kor yak adiabaticlimandspec

arX

iv:m

ath/

0703

785v

1 [

mat

h.D

G]

27

Mar

200

7

ADIABATIC LIMITS AND THE SPECTRUM OF THE

LAPLACIAN ON FOLIATED MANIFOLDS

YURI A. KORDYUKOV AND ANDREY A. YAKOVLEV

Abstract. We present some recent results on the behavior of the spec-trum of the differential form Laplacian on a Riemannian foliated mani-fold when the metric on the ambient manifold is blown up in directionsnormal to the leaves (in the adiabatic limit).

Contents

Introduction 11. Adiabatic limits and eigenvalue distribution 31.1. Riemannian foliations 31.2. A linear foliation on the 2-torus 81.3. Riemannian Heisenberg manifolds 112. Adiabatic limits and differentiable spectral sequence 142.1. Preliminaries on the differentiable spectral sequence 142.2. Riemannian foliations 152.3. A linear foliation on the 2-torus 172.4. Riemannian Heisenberg manifolds 19References 21

Introduction

Let (M,F) be a closed foliated manifold, dimM = n, dimF = p, p+ q =n, endowed with a Riemannian metric g. Then we have a decomposition ofthe tangent bundle to M into a direct sum TM = F H, where F = TF isthe tangent bundle to F and H = F is the orthogonal complement of F ,and the corresponding decomposition of the metric: g = gF + gH . Define aone-parameter family gh of Riemannian metrics on M by

(1) gh = gF + h2gH , h > 0.

By the adiabatic limit, we will mean the asymptotic behavior of Riemannianmanifolds (M,gh) as h 0.

In this form, the notion of the adiabatic limit was introduced by Witten[38] in the study of the global anomaly. He considered a family of Diracoperators acting along the fibers of a Riemannian fiber bundle over the

Supported by the Russian Foundation of Basic Research (grant no. 06-01-00208).

1

http://arXiv.org/abs/math/0703785v1

2 YURI A. KORDYUKOV AND ANDREY A. YAKOVLEV

circle and gave an argument relating the holonomy of the determinant linebundle of this family to the adiabatic limit of the eta invariant of the Diracoperator on the total space. Wittens result was proved rigorously in [6], [7]and [10], and extended to general Riemannian bundles in [5] and [13]. Thisstudy gave rise to the development of adiabatic limit technique for analyzingthe behavior of certain spectral invariants under degeneration that has manyapplications in the local index theory (see, for instance, [8]).

New properties of adiabatic limits were discovered by Mazzeo and Melrose[30]. They showed that in the case of a fibration, a Taylor series analysisof so called small eigenvalues in the adiabatic limit and the correspondingeigenforms leads directly to a spectral sequence, which is isomorphic to theLeray spectral sequence. This result was used in [13], and further developedin [16], where the very general setting of any pair of complementary distri-butions is considered. Nevertheless, the most interesting results of [16] areonly proved for foliations satisfying very restrictive conditions. The ideasfrom [30] and [16] were also applied in the case of the contact-adiabatic (orsub-Riemannian) limit in [18, 36].

In this paper, we will discuss extensions of the results mentioned above tothe adiabatic limits on foliated manifolds. For any h > 0, we will considerthe Laplace operator h on differential forms defined by the metric gh.It is a self-adjoint, elliptic, differential operator with the positive, scalarprincipal symbol in the Hilbert space L2(M,T M,gh) of square integrabledifferential forms on M , endowed with the inner product induced by gh,which has discrete spectrum. Denote by

0 0(h) 1(h) 2(h)

the spectrum of h, taking multiplicities into account.In Section 1, we discuss the asymptotic behavior as h 0 of the trace of

f(h):

tr f(h) =

+

i=0

f(i(h)),

for any sufficiently nice function f , say, for f S(R). The results given inthis Section should be viewed as a very first step in extending the adiabaticlimit technique to analyze the behavior of spectral invariants to the case offoliations.

In Section 2 we study branches of eigenvalues i(h) that are convergentto zero as h 0 (the small eigenvalues) and the corresponding eigenspacesand discuss the differentiable spectral sequence of the foliation, which is adirect generalization of the Leray spectral sequence, and its Hodge theoreticdescription.

We will consider two basic classes of foliations Riemannian foliationsand one-dimensional foliations defined by the orbits of invariant flows onRiemannian Heisenberg manifolds.

ADIABATIC LIMITS AND THE SPECTRUM OF FOLIATIONS 3

We remark that the adiabatic limit is, up to scaling, an example of col-lapsing (in general, without bounded curvature) in the sense of [9]. For adiscussion of the behavior of the spectrum of the differential form Laplacianon a compact Riemannian manifold under collapse, we refer, for instance,to [4, 11, 17, 22, 27] and references therein.

