi isometric,
(3.2)
IgIM\K - f'Mg ' IM\Klg,l,r+1
0
and
2 - 2 e-tD P -e -tD' p'
(3.9)
2 - 2 e-tD D -e -tD' D-'
(3.10)
are of trace class and their trace norms are uniformly bounded on any t-intervall [aD, al]' aD > O. For the proof we make the following construction. Let V c M\K be open, M \ K\ V compact, dist(V, M \ K\(M\K)) 2: 1 and denote by B E L('H) the multiplication operator B = Xv. The proof of 3.2 consists of two steps. First we prove 3.2 for the restriction of (3.9), (3.10) to V, i.e. for B(3.9)B, thereafter for (1 - B)(3.9)B, B(3.9)(1 - B) and the same for (3.10).
226
Relative Index Theory, Determinants and Torsion
Theorem 3.3 Assume the hypotheses of 3.2. Then
B(e- tD2 P - e- uY P')B, B(e- tD2 D - e- uY [Y)B, B(e- tD2 P - e- tlY P')(1 - B), (1 - B) (e- tD2 P - e- uY P')B, B(e-
tD2
D - e- tiY 15')(1 - B),
(1 - B) (e- tD2 D - e- uY [Y)B, 2 tfy2 (1 - B) (e-tD P - eP')(1 - B), 2 tlY (1 - B) (e-tD D - e15')(1 - B)
(3.11) (3.12) (3.13) (3.14) (3.15) (3.16) (3.17) (3.18)
are of trace class and their trace norms are uniformly bounded on any t-intervall lao, all, ao > o. 3.2 immediately follows from 3.3. We start with the assertion for (3.11). Introduce functions n + 2. a) If "V'h E compl,r("V) C C~r(Bk)' "V' T-compatible, i.e. ["V', T] = 0 then
is independent of t. b) If E' E gencomptd~;j,rel(E) is T-compatible with E, ~.e. [T, X .']+ = 0 for X E T M and ["V', T] = 0, then
is independent of t. Proof. below.
a) follows from our IV 1.1. b) follows from our 1.2 0
Proposition 1.2 If E' E gencompi~d~;j,rel(E) and
T( e- tD2 P _ e-t(U*i* D'iU)2 P') T( e- tD2 D - e-t(U*i* D'iU)2 (U*i* D'iU» are for t > 0 of trace class and the trace norm of
241
Relative Index Theory
T( e- tD2 D - e-t(U*i* D'iU)2 (U*i* D'iU)) is uniformly bounded on compact t-intervals [aD, al], aD > 0, then
is independent of t.
Proof.
Let ('Pi)i be a sequence of smooth functions E ::s 'Pi ::s 'PHI and t->oo 1. Denote by Mi the multiplication operator with 'Pi on
C~(M \ K), satisfying sup Id'Pil ~ 0,
'Pi
~
°
t->oo
L2((M\K, EIM\K), g, h). We extend Mi by 1 to the complement of L 2 ((M \ K, E), g, h) in H. We have to show
e- tD2 P _ e-t(U*i* D'iU)2 P' is of trace class , hence
trT (e_tD2p - e -t(U*i*D'iU)2 p ') = lim trTMj (e- tD2 p - e- t(U*i*D'iU)2 P')Mj
.
J->OO
M j restricts to compact sets and we can differentiate under the trace and we obtain
Consider
There holds trT ( M j e-tD2 D 2M) j
= tr M j grad 'Pi' T D e-tD2 .
242
Relative Index Theory, Determinants and Torsion
Quite similar trT(Mj(e-t(i* D'i)2 (i* D'i)2)Mj ) = trTO+
+ ... + bo(D', m) + ....
2
(1.25)
247
Relative Index Theory
We prove in VI 1.1 and 1.2 Lemma 1.7
n
--2 -< i -< 1.
(1.26) D
indtop(D, D') :=
J
bo(D, m) - bo(D', m).
(1.27)
M
According to (1.26), indtop(D, D') is well defined. l,r+l (E) Theorem. 1 8 A ssume E ' E gencomPL,dijj,F,rel a) Then
ind(D, D', K, K')
J
J
K
K'
bo(D, m) -
+
J
bo(D', m)
+
(1.28)
bo(D, m) - bo(D', m).
M\K=M'\K'
(1.29)
b) If E'
E
gencompi~d~Jj,F(E) then ind(D, D') = indtop(D, D').
(1.30)
c) If E' E gen comptd~Jj,F(E) and inf (Je(D2) > 0 then
indtop(D, D') = indaD - indaD'.
(1.31)
Proof. All this follows from 1.1, the asymptotic expansion, (1.26) and the fact that the L 2-trace of a trace class integral operator equals to the integral over the trace of the kernel. D
Relative Index Theory, Determinants and Torsion
248
Remarks 1.9 1) If E' E gencompi~dt;f,rel(E), g and g', V'h and V',h, . and.' coincide in V = M \ L = M' \ L', L 2 K, L' 2 K', then in IV (2.5) - (2.53) a -1 and the rJ's have compact support and we conclude from IV (3.38), (3.39) and the standard heat kernel estimates that
J
IW(t, m, m) - W'(t, m, m)1 dm::; C . e-~
(1.32)
v and obtain ind(D, D', L, L') =
J
J
£
£'
bo(D, m) -
bo(D', m).
(1.33)
This follows immediately from 1.8 a). 2) The point here is that we admit much more general perturbations than in preceding approaches to prove relative index theorems. 3. inf O"e{D2) > 0 is an invariant of gencompi~dt;f,F(E). If we fix E, D as reference point in gencompi~dt;f,F(E) then 1.8 c) enables us to calculate the analytical index for all other D's in the component from indD and a pure integration. 4) inf O"e{D2) > 0 is satisfied e.g. if in D2 = V'*V' + R the operator R satisfies outside a compact K the condition R
~ "-0 .
id,
"-0
> O.
(1.34)
(1.34) is an invariant of gen compi,~t;f,F(E) (with possibly different K, "-0). 0 It is possible that indD, indD' are defined even if 0 E {Ye. For the corresponding relative index theorem we need the scattering index. To define the scattering index and in the next section relative (functions, we must now use spectral shift functions ~(A) which we introduced in III section 2. According to theorem 2.8 of
Relative Index Theory
249
chapter III, ~(A) == ~(A; A, A') exists if A, A' are self-adjoint and V = A - A' is of trace class. Then, with R'(z) = (A' - z)-l, ~(A) = ~(A,A,A') :=
1T-
1
Iimargdet(1 0:--->0
~(A)
exists for a.e. A E lR.
tr(A - A')
=
+ VR'(A + if))
(1.35)
is real valued, E L1 (lR) and
j ~(A) dA,
1~ILl::;
IA - A'h.
(1.36)
I
If I(A, A') is the smallest interval containing O"(A) U O"(A') then ~(A) = 0 for A ~ I(A, A'). Let
Q={f:lR----+lR
I
fELl
jli(p)l(l+lpl)
and
dp 0,
00
a) tr(e- tH - e- tH ') = -t J e-tA~(A) dA. b) For every rp
E
o Q, rp(H) - rp(H') is of trace class and
tr(rp(H) - rp(H')) =
j Rrp'(A)~(A) dA. I
c)
~(A)
= 0 for A < O.
o
250
Relative Index Theory, Determinants and Torsion
We apply this to our case E' E gencompi~d~Jf,rel(E). According to corollary 1.4, D and U*i* D'iU form a supersymmetric scattering system, H = D2, H' = (U*i*D'iU)2. In this case
e21ri((>",H,H') = det S('x),
J
where, according to II (2.5) and (2.6), S = (W+)*W- = S('x) dE'('x) and H~c = J,X dE'('x). Let Pd(D), Pd(U*i* D'iU) be the projector on the discrete subspace in 1i, respectively and Pc = 1 - Pd the projector onto the continuous subspace. Moreover we write
(U*i* D'iU)2 =
(H~+
:,_). (1.38)
We make the following additional assumption.
e- tD2 Pd(D), e- t (U*i*D'iU)2 Pd(U*i* D'iU) are for t > 0 of trace class.
(1.39)
Then for t > 0
is of trace class and we can in complete analogy to (1.35) define ~C('x,H±,Hd):=
-1f
lim argdet[l
e-+O+
+ (e- tH ± Pc(H±)
_e- tH'± Pc(H d )) (e-tHt± Pc(H d ) - e->..t - ic)-l]
(1.40)
According to (1.36),
J~C('x, 00
tr(e- tH ± Pc(H±)-e-tH'± Pc(Hd))
= -t
H±, Hd)e- t>.. d'x.
o (1.41) We denote as after (1.11) fy = D' in the case 'il' E compl,r('il) and fy = U*i*D'iU in the case E' E gencomptd~Jf,rel(E). The assumption (1.39) in particular implies that for the restriction
251
Relative Index Theory
of D and fy to their discrete subspace the analytical index is well defined and we write inda,d(D, fy) = inda,d(D) - inda,d(D') for it. Set (1.42) Theorem 1.11 Assume the hypotheses of 1.1 and {1.39}. Then nC(A, D, 15') = nC(D, 15') is constant and
ind(D, 15') - inda,d(D, 15')
= nC(D, 15').
(1.43)
Proof.
ind(D, 15')
2
-,2
tfT(e- tD P - e- tD P')
=
= trTe- tD2 Pd(D)P - trTe- tD ,2 Pd(D')P' + +trT(e- tD2 Pc(D) - e- tD ,2 Pc(D')) =
J 00
inda,d(D,D')
+t
e-tAnC(A, D, 15') dA.
o
According to 1.1, ind(D,D') is independent of _ t. _ 00 holds for inda,d(D, D'). Hence t
The same
J e-tAnC(A, D, D') dA
is inde-
o
00
pendent of t. This is possible only if
J e-tAnC(A, D, D') dA = t o
or nC(A, D, 15') is independent of A.
o
Corollary 1.12 Assume the hypotheses of 1.11 and addition-
ally
o
VI Relative (-functions, 7]-functions, determinants and torsion In this chapter, we apply our preceding considerations and results to the construction of relative zeta functions and related invariants. We will attach to an appropriate pair of Clifford data a relative zeta function, which is essentially defined by the corresponding pair of asumptotic expansions of the heat kernel. Therefore we must first consider such a pair of expansions.
1
Pairs of asymptotic expansions
Assume E' E gencompi~tJf,F(E). Then we have in L 2 ((E,M), g, h) the asymptotic expansion trW(t,m,m)
rv
t-O+
r~b_!!(m) +r~+lb_!!+l 2
2
+...
(1.1)
and analogously for tm-! (m)W'(t, m, m)a! (m) = tr W'(t, m, m) with b_~+l(m)
b~%+l(m)
= b-~+I(D(g, h, \7), m), = b-%+I(D(g', h, \7'), m).
Here we use that the odd coefficients vanish, i.e. terms with r~+!, r%+~ etc. do not appear. The heat kernel coefficients have for l ~ 1 a representation I
b-~+l =
k
LL k=l q=O il +i2+·+ik=2(l-k)
tr (\7i q + 1 RE ... \7ik RE)Ci),···,i k ,
(1.2)
where Cil, ... ,ik stands for a contraction with respect to g, i.e. it is built up by linear combination of products of the gi j , gij' 252
253
Relative (-functions
1
Lemma 1.1 b-~+l - b'-~+l E L (M, g), 0 ~ l ~
nt3.
Proof. First we fix g. Forming the difference b-~+l - b'-~+l' we obtain a sum of terms of the kind 'ViI Rg ... 'V iq Rg tr ['V iq +1 RE ... 'V ik RE
_ 'V /iq + 1 R,E ... 'V /ik R,EJCil, ... ,ik.
(1.3)
The highest derivative of Rg with respect to 'V g occurs if q = k, i1 = ... = i q - 1 = O. Then we have
(1.4)
('V 9 )21-2k Rg.
By assumption, we have bounded geometry of order :2: r > n+2, i. e. of order :2: n + 3. Hence ('V9)i Rg is bounded for i ~ n + 1. To obtain bounded 'V j RLcoefficients of [... J in (1.3), we must assume n +3 (1.5) 2l- 2 ~ n + 1, l < - -2 - . Similarly we see that the highest occuring derivatives of R E , R'E in [... J are of order 2l - 2. The corresponding expression is R E'V 21 - 2RE _ RIE'V/21-2 R'E = (RE _ R'E) ('V 21 - 2RE) + R'E ('V 21 - 2RE _ 'V 21 - 2R ,E ). (1.6) We want to apply the module structure theorem. 'V - 'V' E n1 ,1,r (Q~'l, 'V) = n1 ,1,r (Q~l, 'V') implies RE - R,E E n2,1,r-1. We can apply the module structure theorem (and conclude that all norm products of derivatives of order ~ 2l - 2 are absolutely Hence, integrable) if 2l - 2 ~ r - 1, 2l - 2 ~ n + 1, l ~ E (1.6) E L1 since R , R'E, Cil ...ik bounded. It is now a very simple combinatorial matter to write [... J in (1.3) as a sum of terms each of them is a product of differences ('Vi RE - 'V,i R'E) with bounded terms 'V j R E , 'Vd' R'E. We indicate this for an expression 'Vi RE'V j RE - 'V'i R'E'V /j R 'E , 'Vi RE'V j RE _ 'V'i RIE'V /j R'E = 'ViRE('V j RE _ 'V /j R'E) + 'ViRE'V j RE _ 'V/iRIE'V/j R'E
nt3.
= 'ViRE('VjRE - 'V/jR'E ) + ('ViRE - 'V/iRIE)'V/jR'E. (1.7)
254
Relative Index Theory, Determinants and Torsion
The general case can be treated by simple induction. Remember \7, \7' E CE(Bk). Admit now change of g. We write \7ilR9 . .. \7 iq R9tr(\7i q + 1 RE ... \7i k RE )Ci l, ... ,ik as R}(g)trR2(h, \7)C(g), similarly R 1 (g')trR 2(h, \7')C(g'). Then we have to consider expressions Rl(g)trR2(h, \7)C(g) - R 1 (g')tr'R2(h, \7')C(g')
= [R 1 (g) - R 1 (g')JtrR 2(h, \7)C(g) +R1 (g')trR 2(h, \7)[C(g) - C(g')J +R1 (g)tr[R 2(h, \7) - R2(h, \7')JC(g').
(1.8)
But each term of [R 1 (g) -R 1 (g')J and [R2(h, \7J - R2(h, \7') can be written as a product of terms of type (1.7) composed with bounded terms (bounded morphisms). The terms [C (g) - C (g') J are E Ll since g' E COmpl,r+l (g). Then the module structure theorem for P = PI = P2 yields again that the whole expression (1.8) is ELI. This proves lemma 1.1. 0
Lemma 1.2 There is an expansion tr ( e _W2 - e -t(U' D
1
U)2)
n r"2a_¥-
+ ... + r"2n+[!!H] a_¥-+[~] 2
3 +o(r¥-+[nt ]+I). Proof.
Set
a_¥-+i
=
J
(b_¥-+i(m) -
b~¥-+i(m))
(1.9)
(1.10)
dm
and use tr W(t, m, m)
n = r"2b_~ + ... + r"2n+[n+3] - 2 b_~+[nt3]
3 +O(m, r¥-+[nt ]+I), tr W'(t, m, m) = r¥-b~!! 2
(1.11)
+ ... + O'(m, r¥-+[nt
3
]+1)
(1.12) 1
tr ( e _tD2 - e-t(U' D U)2)
=
J
(tr W(t, m, m) - tr W'(t, m, m)) dm.
255
Relative (-functions
Using lemma 1.1, the only critical point is
j O(m,r~+[~J+l)-O'(m,r~+[~J+l)
dm
= O(r~+[~J+l).
M
(1.13) (1.11) requires a very careful investigation of the concrete representatives for O(m, C~+[~J). We did this step by step, following [38], p. 21/22, 66 - 69. Very roughly speaking, the m-dependence of O(m,·) is given by the parametrix construction, i. e. by differences of corresponding derivatives of the rfa' r'fa, which are integrable by assumption. D If E' E gencomp}~dt;f,F,rel(E) then we immediately derive from IV, theorem 3.2
is of trace class. The heat kernel and its asymptotic expansion split into its restriction to K and M \ K = V or to K' and M' \ K' = V, respectively. 2 (e-tD
+ Wg,h(t, m, m)lv, = W~I,h(t, m, m)lK' + W~I,h(t, m, m)lv 1,
P)(t, m, m) = Wg,h(t, m, m)IK '2
(e-t(u' D UP')(t,
m, m)
hence tr( e- tD2 P _
e-t(u' D '2 U) P')
= j W(t, m, m)dm - j W'(t, m, m)dm K
+ j(W(t,m,m) v
K'
W'(t,m,m))dm
256
Relative Index Theory, Determinants and Torsion
n (K) =r2a_~ ,g
+· .. +r2n +[.!!.±!.] 2 a_~_[~] (K ,g )
+O(r~+[~]+l)
n (K,g ") +r2a_~
, ') + ... + r"2n+[.!!.±!.] a_~_[~] (K,g 2
+O(r~+[~]+l)
n (V ') +r2a_~ ,g,g
') + ... + r2n+[.!!.±!.]a_~_[~] ( V ,g,g 2
+O(r~+[nt3]+1)
= r~(a_~(K, g) - a_~(K', g') + a_~(V, g, g')) () ( , ') +r"2n+[n+3] -2- a_~_[~] K, 9 - a_~_[nt3] K , 9 +a_~_[~](V, g, g')
+ O(r~+[~]+l),
+ ... (1.14)
where the ai (K, g) or ai (K', g') are the integral terms of the asymptotic expansion on K or K', respectively, and
ai(V, g, g') =
J
(bi(m) -
b~(m))dm.
(1.15)
v The existence of the integrals (1.15) follows from the proof of the lemmas 1.1 and 1.2. Hence we proved
Lemma 1.3 Suppose E' E gencomptd~Jf,F,rel(E). Then there exists an asymptotic expansion tr(e-tD2 P _ e- t (u*D /2 u) P') n n+[n+3] = r2L~ + ... + r2 -2- C_~+[nt3] +O(r~+[nt3]+1).
(1.16)
o
2
Relative (-functions
For a closed oriented manifold (Mn, g), a Riemannian vector bundle (E, h) - - - t (Mn, g) and a non-negative self-adjoint elliptic differential operator A : COO(E) - - - t COO(A) there is a
257
Relative C; -functions
well-defined zeta function
((s,A) =
1 ,V
L
(2.1)
)'Eu(A) ).>0
which converges for Re (s) > ~ and which has a meromorphic extension to C with only simple poles. In particualar, s = 0 is not a pole. Hence
is well defined and one defines the (-determinant of A as det (A :=
(2.2)
e-(,(O,A).
This is the first step to define analytic torsion. On open manifolds, (2.1) does not make sense since O"(A) is not nessecarily purely discrete. (2.1) can be rewritten as
rts) Je- (tre00
((s, A)
=
1
tA
-
dim ker A)dt.
(2.3)
o
But (2.3) has a meaningful extension to open manifolds as we will establish in this section. Definition. Assume E' E gencompi~d~}f,F(E). Define
rts) Je1
(1(S, D2, (U* D'U)2) :=
o
1
tr (e- tD2
-
e- t (U*D ' U)2) dt.
(2.4)
We insert the expansion (1.9) into the integrand of (2.4), thus
258
Relative Index Theory, Determinants and Torsion
obtaining 1
tS-lr~tldt =
J
1
o
(2.5)
-!!:2 +l'
8
1
Jo e-lr~+[~l dt
1
= 8
+!!:2 + [n+3] , 2
1
_1_
f(8)
Je-lo(r~+[~l+l)
dt holomorphic for
o
Re (8)
n
n+3
+ (-2") + [-2-] + 1 >
°
(2.6)
and [nt3] ~ ~+1, we obtain a function merom orphic in Re (8) > -1, holomorphic in 8 = with simple poles at 8 = ~ - l, l ~
°
[nt3].
00
Much more troubles causes the integral
J.
Here we must addi-
1
tionally assume
(2.7) (2.7) implies l1e(D,2) = inf o"e((U* D'U)21(ker(U*D1U)2)-L) > 0. Denote by 110(D 2), 110(D,2) = 110((U* D'U)2) the smallest positive eigenvalue of D2, D,2, respectively and set I1(D2) min{l1e(D2), 110(D 2)}, I1(D,2) min {l1e( D,2), 110 (D,2)}, 2 I1(D , D,2) ._ min{I1(D2) , I1(D,2)} > 0.
(2.8)
If there is no such eigenvalue for D2 then set I1(D2) = l1e(D2), analogous for D,2. D2, D,2, (U* D'U)2 have in ]0, I1(D 2, D,2)[ no further spectral values. We assert that the spectral function ~(A) = ~(A, D2, (U* D'U)2) is constant in the interval [0,I1(D 2,D,2)/2[.
259
Relative (-functions
Consider the function We: (X) = { ce:e -
e{~x2 Ixl S E
o
Ixl > E
and choose
Ce: s. t. J We: (x) dx = 1. Let 0 < 3E < % and X[-6-2e:,H2e:] the characteristic function of [-0 - 2E, 0+ 2E]. Then 'P6,e: := X[-6-2e:,H2e:] * We: satifies 0 S 'P6,e: S 1, 'P6,e:(X) = 1 on [-0 - E, + E], 'P6,e(X) = 0 for x ~] 3E, + 3E[, 'P' ,e: S K· E- 1 and e:~06~O lim lim 'P6,e: = 0- distribution. Assume + 3E < ~. A regular distribution fED' (] - ~,~ [) equals to zero if and only if (f()..) , we:().. - a) sin(k()" - a))) = 0, (f()..), we:().. - a) cos(k()" - a))) = 0 for all sufficiently small E and all a (s.t. lal + E < ~ ) and for all k. ([71], p. 95). This is equivalent to (f, we:().. - a)) = 0 for all sufficiently small E and a and the latter is equivalent to (f()..) , 'P6,e: - 'P61,e:) = 0 for ~ll 0,0' andallE(s.t. 0+3E,O'+3E< ~). Wegetforf-L=f-L(D 2,D' ),0< 3E < %, 0+3E < ~ that 'P6,e:(D 2) -'P6,e:((U* D'U)2) is independent of 0, E, tr('P6,e(D2) - 'P6,e((U* D'U)2)) is independent of 0, E, 0 = tr( 'P6,e:(D 2) -'P6,e( (U* D'U)2)) -tr( 'P61,e:(D2) -'P61,e:( (U* D'U)2)) =
°
°- °
°
I± 2
J('P6,e: - 'P61,e:)'()..)~()..) d).., i.e. the distributional derivative of ~ o equals to zero, ~()..) is a constant regular distribution. We write ~()..)I[o,~[ = tr('P6,e(D2) - 'P6,e:((U* D'U)2)) == -h. Set quite parallel to [49] ~()..) := ~()..) + h which yields ~()..) = 0 for)" < ~ 00
00
and -t J e-t>'~()..) d)" = h - J e-t>'~()..) d)". The latter integral o ~ converges for t > 0 and can for t ~ 1 estimated by
J1~()..)le-ti 00
e-t~
d)" S
Ce-t~.