We are grateful to J. Alvarez Lopez for useful discussions. We also thankthe referee for suggestions to improve the paper.

1. Adiabatic limits and eigenvalue distribution

Let (M,F) be a closed foliated manifold, endowed with a Riemannianmetric g. In this Section, we will discuss the asymptotic behavior of thetrace of f(h) in the adiabatic limit.

1.1. Riemannian foliations. For Riemannian foliations, the problem wasstudied in [25]. Recall (see, for instance, [33, 34, 31, 32]) that a foliationF is called Riemannian, if there exists a Riemannian metric g on M suchthat the induced metric g on the normal bundle = TM/F is holonomyinvariant, or, equivalently, in any foliated chart : U Ip Iq with localcoordinates (x, y), the restriction gH of g to H = F

is written in the form

gH =

q

,=1

g(y),

where H is the 1-form, corresponding to the form dy under theisomorphismH = T Rq, and g(y) depend only on the transverse variablesy Rq. Such a Riemannian metric is called bundle-like.

It turns out that the adiabatic spectral limit on a Riemannian foliationcan be considered as a semiclassical spectral problem for a Schrodinger op-erator on the leaf space M/F , and the resulting asymptotic formula for thetrace of f(h) can be written in the form of the semiclassical Weyl for-mula for a Schrodinger operator on a compact Riemannian manifold, if wereplace the classical objects, entering to this formula by their noncommu-tative analogues. This observation provides a very natural interpretation ofthe asymptotic formula for the trace of f(h) (see Theorem 1.1 below).

First, we transfer the operators h to the fixed Hilbert space L2 =

L2(M,T M,g), using an isomorphism h from L2(M,T M,gh) to L

2defined as follows. With respect to a bigrading on T M given by

kT M =

k

i=0

i,kiT M, i,jT M = iH jF ,

we have, for u L2(M,i,jT M,gh),(2) hu = h

iu.

The operator h in L2(M,T M,gh) corresponds under the isometry h

to the operator Lh = hh1h in L

2.


With respect to the above bigrading of T M , the de Rham differentiald can be written as

d = dF + dH + ,

where

(1) dF = d0,1 : C(M,i,jT M) C(M,i,j+1T M) is the tangen-

tial de Rham differential, which is a first order tangentially ellipticoperator, independent of the choice of g;

(2) dH = d1,0 : C(M,i,jT M) C(M,i+1,jT M) is the transver-

sal de Rham differential, which is a first order transversally ellipticoperator;

(3) = d2,1 : C(M,i,jT M) C(M,i+2,j1T M) is a zeroth

order differential operator.

One can show that

dh = hd1h = dF + hdH + h

2,

and the adjoint of dh in L2 is

h = h1h = F + hH + h

2.

Therefore, one has

Lh = dhh + hdh

= F + h2H + h

4 + hK1 + h2K2 + h

3K3,

where F = dF dF+d

FdF is the tangential Laplacian, H = dHd

H+d

HdH is

the transverse Laplacian, = + and K2 = dF

+dF +F +Fare of zeroth order, and K1 = dF H + HdF + F dH + dHF and K3 =dH

+ dH + H + H are first order differential operators.Suppose that F is a Riemannian foliation and g is a bundle-like metric.

The key observation is that, in this case, the transverse principal symbolof the operator H is holonomy invariant, and, therefore, the first orderdifferential operator K1 is a leafwise differential operator. Using this fact,one can show that the leading term in the asymptotic expansion of the traceof f(h) or, equivalently, of the trace of f(Lh) as h 0 coincides with theleading term in the asymptotic expansion of the trace of f(Lh) as h 0,where

Lh = F + h2H .

More precisely, we have the following estimates (with some C1, C2 > 0):

| tr f(Lh)| < C1hq, | tr f(Lh) tr f(Lh)| < C2h1q, 0 < h 1,

where we recall that q denotes the codimension of F .We observe that the operator Lh has the form of a Schrodinger operator

on the leaf space M/F , where H plays the role of the Laplace operator,and F the role of the operator-valued potential on M/F .


Recall that, for a Schrodinger operator Hh on a compact Riemannianmanifold X, dimX = n, with a matrix-valued potential V C(X,L(E)),where E is a finite-dimensional Euclidean space and V (x) = V (x):

Hh = h2+ V (x), x X,

the corresponding asymptotic formula (the semiclassical Weyl formula) hasthe following form:

(3) tr f(Hh) = (2)nhn

T XTr f(p(x, )) dx d + o(hn), h 0+,

where p C(T X,L(E)) is the principal h-symbol of Hh:

p(x, ) = ||2 + V (x), (x, ) T X.