I± 2
Hence we proved
Proposition 2.1 Assume E' E compYdtf1fF(E), inf O"e(D2 2 " 2 l(kerD2).L) > 0 and set h = tr('P6,e(D ) - 'P6,e((U* D'U) )) as above. Then there exist C > 0 s. t. tr(e- tD2 _ e-t(u' D U)2) = h + O(e- ct ). (2.9) 1
260
Relative Index Theory, Determinants and Torsion
o Define for Re (s) < 0
rts) J 00
(2(S, D2, D,2) :=
t S - 1 [tr(e- tD2 - e- t (U*D'U)2) - h] ds.
1
(2.10) Then (2 (s, D2, D,2) is holomorphic in Re (s) < 0 and admits a meromorphic extension to C which is holomorphic in s = o. Define finally
rts) Je00
((s,D 2,D,2):=
1
[tr(e- tD2
-
e- t(U*D,U)2) - h] dt
o 1
= (l(S, D2, D,2)
+ (2(S, D2, D,2) -
rts) J
+ (2(S, D2, D,2) -
r(s:
t S - 1 h dt
o
= (l(S, D2, D,2)
1)
(2.11)
We proved
Theorem 2.2 Suppose E' E gencompi~d~;f,F,rel(E), inf O"e(D2 l(kerD2).L) > 0 and set h as above. Then ((s,D 2,D,2) is after meromorphic extension well defined in Re (s) > -1 and holomorphic in s = O. 0 We remark that (U* D'U)2 = U* D,2U, but (U*i* D'iU)2 =I-
U*i* D,2iU. The situation is more difficult if there is no spectral gap above zero. Then tr( e- tD2 - e-t(u* D'U)2) must not - up to a constant - exponentially decrease to zero. But one can ask, whether there exists for t ----+ 00 an asymptotic expansion in negative powers of t. The answer is yes, if the spectral function ~(A) has an asymptotic expansion in postive powers of A near A = 0 (cf. [49]). We perform our considerations at first for the general situation of a supersymmetric scattering system H, H' such that for t > 0 e- tH - e- tH' is of trace class.
261
Relative (-functions
Proposition 2.3 Suppose that for an interval [0, c] there exists u sequence o :S 'Yo < 'Yl < 'Y2 < ... ---+ 00 such that for every N E 1N N
~(,\)
= LCixYi +O(XYN +1 ).
(2.12)
i=O
Then there exists for t ---+ 0 an asymptotic expansion 00
tr( e -tH - e -tHI)
f'V
L
C'Yi b'Yi .
i=O
Proof. According to II (2.17),
J~('\)e-t>'d'\ 00
tHI tr(e- tH - e- )
=
-t
o -t
For t
~
[!
1
';(A)e-tAdH l';(A)e-tAdA
1
J~('\)e-t>'d'\ 00
:S C . e- te / 2
e
and
J,\ 00
o
,
'Y e-t>'d,\
~ t'Y+l
J,\ 00
-
'Y e-t>'d,\
e
+ O(e- te / 2 )
~ + O(e- te / 2 ) t'Y+ 1 o
262
Relative Index Theory, Determinants and Torsion
Looking at proposition 2.3, there arises the natural question, under which conditions an expansion (2.12) is available. We assume Hand H' self-adjoint, 2': 0 and e- tH - e- tH' for t > 0 of trace class. Following [49], we can now establish Proposition 2.4 Suppose ]0, c[e aac(H'), ~(A) continuous in ]0, c[ and that there exists mEN such that for A E]O, c[, det S(A) extends to a holomorphic function on Dc,m, where Dc = {z E Clizi < c} and Dc,m ---t Dc is the ramified covering given by z ---t zm. Then there exist c1, 0 < c1 < c~, and an expansion 00
~(A) =
L CkA~,
A E]O, cd·
(2.13)
k=O
Proof.
We have by assumption that det S(A) is smooth on
]0, cr. I det S(A)I = 1 and ]0, c[e aac(H') imply that there exists cP E Coo(]O, cD such that det S(A) = e- 27ficp (A) for A E]O, cr. We know from II (2.8) det S(A) = e- 27fi((A). Hence there exists k E Z such that ~(A) = cp(A) + k, A E]O, cr. This implies ~(A) smooth on ]0, cr. Differentiation of det S(A) = e- 27fi((A) and the unitarity of S(A) yield
d~(A) = __ 1 dA
27ri tr
(s*( ') dS(A)) /\ dA .
(2.14)
We infer from the assumption Dc,m ---t Dc,m and ( ---t )-extension of det S(A) that lim cp(A) exists, hence lim ~(A) too. Thus we A-+O+
A-+O+
obtain from (2.14) A
~(A)
~(O+) - ~Jtr (s*(a)dS(a)) 27r2
da
o
da
.1.
~(O+) - 2~i ]
o
tr (s*(a
m
m )
dSd:
))
da. (2.15)
Relative 00
an
Relative (-functions
265
asymptotic expansion 00
tr(e- tH - e- tH') '"
L ajt
CXj
,
-00 < ao < al < ... -+ 00.
j=O
(2.21) Then (2.20) defines a relative (-function which is meromorphic inC and holomorphic in s = O.
Proof. We start with
J 00
(l(S, H, H') =
ts-1tr(e- tH - e-tH'dt.
(2.22)
o
Inserting (2.21) into (2.22) yields as in (2.5) a meromorphic extension with simple poles at s = -aj which is holomorphic at s = 0 (since we multiply with f(s))' According to (2.19),
J 00
/" (S,' H H') =
':,2
tS-1tr(e- tH - e-tH'dt
JtS-l~(O+)dt + J 00
__1_
f(s)
00
_1_
tS-10(C-ln)dt.
f(s)
c
c
(2.23) Here the first integral is absolutely convergent in the half-plane Re( s) < 0 and equals there to s . f(s)
f -,-;-----=
s(~+'Y+s.h(s))
1 + s . 'Y . s . h( s) ,
which admits a meromorphic extension toC. The second integral yields a holomorphic function in the half-plane Re( s) < ~. The meromorphic extension to C is given by integration of (2.16) D with simple poles at s = i3k, k > O.
266
Relative Index Theory, Determinants and Torsion
Remark 2.8 We see from the proof of theorem 2.7 that we only need (2.18) to have a well-established (2, i.e. the stronger condition (2.21) is not necessary for that. Moreover, using tr(e- tH - e- tH')
1
00
= -t
e-tAe(,\)d,\,
we see that
tr(e- tH - e- tH') = bo + O(r{!) for t
-----t
00
(2.24)
is equivalent to
e(,\) = -bo + O('\{!) for ,\
~ 0+.
(2.25) D
Corollary 2.9 Suppose e- tH -e-tH' of trace class, (2.21) and (2.18) or (2.24). Then (2.20) defines a relative ( -function which is meromorphic in the half-plane Re( s) < {! and holomorphic in s = 0, which is explicitely given by
J 00
I bo 1 (2(s,H,H)=-qs+1)+qs)
(-tH -e -tH') -bodt. ) t S-I( tre
1
We apply this to our Clifford bundle situation.
Theorem 2.10 Let E E CLBN,n(I, B k ), k ~ r + 1 > n + 3, EI E gen comp~4t;f,F,rel(E) n CLBN,n(I, Bk)' Suppose additionally 10, E [c (J ac (D' ) and det S (,\) extends to a holomorphic function on De,I' Then (2.20) defines a relative (-function ((s, D2, (U* D ' U)2) which is meromorphic in C and holomorphic in s = O.
Proof. According to IV, theorem 3.2 for t > 0 e-tD2 P e-t(u* DU)2 pI is of trace class. Hence the wave operators are defined, complete and Sand are well-defined. Lemma 1.3 yields an asymptotic expansion of type 2.21 and we get (1 (s, D2, (U* D* U)2) as in (2.22) and thereafter. The assumptions provide (2.18) and we obtain (2(8, D2, (U* D ' U)2) as above. D
e
Relative (-functions
3
267
Relative determinants and QFT
It is well-known that the evolution of a quantum system is described by the S-matrix of QFT. For the elements of this matrix there exist well-known formulas, given by the Feynman path integrals. The exact mathematical understanding and meaning has occupied up to now many mathematicians. There are several approaches. One essential part of these path integrals is the so-called partition function which can be written (perhaps after a so-called Wick-rotation) as
z:=
J
e-S(A) dA.
(3.1)
Here S(A) is an action functional and A runs through an infinite dimensional space, e.g. a space of connections (in gauge theory), a space of Riemannian metrics and embed dings (in string theory). In many cases S(A) is of the kind (HA, Ah 2 , where H is an elliptic self-adjoint non-negative differential operator. The model, how to calculate (3.1) is now the Gaufi integral (3.2)
This yields a hint, how to attack, better to define the integral in the infinite dimensional case. One simply replaces the determinant in (3.2) by the zeta function determinant, if the latter is defined. Suppose the underlying space M n to be compact. Then (J"(H) is purely discrete and one defines
det(H)
:=
det(H)
:= e-/.-((s,H)ls=o = e-('(O) ,
(3.3)
(
((8, H)
=
L
A-s
>.eCT(H)
>'>0
with meromorphic extension.
(3.4)
268
Relative Index Theory, Determinants and Torsion
In the case of string theory,
00
Z =
L
Zp, I:p closed surface of genus p.
p=O
If the underlying manifold Mn is open then (J(H) is not purely discrete and (3.3), (3.4) do not make sense. We are now able to rescue this situation by considering relative determinants, det(H, H') := e-('(O,H,H'). (3.5) 1 +1
- 2
If E', E" E gencomp/diff,F(E) then we denote as above D' = - 2 (U* D'U)2, D" = (V* D"V)2 for the transformed operators acting in L2((M, E), g, h). Theorem 3.1 Suppose E', E" E gen compt~tf1fF(E) and inf (J 2 ' , (D l(kerD2)J.) > O. -2 -2 -2 -2 a) Then ((s, D2, D' ), ((8, D2, D" ), ((8, D' ,D") are after meromorphic extension in Re (8) > -1 well defined and holo2 morphic in 8 = O. In particular det(D2, D,2) = e-('(O,D ,D'\ 2 det(D 2,D,,2) = e-('(O,D 2,I5 det(D,2,D,,2) = e-('(O,D,2, I5 11 ) are well defined. b) There holds
I1\
(3.6) etc. and
Proof. a) follows from 2.1 and the fact that E, E" E gencomp(E) implies E" E gencomp(E')(= gencomp(E)). tD2 b) immediately follows from the definitions and tr( ee- tJ5112 ) = tr(e- tD2 _ e- tD ,2) + tr(e- tD ,2 _ e- tI5l1 \ D
Relative (; -functions
4
269
Relative analytic torsion
If we now restrict to the case E = (A*T*M 0C,gA) then g' E gencompl,r+l(g) does not imply E' = (A*T*M 0C,g~) E gen comp};~dtJf,F(E) since the fibre metric changes, gA ----t g~. Hence the above considerations for constructing the relative (function are not immediately applicable, since they assume the invariance of the fibre metric. Fortunately we can define relative (-functions also in this case. We recall from [38], p. 65 - 74 the following well known fact which we used in (1.1), (1.2) already. Let P be a self adjoint elliptic partial differential operator of order 2 such that the leading symbol of P is positive definite, acting on sections of a vector bundle (V, h) ----t (Mn, g). Let Wp(t,p,m) be the heat kernel of e- tP , t > O. Then for t ----t 0+ trWp(t,m,m)
rv
r%_!j(m)
+ r!j+lb_!j+l(m) + ...
and the bv(m) can be locally calculated as certain derivatives of the symbol of P according to fixed rules. As established by Gilkey, for P = D. or P = D2 the b's can be expressed by curvature expressions (including derivatives). This is (1.1), (1.2), (1.3). We apply this to e-tL:;. and e-t(U*i*L:;.'iU) but we want to compare the asymptotic expansions of WL:;.(t, m, m) and WL:;.,(t, m, m). The expansion of (4.1)
and
(4.2) coincide since
The point is to compare the expansions of (4.4) and
(4.5)
270
Relative Index Theory, Determinants and Torsion
i.e. we have to compare the symbol of i* b.'i = i* b.' and b.'. For q = 0 they coincide. Let q = 1, m E M, WI, ... ,Wn a basis in T:nM, E nl(M), b.'lm = + ... + ~nwn' Then, according to IV (2.55) i*(b.'i)lm = gklg~k~iwl' i.e.
eWl
((i* - l)b.~ n + 2, g' E compl,r+l(g), l ::; b_~+l(b.(g, gA*), g, gA*, m) and b_~+I(U*i* b.'(g', g~* )iU, g, gA*) the coefficients of the asymptotic expansion of tr gAo W ~ (t, m, m) and tr gA * WU*i*~/iU(t, m, m) in L 2 (M, g), repectively. Then
b-~+I(b.,g,gA*,m)
Proof.
- b_~+I(U*i*b.'iU,g,gA*,m) E L1(M,g). (4.8)
Write
b-~+l(b.,
g, gAo, m) - b-~+I(U*i* b.'iU, g, gAo, m) = b-~+l(b., g, gAo, m) - b_~+l(b.', g', g~*, m) + +b-~+l(b.', g', g~*, m) - b-~+l(i* b.'i, g', gAo, m) + +b-~+l(i* b.'i, g', gAo, m) - b-~+I(U*i* b.'iU, g, gAo, m)
(4.9) (4.10)
(4.11)
where b-'!!:.+l(b.', g', g~*, m), b_'!!:.+l(i* b.'i, g', gA*) are explained in 2 2 (4.5), (4.4), respectively. (4.11) vanishes according to (4.3). (4.9) E L1(M,g) according to the expressions (1.3) and g' E compl,r (g). We conclude this as in the proof of 1.1. Finally (4.10) ELI according to i*b.'i = (i* -l)b.' + b.', coeff (i*b.'i) = coeff (i* - 1) + coeff (b.'), (i* - 1) E no,r(End (A*)), the rules for calculating the heat kernel expansion and according to the module structure theorem. 0
Relative (-functions
271
Theorem 4.2 Let (Mn,g) be open, satisfying (1), (B k ), k ~ r+1 > n+3, g' E compl,r+1(g),.6. = .6.(g,gA'),.6.' = .6.(g',g~,) the graded Laplace operators, U, i as in (2.54), and assume inf O"e (.6.1 (kerA).L) > O. a) Then for t > 0 e-tA -e -tU'i' A'iU is of trace class.
b) Denote h = tr(ip,5,c(.6.) - ip,5,e(U*i*.6.'iU)) for 0 < 3E < ~,
8 + 38
-1 which is holomorphic in s = O.
c) The relative analytic torsion r a(Mn , g, g') , 1
log ra(Mn, g, g') :=
n
2 2)
-l)qq . (~(O,.6., .6.')
(4.12)
q=O
is well-defined.
Proof. a) is just IV theorem 3.8. b) immediately follows from theorem 2.2 and the proof of theorem 2.2. c) is a consequence of b). 0 We defined in II 4 gencompz~t}f,rel(g) which induces at the level of A*T*M gencompZ~dt}f,rel(A*T*,gA)' According IV 3.9, for t > 0 e-tA P _ et(U'i' A'iU) p' is of trace class. Then we can apply the asymptotic expansion (1.16) and obtain as above relative (-functions
(q(s,.6., .6.')
= (q(x, (M, g, .6. 9 ), (M', g', .6.~)),
which are holomorphic at s = O. Hence we got
Relative Index Theory, Determinants and Torsion
272
Theorem 4.3 Let (Mn,g) be open, satisfying (1), (B k ), k 2: l,r+lJ,reZ (Mn) r > n + 3 ,g' E gen comp L,diJ , 9 an d suppose inf (Te(~I(ker~)l.) > O. Then there is a well-defined relative analytic torsion Ta((M,g), (M',g')),
logTa((M,g), (M',g')):=
~ t(-l)qq. (~(O,~,~').
(4.13)
q=O
o The last step ist to give up assumption inf (Te(~I(ker~)l.) > O. Theorem 4.4 Let (Mn,g) be open, satisfying (1), (B k ), k 2: r > n + 3, g' E gencomp;;4:fJ,rez(Mn,g) and suppose additionally ]0, E[e (Tac(~') and that det S(.\) extends to a holomorphic function on De,ffi' Then there is a well defined relative analytic torsion Ta((M,g), (M',g')), logTa((M, g), (M', g')) :=
~ t ( -l)qq(~(O,~, ~'),
(4.14)
q=O
where
(q(s,~,~')
is defined by (2.20) and theorem 2.10.
0
Remark 4.5 The assertion of theorem 4.4 remains valid if we replace the assumption det S(.\) extends to a holomorphic func0 tion on De,m by (2.18).
5
Relative 1]-invariants
Finally we turn to the relative 7]-invariant. On a closed manifold (Mn, g) and for a generalized Dirac operator the 7]-function is defined as
J 00
7]D(S)
:=
'" Lt
sign .\
1
~ = r (81 1 )
0
t
8-1 2
tr(De
-tD2
) dt.
A E cr(D)
AiO
(5.1)
273
Relative n, it has a meromorphic extension to C with isolated simple poles and the residues at all poles are locally computable. r (Btl) . 'rJD( s) has its poles at n+~-v for /J E IN. One cannot conclude directly 'rJ is regular at s = 0 since r(u) is regular at u = ~,i.e. r (stl) is regular at s = O. But one can show in fact using methods of algebraic topology that 'rJ( s) is regular at s = O. A purely analytical proof for this is presently not known (cf. [38], p. 114/115). (5.1) does not make sense on open manifolds. But we are able to define a relative 'rJ-function and under an additional assumption the relative 'rJ-invariant. De- tD2 is an integral operator with heat kernel DpWD(t,m,p) which has at the diagonal a well defined asymptotic expansion (cf. [16], p. 75, lemma 1.9.1 for the compact case) 'rJD(S)
(5.2) In [3] has been proved that the heat kernel expansion on closed manifolds also holds on open manifolds with the same coefficients (it is a local matter) independent of the trace class tD2 property. The (simple) proof there is carried out for e- , trW(t, m, m), but can be word by word repeated for De- tD2 , DW(t, m, m). The rules for calculating the b-n±l (D2, D, m) are 2 quite similar to them for b-n±l (D, m) (cf. [38], Lemma 1.9.1). 2 We sum up these considerations in Proposition 5.1 Let E' E gencompi~dt;f,F(E), r Then for t > 0
+ 1 > n + 3. (5.3)
is of trace class, for t
---t
0+ there exists an asymptotic expansion
tr( e- tD2 D - e-t(u* D'U)2 (U* D'U))
=
I: Ja-~±l /=0 M
(m) dvolm(g)t
-~±l + O(d).
(5.4)
274
Relative Index Theory, Determinants and Torsion
Proof. The first assertion is just IV theorem 2.9. We recall from [38] the existence of an asymptotic expansion for the diagonal of the heat kernel
2D 1 D- , = (-tD e - e -t1512 15 ) (m,m )
=
~[2 ~ b_!!fl (D ,D, m) -
-2 - ]
b_!!fl (D' ,D', m) C
ill 2
+ O(t2). 4
1=0
(5.5) It can be proved, absolutely parallel to lemma 1.1, that [ ] E L 1 , O(d) ELI. Integration of (5.5) yields (5.4). D We recall from [49] the following
Proposition 5.2 Assume that D and fy = U*i* D'iU satisfy
(5.3), (5.4) and that the spectra of D and D' have a common gap [a, b], (CJ(D) U (J(fY)) n [a, b] = 0. Then there exists a spectral shift function ~(A) = ~ (A, D, D') having the following properties. 1) ~ E L 1,loc(lR) and ~(A) = 0 for A E [a, b]. (5.6) 2) For all
0 s. t. (J(D) U (J(i* D'i) n ([-/1, -~] U [~, /1]) = 0 for all
275
Relative C; -functions
1/
~
•
I/O.
Hence, accordmg to (5.5),
J ft(>.e- tA )~(>.) d>' = 0 and J1,
2
-J1,
-00
J 00
+
e-
tA22
11_ 2t>'II~(>')1 d>.]
J1,
= C· e -t!!c"2. o
Theorem 5.4 Assume E' E gencompi~d~}f,F(E), k ~ r + 1 > n + 3 and inf o"e( D21 (ker D2).l.) > O. Then there is a well defined relative ",-function 00
'1l(s D D'):= 'I
"
r
1
Jt
(S~l) 0
S
;l
tr(De- tD2 _U*D' Ue- t (U*D ' U)2) dt
(5.10) which is defined for Re (s) > ~ and admits a meromorphic extension to Re (s) > -5. It is holomorphic at s = 0 if the coefficient J a_!(m) dvolm(g) of t-! equals to zero. Then there is a well 2 defined relative ",-invariant of the pair (E, E') ,
Proof.
We write again U* D'U
= fy. Then according to
276
Relative Index Theory, Determinants and Torsion
proposition 5.1, 00
~(s, D, fy) ~ r (~) [
I
[1',' tr( e~w' D -
1
e- HY ' 15') dt
00
+r (~) 1
t ',' tr(e-
W
'
1
n+3
D - e-H'''fj')dt
J
= r (s+1) s n I 1 a_!!±l dvolm(g) - 2 '" ~---+-+2 1=0 2 2 22M
J 1
+r
1
(s~1)
0
8-1
4
t-2 O(t2)dt
(5.11)
(5.12)
We infer from (5.9) that (5.13) is holomorphic in C. (5.12) is holomorphic in Re (s) > -5. (10.45) admits a meromorphic extension toC. T/(s, D, D') is holomorphic at s = 0 if the coefficient J a_! (m) dvolm(g) equals to zero. 0
M
2
Theorem 5.4 immediately generalizes to the case of additional compact perturbations. Theorem 5.5 Assume E' E gencompi~dtJf,F,rel(E), k ~ r+1 > n+3
Relative r:;, -functions
277
Then there is a well defined relative 'T/-function
J 00
D fy) :=
71(S '1
"
r
e! 1
1
(;1 tr(De- tD2 P - fYe- tfy2 Pl)dt
) 0
'
which is defined for Re (s) > ~ and admits a meromorphic extension to Re (s) > -5. Here fy = U* D'U as above. 'T/(s, D, fy is holomorphic at s = 0 if the integrated coefficient a_!2 =
-J
J
b_!2 (D2, D, m) dvolm(g)
K
b_ frac12 (fy2, fy, m/) dvolm, (g')
K'
+
J
M\K=M'\K'
equals to zero. We repeat for the proof the single arguments from the proof of 5.4 which remain valid in the case of 5.5. 0
6
Examples and applications
In this section, we present examples of pairs of generalized Dirac operators which satisfy the assumptions of sections 1-5 and present applications of some theorems of these sections. Let (Mn,g) be open with finitely many collared ends Ci, the collar [0, oo[ XN;-l of Ci endowed with a warped product metric, i.e. glC:i ~ dr2 + fi(r)2da}vi' Ni closed, hi = da7vi, i = 1, ... ,m. We consider one end C with collar [0, oo[xN with the warped product metric ds 21c: = dr2 + f(r)2da 2 and we first calculate the curvature. Let Uo, U1 , ... , Un - 1 be an orthogonal basis in T(r,u) ([0, oo[xN) with respect to ds 2, U = :" U1 , ... , Un - 1 orthonormal in TuN
278
Relative Index Theory, Determinants and Torsion
with respect to d(J"2 = h. Then, in coordinates (r,u\ ... ,un-I), we get for the Christoffel symbols r~,,8(g), Ct, (3, '"Y = 0, ... , n -1, the following expressions
rgo = 0,
rg
c k > 0, r k00 = 0 lor
r OJk
r?j = -1' fh ij for i, j
> 0,
j
= 0 for --
j
> 0,
!Ls:k c . k 2 f' U j lor J,
(6.1)
> 0,
r~j = r~j(h) for i, j, k
>
o.