Now we demonstrate how the asymptotic formula for the trace of f(h)in the adiabatic limit can be written in a similar form, using noncommuta-tive geometry. (For the basic information on noncommutative geometry offoliations, we refer the reader to [26] and references therein.)

Let G be the holonomy groupoid of F . Let us briefly recall its definition.Denote byh the equivalence relation on the set of piecewise smooth leafwisepaths : [0, 1] M , setting 1 h 2 if 1 and 2 have the same initial andfinal points and the same holonomy maps. The holonomy groupoid G is theset of h equivalence classes of leafwise paths. G is equipped with the sourceand the range maps s, r : G M defined by s() = (0) and r() = (1).Recall also that, for any x M , the set Gx = { G : r() = x} is thecovering of the leaf Lx through the point x, associated with the holonomygroup of the leaf. We will identify any x M with the element of G givenby the constant path (t) = x, t [0, 1].

Let L denote the Riemannian volume form on a leaf L given by theinduced metric, and x, x M , denote the lift of Lx via the holonomycovering map s : Gx Lx.

Denote by : NF M the conormal bundle to F and by FN thelinearized foliation in NF (cf., for instance, [32, 26]). Recall that, for any G, s() = x, r() = y, the codifferential of the corresponding holonomymap defines a linear map dh : N

yF NxF . Then the leaf of the foliation

FN through NF is the set of all dh() NF , where G, r() =().

The holonomy groupoid GFN of the linearized foliation FN can be de-scribed as the set of all (, ) GNF such that r() = (). The sourcemap sN : GFN NF and the range map rN : GFN NF are defined assN(, ) = dh

() and rN (, ) = . We have a map G : GFN G given

by G(, ) = . Denote by L(T M) the vector bundle on GFN , whosefiber at a point (, ) GFN is the space of linear maps

(T M)sN (,) (T M)rN (,).


There is a standard way (due to Connes [12]) to introduce the structureof involutive algebra on the space Cc (GFN ,L(T M)) of smooth, com-pactly supported sections of L(T M). For any NF , this algebrahas a natural representation R in the Hilbert space L

2(GFN , sN (

T M))that determines its embedding to the C-algebra of all bounded operatorsin L2(GFN , s

N (

T M)). Taking the closure of the image of this em-bedding, we get a C-algebra C(NF ,FN , T M), called the twistedfoliation C-algebra. The leaf space NF/FN can be informally consideredas the cotangent bundle to M/F , and the algebra C(NF ,FN , T M)can be viewed as a noncommutative analogue of the algebra of continuousvector-valued differential forms on this singular space.

Let gN C(NF) be the fiberwise Riemannian metric on NF inducedby the metric on M . The principal h-symbol of h is a tangentially ellipticoperator in C(NF , T M) given by

h(h) = FN + gN ,

where FN is the lift of the tangential Laplacian F to a tangentially el-liptic (relative to FN ) operator in C(NF , T M), and gN denotes themultiplication operator in C(NF , T M) by the function gN . (Ob-serve that gN coincides with the transversal principal symbol of H .) Con-sider h(h) as a family of elliptic operators along the leaves of the folia-tion FN and lift these operators to the holonomy coverings of the leaves.For any NF , we get a formally self-adjoint uniformly elliptic oper-ator h(h) in C

(GFN , sN (

T M)), which essentially self-adjoint in

the Hilbert space L2(GFN , sN (

T M)). For any f S(R), the family{f(h(h)), NF} defines an element f(h(h)) of the C-algebraC(NF ,FN , T M).

The foliation FN has a natural transverse symplectic structure, which canbe described as follows. Consider a foliated chart : U M Ip Iqon M with coordinates (x, y) Ip Iq (I is the open interval (0, 1)) suchthat the restriction of F to U is given by the sets y = const. One hasthe corresponding coordinate chart in T M with coordinates denoted by(x, y, , ) Ip Iq Rp Rq. In these coordinates, the restriction of theconormal bundle NF to U is given by the equation = 0. So we havea coordinate chart n : U1 NF Ip Iq Rq on NF with thecoordinates (x, y, ) Ip Iq Rq. Indeed, the coordinate chart n is afoliated coordinate chart for FN , and the restriction of FN to U1 is given bythe level sets y = const, = const. The transverse symplectic structure forFN is given by the transverse two-form

j dyj dj .