For the curvature tensor and the sectional curvature holds
1"
R(Uo, Ui)UO = jUi
(6.2)
R(Uo, Ui)Uj = -1" fhijUO (6.3) R(Ui , Uj)Uo = 0 (6.4) R(Ui , Uj)Uk = - f'2(h jk Ui - hikUj ) + RN(Ui , Uj)Uk , (6.5) which implies immediately
(6.6) (6.7) Here i, j, k = 1, ... ,n - 1. The easy calcualtions are performed in [28]. It is now easy from (6.2) - (6.7) to calculate the general curvature K(V, W).
Examples 6.1 1) Take f(r) = e- r , N fiat, then K == -1, E satisfies (Bo) but rinj(E) = O. 2) Choose f(r) = e- r , KN =1= 0, then E does not satisfy (Bo) and again rinj(E) = O. 3) If f(r) = er , N fiat, then E satisfies (Bo) and rinj(E) > O. 4) Finally take f(r) = er2 , then E does not satisfy (Bo) but (I). Hence all good and bad combinations of properties are possible. D
279
Relative (-functions
Proposition 6.2 Suppose f(r) such that
inf f(r) > 0 or f monotone increasing
(6.8)
If(lI)1 ::; cllf, v = 1,2, ....
(6.9)
r
and Then 9 Ie satisfies (/) and (Boo).
Proof. rinj(C:)
inf f (r) > 0 and
h) > 0 immediately imply > O. (6.6) and (6.7) immediately imply (Bo). (Bd is r
rinj (N,
equivalent with (B o) and
IV'eJR(eiJ,e),)e,,)lx::; C IV'e",eiJlx ::; c,
(6.10) (6.11)
eo, ... ,en-l tangential vector fields, orthonormal in TxM. We apply this to x = (v,y) E [O,oo[xN, Uo = Ui = a~i' eo = Uo, Un-I Then accor d'mg t 0 (6 . 1) , el -- UI f ' ... , en-l -- -1-'
tr,
V'ei eO
We see, each term on the r.h.s. of (6.12) - (6.15) is - up to a constant or bounded function - a sum of terms
l'
l'
1
jei, jeo, 7el
(6.16)
with pointwise norms (w.r.t. gle)
I'll'7'
Ij
(6.17)
280
Relative Index Theory, Determinants and Torsion
i.e. (6.11) is satisfied. Next we establish (6.10)
f 1'" - ~f' 1"
P
(6.18)
ei,
1
P "Vuj(R(Uo, Ui)Uo) =
1" 1 p "VUjjUi
f"
k J3 "VUjUi = f J3" (-,f fhijUO+ rij(h)Uk)
1" f' 1" k -yhijeO + p rij(h)ek,
(6.19)
1
"V Ua pR(Uo, Ui)Uj ) -
~' R(Uo, Ui)Uj ) + )2 "V Ua ( - 1"fhijUO)
-
~' (-1"fhijUO) + )2 (-1"'f -
2f'1" (fill phijeO- T (
1"f')hijUO
1"f') +Y hijeo
f'1" Y - Tfill) hijeo,
(6.20)
)3 "Vuk(R(Uo,Ui)Uj) = )3 "VUk (-f"fhijUO) 1"
I
- phijrkOUl I1"f'
1"
If'
I
= - phij '2j[)kUl
-'2phij Uk ,
(6.21)
Relative (-functions
1
J2 "VUo(R(Ui , Uj)Uo) =
281
0
1
J3 "Vuk(R(Ui, Uj)Uo) "Vek(R(ei,ej)eO),
(6.22)
r1 "Vul(R(Ui, Uj ), Uk) =
1 "V Ul [/2( f4 - f hjkUi - hikUj
+
RN(Ui , Uj)UkJ
1'2
-]4(hjk "VUIUi - hik "VUIUj )
+
1 f4 "Vul(RN(Ui, Uj)Uk)
1'2
-]4(hjkrr:(h)Um - hikrlj(h)Um)
+
r1 "Vul(RN(Ui , Uj)Uk).
(6.23)
If we take the pointwise norm of the r.h.s. of (6.18) - (6.23) and apply the triangle inequality, then we obtain on the r.h. sides a finite number of terms, each of which is - up to a constant or a bounded function - of the type
1
If(Vl)I·lf(V2)1
fa
J2
a 2: O.
(6.24)
But according to (6.9), each term of the kind (6.24) is bounded on c, i.e. we established (Bl)' To establish (B2)' we have at the end to estimate expressions of the kind
(j).(j)',
( f').~ f f' (6.25)
282
Relative Index Theory, Determinants and Torsion
Again, according to (6.9), each term of the kind (6.25) is bounded. A very easy induction now proves (Bk) for all k, i.e. (Boo). 0 Collared ends are isolated ends. Hence, if all ends of an open manifold Mn are collared, then Mn can have only a finite number of ends. If an open manifold has an infinite number of ends, then at least one end is not isolated.
Theorem 6.3 Let (Mn,g) be open. If each end E of Mn is collared then M has only a finite number of ends, El,.··, Em. Suppose glc; ~ dr2 + Jl(r)dO"Jv; such that each fi satisfies {6.8} and {6.9}. Then (Mn, g) satifies (1) and (Bo). This follows immediately from proposition 6.2. 0 Interesting examples for the fs are f(r) = e9 (r) , g(r) > 0 and g(v)(r) bounded for all v. We consider here the special case g(r)
= b· r, b> 0 i.e. (6.26)
In the sequel, we need the knowledge of the essential spectrum O"e of such manifolds.
Theorem 6.4 Suppose (Mn,g) has only collared ends Ei, i = 1, ... , m, each of them endowed with a metric of type {6.26}. Then there holds
O"e(t:.q(Mn , g)) \ {O}
~ [m,m ( min { (n - ~q 00 [ \
{O} for q =I ~
1)\~, (n - ~q + 1) \; }) , (6.27)
and (6.28)
Relative (-functions
283
We refer to [3J for the proof which essentially relies on [28J.
0
Corollary 6.5 Suppose the hypotheses of 6.4, n even and minb~ ~
> O.
(6.29)
Then the graded Laplace operator D. = (D.o, ... ,D.n ) has a spec0 tral gap above zero.
A special case of theorem 6.4 is the case of a rotationally symmetric metric at infinity, i.e. (Mn \ K M, gIM\K) ~ (lRn \ KlRn, dr2 + e2br dO'~n-l)' Then for b > 0
for q -:F
n
2'
If we replace e2br by (sinh 1')2 and set K real hyperbolic space H:!:.l and 0'
e
(D. (Hn )) = q
-1
(6.30)
= 0, then we get the
(~)2b2} [ [{( ~)2 2' 2 ' 00 } [1 [ { {O U 4,00
for q -:F % for q = % (6.31)
Corollary 6.6 In the case (Mn,g) ~ (lR 2k ,dr2 + e2brd~n_l) or (Mn, g) = H~1, the graded Laplace operator D. = (D.o, ... , D. n ) has a spectral gap above zero. 0 Corollary 6.7 In the following cases, the graded Laplace operator D. = (D.o, ... ,D.n ) has a spectral gap above zero. a) (Mn,g) is a finite connected sum of manifolds with collared ends, warped product metrics {6.26} satisfying {6.29} and n is even, b) any compact perturbation of manifolds of a}, c) any finite connected sum of manifolds of type (Mn \ K M, gIM\K) ~ (lRn \ KlRn,dr2e2brdO'~n_l)' b> 0 and n even,
284 d) e) f) g)
Relative Index Theory, Determinants and Torsion any compact perturbation of c), any finite connected sum of the hyperbolic space H'!.-1, any compact perturbation of e), any (M2k, g'), g' E comp~~+1(g), 9 of type a) - f).
0
Remark 6.8 Compact perturbations and connected sums of collared manifolds with (6.26) are again of this type. We introduced 6.6. a), b) to indicate how to enlarge step by step a given set of such warped product metrics at infinity by forming connected sums and compact perturbations. 0 We apply the facts above to the case E
=
(A*T* M ®C, gA*), V 9A *), D
= d + d*, D2 = D. = (D.o, ... ,D. n ).
Theorem 6.9 Let (M2k,g) be one of the manifolds 6.6 a) - f), g' E comp~~+l(g), g' smooth, r + 1 > 2k + 3. Then the relative (-function (q(s, Do, Do') as in section 4 and the relative analytic torsion, Ta((M, g), M', g')),
2k
log Ta((M, g), (M', g')) =
2:) -l)qq· (~(O, Do, Do') q=O
are well defined.
Proof. According to proposition 6.2, (Mn,g) and (M',g') satisfy (1) and (Boo), and the general Laplace operator has a spectral gap above zero. The assertion then follows from theorem 4.4. 0
Corollary 6.10 Let (M,g) be as is 6.6 a) - g). Then the attachment (M,2k, g') ----t Ta((M, g), (M', g')) yields a contribution to the classification of the elements of gen compi~~Jf,rel (A*T*,gA*)'
285
Relative (, -functions
Remark 6.11 If n = 2k + 1 and (Mn, g) belongs to one of the classes 6.6 a) - g) then the relative (-functions (( s, !::l.q, !::l.~) are for (Min, g') E gencompi~dtJf,rel(Mn, g) n C r +l and q 1= k, k + 1 well defined. D
Another very special case is given by b = 0 in (6.26), i.e. cylindrical ends E, (6.32) ds 2 = dr 2 + da'iv. Suppose, we have m cylindrical ends
Ei,
i
= 1, ... , m,
and let {Xl (i)} k be the (purely discrete) spectrum of !::l.q (Nr- l , hi), i = 1, ... , m. Proposition 6.12 Then
ae(!::l.q(M, g» = U U([Ak(i), OO[U[Arl(i), oo[). i
(6.33)
k
We refer to [3] and [28] for the proof.
D
Corollary 6.13 a) If Hq(Ni ) = Hq-l(Ni ) = (0), i = 1, ... , m, then a(!::l.q(M)) has a spectral gap above zero. b) If for at least one i Hq(Ni ) 1= 0, then a(!::l.q+l(M)) = a (!::l.q(M) = [0,00[. e) In the case of cylindrical ends, the graded Laplace operator never has a spectral gap above zero. Proof. a) and b) immediately follow from (6.34). c) follows from HO(Ni ) 1= 0 for all i, hence a(!::l.o(M)) = a(!::l.l(M)) [0,00[= a(!::l.o, ... ,!::l.n). D
Corollary 6.14 Suppose (Mn, g) with cylindrical ends El,'" ,Em and let (M,n,g') E gencompL,diff,rel(Mn, g). If Hq (Ni ) = Hq-l(Ni ) = (0), i = 1, ... , m, then ((s, !::l.q, !::l.~) and det(!::l.q, !::l.~) are well defined. D
286
Relative Index Theory, Determinants and Torsion
A special case is given by the pair
(Mn,g) and
(Q
N, x [0,
Q
oo[~ N x [0,00[' dr' + INr )dO"~,) ~ (M'", g').
Here M'n is a manifold with boundary 8M'n = N = UN i , and the latter falls out from our considerations. But if we consider the case q = 0 and b.. o(Min, g') with Dirichlet boundary conditions at 8M' = N = U N i , then we get an essentially selfadjoint operator b..~, O"(b..~) = [0,00[= [O,oo[U U [Aj, 00[, and Aj>O
e- tLlo -
e-tLl~pl is of trace class.
The latter fact is an immediate consequence of the proof of theorem 3.9. Hence the wave operator W±(b.. o, b..~) exist, are complete and the absolutely continuous parts of b.. o and b..~ are unitarily equivalent. We intend to present an explicit representation of the scattering matrix S(A), of tr(e- tLlo - e-tLl~) and of the relative (-function. Here we essentially follow [49]. Then
and O"p(b.. o) consists of eigenvalues
o < /-ll
::; /-l2 ::; /-l3 ::; . . . --+
00
of finite multiplicity without finite accumulation point and (6.34) (6.35) immediately implies for t > 0 (6.35) and (6.36)
287
Relative C. -functions
where ~OIL2,O"p is the restriction of ~o to the subspace spanned by the L 2-eigenfunctions. If we apply (6.34) to the case q = 0 then we obtain (6.37) i
k
j
where the AjS are the eigenvalues of ~o(N) and simultaneously the thresholds of the (absolutely) continuous spectrum. We describe the continuous spectrum in terms of generalized eigenfunction. Consider ~o(N)(= (~O(Nl)"'" ~o(Nm)), an eigenvalue Aj E O'(~o(N)) and let E(Aj) be the corresponding eigenspace. For f../, > Aj and different from all thresholds and for 'I O.
Proposition 6.18 Under the above assumptions, the relative (-function ((8, ~o + z, ~~ + z) for ~(z) > 0 is given by
j
k
(6.40)
Proof. At first we remark that for 0 < A < AI, S(A) is a scalar, S (A 2 ) extends to a meromorphic function on {z E ell z I < AI}, which is holomorphic at A = 0 and, in particular, S(A) is holomorphic in Uc(O), E small. Using this and equation (6.39), we see that for t ---> 00 we have a representation tr( e- tllo e-tll~PI) = bo + O(re), (2 > 0, which is equivalent to ~(A) = -bo - O(Ae) for A ---> 0+. Hence (2.24) and both assumptions
289
Relative (, -functions
of proposition 2.4 are satisfied. We obtain from corollary 2.5 a) and (6.39) an asymptotic expansion
1 tr(e-t~o-e-t~~ PI) "" dim ker ~+4trS(0)+
z=
00.
Cjr~,
t --+ 00.
j=l
Moreover, for t --+ 0+, we have the standard expansion 00
tr(e-t~o - e-t~~pl)
"" Z=ajr~+j,
t --+ 0+ ,
j=O
and we get the existence of the relative (-function ((8, ~O, ~~) as a meromorphic function. According to [49]
J:z A
log det S(A)dA = O(A n) as A --+ 00.
(6.41)
o
Using this and inserting (6.39) into (,), we obtain finally (6.40). D
It is clear that the product geometry of cylindrical ends is an extremly special case of possible geometries on E = N x [0,00[, and one should admit much more general bounded geometries on M, e.g. bounded geometries of the type gc = gINX[O,oo[ = dr 2 + (e 9 (rl)2dc/iv, g(r) > 0, g(l/l(r) bounded for all 1/. Again we get a pair (~o = ~o(g), ~~ = ~~(gc))' where (E, gc) E l,r J,rel (M) If gen comp L,diJ , g, e -t~o - e _t~10 pI 0 f t race c1ass. inf O"e(~O) > 0 or for t --+ 00 (6.42) then the relative (-function ((8, ~O, ~~) and relative determinant are defined. Explicit g(r) leads to explicit calculations. Similarly for q > O. If e.g. n = 2m, 9 on M is such that glc = dr2+e rdO"'iv and 9 = m then inf O"e(~ml(ker~mlJ.) > O. Here we take for ~~ the Friedrichs' extension of ~m on M x [O,oo[
290
Relative Index Theory, Determinants and Torsion
with zero boundary condition on M x {O}. Hence ((8, .6. m , .6.~) and det (.6. m , .6.~) are well defined. In the case of a manifold M = M'UN x [0,00[,9) with cylindrical ends, the main and interesting part of the geometry is contained in the compact part M', 8M' = N. At the boundary we assume product geometry. X = (M'UM', 9x = 9M,U9M' is then a closed manifold. It is now N a natural and interesting question, how are the .6.-determinant for X and the relative .6.-determinant for M related? The answer would also give a meaning, an interpretation of the relative determinant. A certain answer is contained in [53], and we give an outline of the corresponding result. Consider the following situation. Given (Mn, 9) closed, oriented, connected, Y c M a hypersurface, separating M into two components MI , M 2 . Set Mi = Mi. i = 1,2. Mi are compact with boundary Y, M = MI U M 2 , Y = 8MI = 8M2 . Morey over, let E ---+ M be a Hermitean vector bundle and .6. = .6. M : COO(M, E) ---+ COO(M, E) a Laplace type operator, i.e . .6. is symmetric, non-negative with principal symbol 0' 0, we have the asymptotic expansion (1.16),
c¥ c_!!2
+
+ ... + c¥+[~]c -2+ [!!H] 2 n
O(C¥+[n;3])
and can express the logarithm of the determinant more explicitely. First we remark that in U(s = 0) 1 f(s) = - +1" s· h(s), s where h( s) is a holomorphic function near s = 0 and I' denotes the Euler constant. Hence
~I
_1_
ds s=of(s)
~I
ds 8=0 1
=1
,
(_1_~) r(s) S
r(s) 18=0 = O.
_~I
1
_
- ds 8=of(s + 1) -
-f'(l)
,
292
Relative Index Theory, Determinants and Torsion
(1.16), (2.4) and (2.11) then imply -10gdet(D2, D,2) = :8 Is=0(1(8, D2, D'2) - h· f'(l)
J 00
+
r1[tr(e-
tD2
P - e-
tIJl2
P') - h]dt
1
[~]
L
J 1
C-~+l -
cof'(l)
+
r1[tr(e-
tD2
P - e-
tD'2
P')
0
1=0
[~]
-L
C_~+lr~+l]dt - hf'(l)
1=0
J 00
+
r1[tr(e-
tD2
P - e-
tDl2
P' - h]dt.
(6.44)
1
Let as in section 1 Pd(D) be the projector on the discrete subspace and Pc = 1 - Pd the projector on the continuous subspace and denote by D d and Dc the corresponding restrictions of. These subspaces are orthogonal and remain invariant under Dd or Dc, respectively. Suppose that e-tD~ is of trace class. Then e-tD~ - e- tDl2 is of trace class too. If for e-tD~ there exist an asymptotic expansion of the type (1.1) then there exists a well defined zeta function ((8, D d ) and det DJ can be defined by det D~ := e-('(O,D~).
(6.45)
Then (3.7), (6.45) and
+ e-tD~p _ e- tfy2 P') tre-tDJp + tr(e-tD~ P _ e- tfy2 P') tr(e-tDJP
yield
(6.46) If a > 0, then inf a"e(D2 + a) > 0 then h = 0 in (2.9), (2.24) for the pair D2 + a, iy + a is satisfied and the corresponding relative
Relative (-functions
293
(-function is given by
rts) J 00
«(s, D2 + a, fy2
+ a) =
tS-le-tatr(e-tD2 - e-
tfy2
)dt,
o (6.47) Re( s) > -1. The r.h.s. of (6.47) is also well defined if we replace a by any z E C with Re(z) > O. Then the corresponding function «(s, z, D2, fy2 admits as function of s a merom orphic extension to C which is holomorphic at s = 0 and we finally define
det(D2
+ z, fy2 + z)
:=
e-/s ls =o((s,z,D 2,D/2).
An important property concerning the z-dependence is expressed by l,r+l (E) an d Proposition 6.21 Suppose E' E gen comp L,diJ J,rel {2.24}. Then det(D 2 + z, fy2 + z is a holomorphic function of z E C\]O, 00[.
Proof. Here we essentially follow [49]. According to V, lemma 1.10 a),
Je-tA~(A)dA Je-tA~(A)dA. 2c
tr(e-
tD2
Pe- tD/2 PI)
=
-t
00
- t
o
2c
The second integral on the r.h.s. is O(e- tC ) for t e-tA~(A) E L1IR for t > O. Moreover.
Je-tA~(A)dA
----+
(6.48) 00 since
N
2c
=
o
I:
Ck tk
+ O(t N +1 )
for t
----+
0,
(6.49)
k=O
N E IN arbitrary. (1.16) and (6.49) imply (by taking the dif00
ference) that
J e-tA~(A) dA has a similar asymptotic expansion as 2c
(1.16). We infer from this that the integral
rts) J Je-tA~(A)dAdt 00
F(s, z) =
00
tSe- tz
o
2c
294
Relative Index Theory, Determinants and Torsion
in the half planes Re(s) > -~ and Re(z) > -c absolutely converges and, as function of s, it admits a meromorphic extension to C which is holomorphic at s = o. The first integral on the r.h.s. of (6.48) can be discussed as follows. We obtain for Re(z) > 0
J Je-t>'~()")d)"dt J~()..) J J + )..)-(s+l)~()")d)". 2c
00
tSe- tz
__1_
f(s)
o
0
2c
00
= _1_ f(s)
ee-t(zH)dtd)..
o
0
2c
= -s
(z
(6.50)
o
Hence, for Re(z) > 0, 2c
2
J (zH)-l~(>')d>'+ ~F (O,z).
- 2
det(D +z,D' +z)=e o
s
(6.51)
The r.h.s. of (6.51) has an obvious extension to an analytic (0, z) is holomorphic in Re(z) > function of z E C\] - 00,0], -c. c > 0 was arbitrary, hence we get an analytic extension to
c::
C\]- 00,0].
0
It is an interesting question, how the relative determinant and the relative torsion change under I-parameter change of the metric. We could consider e.g. the most natural evolution of the metrix which is given by the Ricci flow,
o
07 g(7)
.
= -2RIC (g(7)),
g(O)
=
go·
(6.52)
If (Mn, go) is complete and has bounded curvature then, according to [69], [25], there exists for 0 :::; 7 :::; T in the class of metrics with bounded curvature a unique solution of (6.51).
295
Relative (, -functions
Denote ~(T) diagram
~q(g(T)).
-
A(T)
is defined as before by the
~(O) ----t
L2(g(0)) :) 'D{),.(O)
~
L2(g())
~(T) • dvol(O) dvol(r)
i . dvol(r)
1
dvol(O)
~(T)
L2(g(T)) :) 'D{),.(r)
----t
L2(g( T))
It is not yet clear, whether g(T) E compl,r+l(g(O)). We proved in [25] that g(T) E bcomp2(g(0)) , but g(T) E compl,r+1(g(O)) is still open. Therefore we make the following · E' A ssumpt Ion.
l,r+lf (E) C gen comp L,dif l,r+lf (E) . Here comp L,dif comp(·) denotes the arc component. Hence there exists an arc {E(T)}-e~r~e connecting E and E', we assume in the sequel to the arc to be at least C l . E
Then we get a Cl-arc {D(T)}rl and we will study the behaviour of the relative determinant det(D2(0), D2(T)) under variation of T. Additionally, we suppose again (2.24). As before and in the sequel, D2 (T) denotes the transformed to the Hilbert space for
T = 0 D2(T). Denote D2(T)
= trD2(T). By Duhamels principle,
d~ e- tjj2 (r)
J t
=
e- sjj2 (r)
D (T)e-(t-s)jj2(r)ds. 2
(6.53)
o
If D2(T)e- tD2 (r) is for i > 0 of trace class with trace norm uniformly bounded on compact i-intervals lao, al], ao > 0, then according to (6.53), d~e-tjj2(r) is also of trace class for i > 0 and (6.54)
296
Relative Index Theory, Determinants and Torsion
To establish in the sequel substantial results, we must make two additional assumptions.