The corresponding canonical transverse Liouville measure dy d is holo-nomy invariant and, by noncommutative integration theory [12], defines thetrace trFN on the C

-algebra C(NF ,FN , T M). Combining the Rie-mannian volume forms L and the transverse Liouville measure, we get avolume form d on NF . For any k Cc (GFN ,L(T M)), its trace is


given by the formula

trFN (k) =

NFk()d.

The trace trFN is a noncommutative analogue of the integral over the leafspace NF/FN with respect to the transverse Liouville measure. One canshow that the value of this trace on f(h(h)) is finite.

Replacing in the formula (3) the integration over T X and the matrixtrace Tr by the trace trFN and the principal h-symbol p by h(h), weobtain the correct formula for tr f(h) in the adiabatic limit.

Theorem 1.1 ([25]). For any f S(R), the asymptotic formula holds:(4) tr f(h) = (2)

qhq trFN f(h(h)) + o(hq), h 0.

The formula (4) can be rewritten in terms of the spectral data of leaf-wise Laplace operators. We will formulate the corresponding result for thespectrum distribution function

Nh() = {i : i(h) }.Restricting the tangential Laplace operator F to the leaves of the folia-

tion F and lifting the restrictions to the holonomy coverings of leaves, we getthe the Laplacian x acting in C

c (G

x, sT M). Using the assumptionthat F is Riemannian, it can be checked that, for any x M , x is for-mally self-adjoint in L2(Gx, sT M), that, in turn, implies its essential self-adjointness in this Hilbert space (with initial domain Cc (G

x, sT M)).For each R, let Ex() be the spectral projection of x, corresponding tothe semi-axis (, ]. The Schwartz kernels of the operators Ex() definea leafwise smooth section e of the bundle L(T M) over G.

We introduce the spectrum distribution function NF () of the operatorF by the formula

NF () =

MTr e(x) dx, R,

where dx denotes the Riemannian volume form onM . By [24], for any R,the function Tr e is a bounded measurable function on M , therefore, thespectrum distribution function NF () is well-defined and takes finite values.

As above, one can show that the family {Ex() : x M} defines anelement E() of the twisted von Neumann foliation algebra W (G,T M),the holonomy invariant transverse Riemannian volume form for F defines atrace trF on W

(G,T M), and the right hand side of the last formula canbe interpreted as the value of this trace on E().

Theorem 1.2 ([25]). Let (M,F) be a Riemannian foliation, equipped witha bundle-like Riemannian metric g. Then the asymptotic formula for Nh()has the following form:

Nh() = hq (4)

q/2

((q/2) + 1)

( )q/2dNF () + o(hq), h 0.


1.2. A linear foliation on the 2-torus. In this Section, we consider thesimplest example of the situation studied in the previous Section, namely,the example of a linear foliation on the 2-torus. So consider the two-dimen-sional torus T2 = R2/Z2 with the coordinates (x, y) R2, taken modulointeger translations, and the Euclidean metric g on T2:

g = dx2 + dy2.

Let X be the vector field on R2 given by

X =

x+

y,

where R. Since X is translation invariant, it determines a vector fieldX on T2. The orbits of X define a one-dimensional foliation F on T2. Theleaves of F are the images of the parallel lines L(x0,y0) = {(x0 + t, y0 + t) :t R}, parameterized by (x0, y0) R2, under the projection R2 T2.

In the case when is rational, all leaves of F are closed and are circles,and F is given by the fibers of a fibration of T2 over S1. In the case when is irrational, all leaves of F are everywhere dense in T2.

The one-parameter family gh of Riemannian metrics on T2 defined by (1)

is given by

gh =1 + h22

1 + 2dx2 + 2

1 h21 + 2

dxdy +2 + h2

1 + 2dy2.

The Laplace operator (on functions) defined by gh has the form h = F +h2H , where

F = 1

1 + 2

(

x+

y

)2, H =

h2

1 + 2

(

x+

y

)2

are the tangential and the transverse Laplace operators respectively.The operator h has a complete orthogonal system of eigenfunctions

ukl(x, y) = e2i(kx+ly), (x, y) T2,

with the corresponding eigenvalues

(5) kl(h) = (2)2

(1

1 + 2(k + l)2 +

h2

1 + 2(k + l)2

), (k, l) Z2.

The eigenvalue distribution function of h has the form

Nh = #{(k, l) Z2 : (2)2(

1

1 + 2(k + l)2 +

h2

1 + 2(k + l)2

)< }.

Thus we come to the following problem of number theory:

Problem 1.3. Find the asymptotic for h 0 of the number of integerpoints in the ellipse

{(, ) R2 : (2)2(

1

1 + 2( + )2 +

h2

1 + 2( + )2

)< }.