Assumption 1. D2(T) is invertible for T E [-s,s]. Assumption 2. There exists for t pansion 00
tr(D 2 (T)b- 2(T)e-
tjj2
(T))
0, Vi . V2 too. The same holds for subexponential decay. 3) V(x) = X-I and e- X , 0 < a < 1, are of sub-exponential ~~ D Denote by M = M(M) the set of all complete (smooth) metrics on M and let g, hEM. Then we define
m-I
mig - hlg(x) := Ig - hlg(x)
+ L I(\7 9 )i(\79 -
\7 h)lg(x) (1.4)
i=O
and
b,ml g _ hl g := sup mig - hlg(x). xEM
Remark 1.2 The conditions b,Olg - hl g == big - hl g < 00 and big - hlh < 00 are equivalent to g and h quasi-isometric. D Lemma 1.3 Suppose big - hl g, big - hl g < for every m ~ 0 a polynomial
00.
Then there exists
with non-negative coefficients and without constant term such that mig - hlh(X) ::; Pm(lg - hlg(x), ... , 1(\7g )m-I(\7 g
-
\7h)lg(x))' (1.5)
Proof. The proof is essentially contained in that of II, proposition 2.16. There we work with b,ill, but all steps remain valid without "b". D
The Case Injectivity Radius Zero
301
Lemma 1.4 Suppose gl, g2, g3 quasi-isometric. Then there exists for every m 2: 0 a polynomial
with non-negative coefficients and without constant term such that
o Proof. This proof is also contained in that of II, 2.16, replacing there b,ill by ill(x). 0
Lemma 1.5 Let V be a function of moderate decay, g, hEM, Xo E M, c > 0, and suppose
(1. 7) Then there holds a) 9 and hare quasiisometric,
b) there exist constants
C1, C2
> 0 such that
and C1 V(I+d g(x,
xo)) :::; V(I+d h (x, xo)) :::; c2V(I+dg(x, xo)), x E M.
(1.9) We refer to [54] for the simple proof.
o
Remark 1.6 If V(x) is of moderate decay then there exist constants c, C1, C 2 such that
(1.10)
302
Relative Index Theory, Determinants and Torsion
o We refer to [54] for the proof. Define now for m ~ 0, V a function of moderate decay, g, hEM m,vlg _ hl9 = inf{c> 01 mig - hI 9 (x) ~ c· V(1 for all x E M}
+ d9 (x, xo))
if { ... } i= 0 and m,vlg - hl9 = 00 otherwise. Let m ~ 0, 6 > 0, C(n,6) = 1 + 6 + 6y''-C" 2n---C('--n-_-l::-;-) function of moderate decay and set
M 2 IC(n, 6)-1g ~ h ~ C(n,6)g and m,vlg - hl9 < 6}. (1.11)
V8 = {(g, h)
E
Proposition 1. 7 ~ = {V8} 8>0 is a basis for a metrizable uniform structure m,v,U( M) . Proof.
Certainly holds
nV8 =
diagonal = {(g, g) Ig EM}.
8
For the symmetry, we have to show m,vlg - hl9 < 6 implies m,vlg - hlh < 6'(6) such that 6'(6) ----t O. But (1.5), (1.10), (1.11) and C(n, 6)-1g ~ h ~ C(n,6)g immediately imply
mig - hlh(X)
~
cV(1
+ dh(x, xo))
for all x E M,
c = c(C(n,6),c1(6),c2(6),6), limC(n,6) = 1, limC1(6),C2(6) = 8->0 8->0 1 and This yields the symmetry. Similarly we get the transitivity from 0 (1.6). Denote by ~M the pair (M, m,v'u(M)) and by m,V M the com--u
pletion ~M . The elements ofm,v Mare Cm-metrics (cf. [67]). There is an equivalent metrizable uniform structure, based on Ig - hi, 1\79 (g - h)I,· .. , 1(\7 9 )m(g - h)1 instead of Ig - hi, 1\79 \7 hl, 1\79(\7 9 - \7h)I, ... , I(\7 9 )m-1 (\7 9 - \7 h )1, i.e. a uniform structure giving the same (metrizable) topology.
The Case Injectivity Radius Zero
303
Suppose Ig - hI 9(x) :::; C9 . V(1 + d9(x, xo)), hence Ig - hlh(X) :::; ChV(1 + dh(x, xo)). According to lemma 1.5, for any (p, q) E (Z+)2, there exist constants C 1 (p, q), C 2 (p, q) > 0, such that for any (p, q)-tensor t
The equivalence of the uniform structures would follow from an inequality
Cd (\79)i(g -
h)19(X) < 1(\79)i-1(\7 9 - \7 h)19(X) < C2,il(\79)i(g - h)19(X), (1.13)
since then
if and only if
1(\7 9)i-1(\7 9 - \7 h)19(X):::; E2V(1 +d9(x,xo)). We prove (1.13) and set B
= h - g,
D
= \7 h
-
\7 9 . Then
1(\79)iBI9(X) = 1(\79)ihI9(X), 1(\7 h)iBlh(X) = 1(\7 h)iglh(X), h(D(X, Y), Z) = -
(1.15)
(1.16)
~{\7~B(Y, Z) + \7~B(X, Z)
\7~B(X, Y)}
(1.17) \71B(Y, Z) = g(D(X, Y), Z) + g(Y, D(X, Z)). (1.18) If we insert into (1.18) for X, Y, Z local (with respect to h) orthonormal vector fields ei, ej, ek, square, sum up and take the square root, then we get
l\7 h(h - g)lh(X) < v'2 (?=g((\7Zi -
\7~Jej, ek)2)
1
2"
t,),k
v'2lg((\7h - \79)00, Olh(X),
(1.19)
304
Relative Index Theory, Determinants and Torsion
According to (1.12)
Ig((\7 h - \79)0(-), (·))lh(X)
:S
C1(~' 0) Ig((\7 h -
C1(~' 0) l\7
h -
\79)0(-), ('))19(X)
\7919(X)
< C1 (2, 1) l\7h _ \791 (x) - C1 (3,0) h,
(1.20)
together with (1.19)
l\7 h(h - g)lh(X) :S d~,ol\7h - \79Ih(X), or, exchanging the role of 9 and h,
i.e. we have the first inequality (1.13) for i same manner from (1.17)
= 1. We get in the
together with
finally
1\79 - \7hI 9(x) :S C2,il\79(h - g)19(X) for i = 1, (1.13) is done. For i > 1, the proof follows by a simple but extensive induction, differentiating (1.17), (1.18) and applying (1.16). 0 We now set m
m,metl g _ hI 9(x) :=
L 1(\79)i(g -
h)19(X),
i=O
m,met,Vlg _ hl9 := inf{E > 0lm,metl g - hI 9(x) :S EV(1 + d9 (x, xo)) for all x E M}
305
The Case Injectivity Radius Zero
if {- .. }
I- 0 and m,met,vlg -
hl g =
00
otherwise and define
M 2 IC(1, 0 such
308
Relative Index Theory, Determinants and Torsion
that rinj(h, x) 2: min{ c . rinj(g, x), c'},
x E M.
We refer to [54] for the proof. D A special case is given by h E comp°,v (g), V of moderate decay. Set for x E M
hnj(x) := min { 12JK, rinj(X) } . Then, under the assumptions of lemma 2.1,
hnj(h, x) 2: C2i\nj(g, x). Using the standard volume comparison theorem
27r~
r(~)
_ Ce-(n-l)vKd(x,y) ,
X
E M.
(2.4)
Lemma 2.2 There exists a constant C = C(K), such that
(2.5)
309
The Case Injectivity Radius Zero
D We refer to [54] for the proof. We recall from II, lemma 1.10 the existence of appropriate uniformly locally finite covers of (Mn, g) if Ric (g) ~ k. Set for s > E ~ 0, /'i,c:(M, g, s) E IN U {(X)} = smallest number such that there exists a sequence {Xi}~l such that {Bs-c:(Xi)}~l is an open cover of M satisfying
sup #{i E INlx E B 3s+c:(Xi)} ~ /'i,c:(M,g,s)
(2.6)
xEM
and set /'i,(M, g, s) := /'i,o(M, g, s), /'i,(M, g, 0) := 1. We need for norm estimates of cos V75. in the forthcoming section 3 the following Lemma 2.3 /'i,c:(M, g, s) is finite for all s > E and there exist constants C, c > 0 depending on K such that for s > ,(R + E
/'i,c:(M, g, s) ~
c· eCs •
(2.7)
D We refer to [54] for the proof. Next we collect some facts concerning weighted Sobolev spaces. Let (E, hE, '\7 E) ---+ (Mn, g) be a Riemannian vector bundle, ~ a positive, measurable function on M which is finite a.e .. Define the associated weighted Sobolev spaces as follows
n~,r(E, '\7)
= {a)}a is a locally finite cover by normal charts, then there exist constants C(3, C~, C~, multiindexed by (3, "y such that
and
(2.9) all constants are independent of a.
b) If (E,hE' \i'E)
- - - t (Mn,g) is a Riemannian vector bundle satisfying (Bk(M,g)), (Bk(E, \i'E)), then additionally to (2.8), there holds for the connection coefficients r;J.t defined by
The Case Injectivity Radius Zero
V -LC{J).. = 8u'
rf).. C{JJL ,
311
a local orthonormal frame defined by
{C{JJL}JL
radial parallel translation,
(2.10)
o Consider {] > 0, B{} = B{}(O) c IRn, m, k E IN, K, A > 0 and denote by Ell~({], K, A) the set of elliptic operators
P
=
L
aa(x)Da,
aa(x)
= (aa,i,jh:5,i,j:5,k
lal:5,m
satisfying 1) aa,i,j E Cm(B{}), 2)
L
lal<m
3)
laalco(Bu):::; K,
L
laaIc 1 (Bu):::; K,
lal<m
A-ll~lm:::; lal=m L ( L aa'i,j(X)) ~a :::; AI~lm, l:5,i,j:5,k
where laaict(B u) = sup sup sup ID.6aa,i,j(X) I. i,j 1.61:5,1 xEBe We recall a standard elliptic inequality for Euclidean balls B{} and refer to [27], II (1.52), p. 75. Lemma 2.6 For given k, K, A, there exist {] = {](K, A) > 0 and C(A) > 0 such that for all {] :::; (]o, P E Ell~({], K, A)
lulwm(Bu~k :::; C(IPuIL2(Bg,Ck) for all u E C':(B{}).
+ luI L2 (B e,C k)) o
Using 2.5 and 2.6, we immediately get a generalization to balls in Riemannian manifolds and bundles satisfying (B k ) (as in [27], II (1.52)). Lemma 2.7 Suppose (E,hE,VE) - - t (Mn,g) being a Riemannian vector bundle satisfying B 2k (M, g), B 2k (E, VE), IVi Rgl,
312
Relative Index Theory, Determinants and Torsion
l\7j REI :s; K, i = 0, ... ,2k. Then there exist constants Qo(K) > 0, C(K) > 0 such that for all Xo E M and Q :s; min{Qo,i\nj(xo)} there holds
o We refer to [27] or to [54]. If V : [1,00[---t IR+ is of moderate decay, then we define the associated V : Mn ---t IR+ of moderate decay by V(x) = V(1 + dg(x, Xo)). Lemma 2.8 Let (E, hE, \7 E ) ---t (Mn,g) beaRiemannianvector bundle satisfying (Bk(Mn,g)), B 2k (E, \7 E ), k even, and let f3 : M ---t IR+ be of moderate decay. Then there exist bounded inclusions (2.12) and
(2.13)
Proof.
We apply (2.6) to get a cover {Brin;kXi) (Xi)
}:1
by
balls satisfying for all x E M (2.14) Set for u E GOO(IR) with u = 1 on [0,1] and = 0 on [2, oo[ and l:S;j:S;k . d(x,y)
Uj,x(Y)
=
u(2 {
o
J' f· ·(x) InJ
,
Y E Bi'i~j(X)(x), otherWIse.
Hence Uj,x E Cgo(M) and for «J E Hk(E, b..), Uj,x«J E Hk(Bi'inj(X)' E).
313
The Case Injectivity Radius Zero
We infer from (2.11)
Uj,x'P
E W k (B finj (x) (x), E and
Moreover, we infer from (2.11) IUk,x'PIWk(E)
~ C 1 1'Plwk(B
+ c,
f.
2k~ll (x)
t G)
(x),E)
(fi"j(x)t'
< C1 1'PIHk(B f . .
2k'~ll (x)
·I U k-l,.l"lw'-'(E)
(x))
k
+
C3
~ (~) (rinj(x)t 1 u
I k-1,x'PIHk-I(E)'
(2.15) We conclude from (2.14) by an easy induction
o
Lemma 2.9 Let V : [1,00[----+ IR+ be a function of moderate decay. Then for all x, y, q E M there holds C v . V(1
V(1 + d(x, q)) + d(x, y)) ~ V(1 + d(y, q)) ~
1 C v . V(1
+ d(x, y))'
and for every q' E M there exists C = C(q, q') > 0' such that
C- 1 . V(1
+ d(x, q'))
~
V(1
+ d(x, q))
We refer to [54J for the simple proof.
~ C·
V(1
+ d(x, q')). o
314
Relative Index Theory, Determinants and Torsion
Let now c.p E Hk(E, ~). Then lemma 2.9, lemma 2.7, (2.14) and (2.16) imply 00
Ic.pl Wt(E,V')
< C
I: V! (xi)luk,Xic.plwk(E,V') . hnj(Xi)k i=1 00
< C
I: V! (Xi) IUk,Xic.pIHk(E,~) i=1 00
c'" ~ V! (xi)hnj(Xi)-k ·1c.pIHk(B-. .( .) E)'
0 such that
f\nj(Xi)-ki\nj(x)kn ::; Db hence 00
'" V! (Xi) . hnj(Xi)-klc.pIHk(B_. .(x,.)(xi),E) ::; D21c.pIHkf:-2knV(E) , ~ rinJ
bl
~
which yields together with (2.17) the inclusion (2.12). The proof of (2.13) is quite analoguous.
o be continuous. Then W[(E, \7) n cOO(E) n COO (E) are dense in W[(E, \7) or Hr(E, ~), re-
Lemma 2.10 Let
or Hf(E,~) spectively.
~
We refer to [54], lemma 3.1 for the simple proof. We conclude this section with the invariance of weighted Sobolev spaces W[(E, \7 E , g), Hr(E, ~E, g) under change of the metric 9 inside compr,V(g). Proposition 2.11 Let V be of moderate decay, (E, hE, \7 E )
----t
(Mn, g) be a Riemannian vector bundle satisfying (Bo (M, g)) and suppose h E compr,v (g). Then W€1!(E, \7 E, g) ~ Wt(E, \7 E, h), 0::; (] ::; r, (2.18) as equivalent Sobolev spaces.
The Case Injectivity Radius Zero
315
For r = 0, the assertion is clear since, according to (1.8), 9 and h are quasi-isometric. The case r = 1 is also clear, since into the derivative \IE the derivatives \19 or \l h do not enter. In the case r = 2, (M, h) satisfies (Bo(M, h)) too, and h is a C r -metric. We set
Proof.
(2.19) (2.20) !?-1
\I!?
= (\IE ®
® \19)
0 •.. 0
(\IE ® \19)
0
\IE, (2.21)
1
\1 /2
= (\IE Q9 \l h) 0 \IE,
(2.22) (2.23)
!?-1
\II!?
= (\IE ®
® \l h)
0 . '.0
(\IE ® \l h) 0 \IE. (2.24)
1
According to II (1.34), for t.p E wt(E, \IE, g)
n COO (E)
!?
\l1!?t.p
=
L \li-1(\l1 - \I)\lI!?-it.p + \I!?t.p.
(2.25)
i=1
By assumption ~~\\I!?t.p\ E L2(M,g). There remains to consider the terms (2.26) Iterating the procedure, i.e. applying it to \l1!?-i and so on, we have to estimate expressions of the kind (2.27) with
i1
+ ... + i!? = {l,
i!?
0 such that n(n+1) cd 2 ( X,y ) e- tD. 0 (9 ) (x , y) < CfInJ . .(x)--2-e'
t E lao, ad·
(4.3)
The Case Injectivity Radius Zero
323
Lemma 4.2 Let V be a function of moderate decay. Suppose that a, b E IR are such that a) a + b = 2, b) Vb E Ll (M, g), _
n(n+l)
c) Varinj(xt-2-
Loo(M, g).
E
Let Mv the operator V" p E IN. Then the operator Mvl::!,.o (g )Pe-tAo(g) is Hilbert-Schmidt, and for any compact intervall lao, all, ao > 0, the Hilbert-Schmidt norm is uniformly bounded. Proof. Write
Mvl::!,.o(g)Pe- tA
(Mve-~Ao(g»)(l::!,.o(g)Pe-~Ao(g».
=
(4.4)
The second factor on the r.h.s. of (4.4) has bounded operator norm on any intervall lao, al], ao > O. Hence we can restrict to the case p = O. According to corollary 3.3, e-tll.o(g) extends to a bounded operator in L 2 ,vb (M, g) with uniformly bounded norm in 0 < a :::; t :::; b. We infer from Vb E Ll(M,g) that 1 E L 2,vb(M, g) and e- tAo (g)1 E L2,vb(M, g). Hence
b=
(1, e-tll.o(g) 1) L 2,V
JJ
V(X)be-tAo(g) (x, y)dydy
MM
is well defined, and we obtain for the Hilbert-Schmidt norm of Mve-tAo(g)
IMve-tAo(g) Its = =
JJ
lV(x)e-tAO(g) (x, y)1 2dydx
MM
JJ
V(x)2e- 2tll. O(g) (x, y)dydx
MM
:::; sup lV(ute-tll.o(g)(u, v)I' u,vEM
JJ J
V(X)be-tll.O(g) (x, y)dydx
MM
n(n+l)
:::; C· sup lV(u)ali:;~j-2-(u)1
lV(x)b(e- tAo (g)I)(x)dx
uEM M
:::; C l
.
le-
tAo
(g)liL
2
Vb (M,g)
O. E
2
Proof. Write
M r :-2n MvD..o(g )P e-t6. o(g) InJ
= [MTInJ __ 2nMve-~6.o(g) M _1]' [M 1D.. o(g)pe- t6. o(g)].(4.5) v J VJ The properties of V and b ~ 1 immediately imply V~ ~ C· V~, hence according to assumption b) V ~ E Ll (M, g). According to c) and the proof of lemma 4.2, the second factor on the r.h.s. of (4.5) is Hilbert-Schmidt, and the HS-norm is bounded for t E lao, al]' ao > O. We have to show this for the first facn(n+1) 2 tor. c) immediately implies V af:;j-2-- n E Loo(M, g). Hence, together with (4.3),
JJ
Ifinj(xt2nV(x)e-t6.o(g) (x, y)V(y)-~ 2 dxdy 1
MM
~
n(n+1) 2 C sup ifinj(ut-2-- nV(u)al uEM
.JJV(x)be-t6.o(g)(x,y)V(y)-~dxdy. MM b
•
2
2
b+4
2
By assumption b), Va ELI, l.e. V-aV-aV-3 ELI, V-a E b+4 b L ,,¥(M,g). Moreover, we get from V-3 ~ C· Va and
2,v
325
The Case Injectivity Radius Zero
b+4 • ~ ( ) b) that V-3 ELI. Accordmg to corollary 3.3, e- t 0 9 has an extension to a bounded operator in L 2¥ (M, g), and we
get
J e-t~o(g)(x,y)V(y)-~dy AI
2,V
E
2¥(M,g) with uniformly
L 2,V
bounded norm in t E lao, all. ao > O. Write Vb . VbV-£¥ = V~ . V~(b-1), from which follows Vb E L2 v-2¥ (M, g). Between
L2,v(M, g) and L 2,v-l (M, g) there is a st~ndard pairing (,) (cp,7jJ) =
J
cp(x)7jJ(x)dx,
cp E L 2 ,v, 7jJ E L 2 ,v-1.
(4.6)
(4.6) allows to rewrite
JJV(x)be-t~o(g)(x, y)V(y)-~dxdy
= (Vb, e-t~o(g)V-~)
0, e-tD. g(a - 1) is of trace class and the trace norm is uniformly bounded on compact tintervalls lao, all, a>O. Proof. We write
e-tD.g(a -1) = (e-~D.gMvk)· (Mv-ke-tD. g(a -1))
(4.8)
and estimate, using (4.1), the proofs of 4.2, 4.3., lemma 4.5,
(4.6)
le-~D.g Mvk I~s =
JJ
l(e-tD.g(x, y),
.)V~ (xWdxdy
MM
:S Cl
JJle-tD.O(9)(x,YWV~(x)dxdy
O. Quite analogous, we handle the other 5 integrals and indicate this by the decompositionn of the integrands and their estimate: t
"2
J(4.11)ds: o
e- s 6. g (l _ a-~)~he-(t-S)6.ha~
= e- s 6. [{(l_ a-~)e-t"4S6.hMv_~}{Mv~e-t"4S6.h}J g
t-8
1
'~he-2""" ~ha"2,
1(1 - a-~)e- t"4 6.h Mv_~ I s
:s;
c ·1V(x)e-t"4S6.o(g)(x,y)V(Y)-~1
IMv~e-t4S6.hl
:s; C
·1V(x)~e-t"4S6.o(g)(x, Y)I,
t
J(4.10)ds: t
"2
t
"2
J(4.10)ds: o
t
everything is done, J(4.9)ds : t
"2
330
Relative Index Theory, Determinants and Torsion
t
2
everything is done, J(4.9)ds : o
everything is done. This finishes the proof of 4.7.
o
Remarks 4.8 1) Theorem 4.7 and the theorems in chapter IV are neither disjoint nor there is an inclusion. 2) In 4.7 we only permitted a perturbation of the metric of Mn. It would be also interesting to consider additionally appropriate perburbations of the fibre metric and the fibre connection. The advantage of this chapter is that we do not restrict to the case nnj(M, g) > O. Hence we admit e.g. locally symmetric spaces of finite volume. 0 As an immediate consequence of theorem 4.7 we have Theorem 4.9 Suppose the hypotheses of theorem 4.7. there exist the wave operators
Then
and they are complete. Hence the absolutely continuous parts of and /j.h are unitarily equivalent. 0
/j.g
We have in mind still other admitted perturbations to establish a scattering theory. This will be the topic of a forthcoming treatise.
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List of notations page b ,2(M) .................... 30 d!i,~iff,F,rel, (El , E2) ....... 136 il Q ,2(M) ................... 31 det(D2, D,2) .............. 268 q
(Bk) ...................... 65 DP,r(M, N) ............... 123 BZ~ff(I, B k) ............. 131 DP,r(M) .................. 123 Bz~r/(J, B k) ............ 131 D~{(El' E 2) .............. 124 BZ'~r[,F(J, B k) ........... 135 ~b,rel(El' E 2) ............ 124 BZ~r[,rel(I, B k) .......... 134 TJ(s,D,D') ............... 275 compp,r(g) ............... 109 gen compj;~diff,rel(M, g) ... 129 b,mcomp(g) ............... 105 gen compj;~diff,rel(E) ...... 134 b,2 com pp,2(g) .............. 16 gencompj;~diff,F,rel(E) .... 136 compp,r (