In the case when is rational, this problem can be easily solved by el-ementary methods of analysis. In the case when is irrational, such anelementary solution seems to be unknown, and, in order to solve the prob-lem, the connection of this problem with the spectral theory of the Laplaceoperator and with adiabatic limits plays an important role.

Theorem 1.4 ([40]). The following asymptotic formula for the spectrumdistribution function Nh() of the operator h for a fixed R holds:

1. For 6 Q,

(6) Nh() =1

4h1+ o(h1), h 0.

2. For Q of the form = pq , where p and q are coprime,

(7) Nh()

= h1

kZ|k| 0, consider the nestedsequence of graded subspaces

H1(h) H2(h) H3(h) H(h) ,

where Hrk(h) is the space generated by the eigenforms of h correspondingto eigenvalues ri (h) with i mrk; in particular, we have Hk(h) = H(h) =kerh for k large enough. Set also Hk(0) = Hk. We have dimHrk(h) = mrkfor all h > 0, so the following result is a sharpening of Theorem 2.1.

Corollary 2.3. For any Riemannian foliation on a closed manifold witha bundle-like metric and k = 2, 3, . . . ,, the assignment h 7 Hrk(h) de-fines a continuous map from [0,) to the space of finite dimensional linearsubspaces of L2r for all r 0. If dim Er1 < , then this also holds fork = 1.

By the standard perturbation theory, the map h 7 Hrk(h) is, clearly, Con (0,) for any Riemannian foliation on a closed manifold with a bundle-like metric, k = 2, 3, . . . , and r 0. As shown in [30], this map is Cup to h = 0, if the foliation is given by the fibers of a Riemannian fibration.In the next section, we will see an example of a Riemannian foliation and abundle-like metric such that the map h 7 Hrk(h) is not C at h = 0.

2.3. A linear foliation on the 2-torus. In this Section, we consider thesimplest example of the situation studied in the previous section, namely the example of a linear foliation on the 2-torus. So, as in Section 1.2,consider the two-dimensional torus T2 = R2/Z2 with the coordinates (x, y),the one-dimensional foliation F defined by the orbits of the vector fieldX = x +

y , where R, and the Euclidean metric g = dx2+dy2 on T2.

The eigenvalues of the corresponding Laplace operator h (counted with


multiplicities) are described as follows:

spec0h = spec2h = {kl(h) : (k, l) Z2},

spec1h = {k1l1(h) + k2l2(h) : (k1, l1) Z2, (k2, l2) Z2},

where kl(h) are given by (5). So, for / Q, small eigenvalues appear onlyif (k, l) = (0, 0) and (k1, l1) = (k2, l2) = (0, 0) and have the form

(10) 00(h) = 20(h) = 0,

10(h) =

11(h) = 0.

For Q of the form = pq , where p and q are coprime, small eigenval-ues appear only if (k, l) = t(p, q), t Z, and (k1, l1) = t1(p, q), (k2, l2) =t2(p, q), t1, t2 Z. So there are infinitely many different branches of eigen-values h with h = O(h

2) as h 0, and all the branches of eigenvalues hwith h = O(h

4) as h 0 are given by (10).Now let us turn to the differential spectral sequence. By a straightforward

computation, one can show that

Eu,v2 = Eu,v = R, u = 0, 1, v = 0, 1,

that agrees with the above description of small eigenvalues.The case of E1 is more interesting. First of all, it depends on whether

is rational or not. For Q, F is given by the fibers of a trivial fibrationT2 S1, and, therefore, for any u = 0, 1 and v = 0, 1, we have

Eu,v1 = Eu,v1 =

u(S1)Hv(S1) = C(S1).

For / Q, we have

Eu,v1 = R, u = 0, 1, v = 0, 1.

The description of E1 is more complicated and depends on the diophantineproperties of . Recall that / Q is called diophantine, if there exist c > 0and d > 1 such that, for any p Z \ {0} and q Z \ {0}, we have

|q p| > c|q|d .

Otherwise, is called Liouville. It is easy to see that E1,01 = E0,01 and

E1,11 = E0,11 . As shown in [21] and [35], we have

E0,01 = R; E0,11 = R if is diophantine and E

0,11 is infinite dimensional if is

Liouville.

So when is a Liouvilles number, 00,11 = 01,11 6= 0. As a direct consequence

of this fact and [1, Theorem D], we obtain that, when is a Liouvillesnumber, there exists a bundle-like metric on T2 such that the associatedmap h 7 H1(h), which is continuous on [0,) by Corollary 2.3 and Con (0,), is not C at h = 0.


2.4. Riemannian Heisenberg manifolds. In this Section, we discusssimilar problems for adiabatic limits associated with the Riemannian Heisen-berg manifold (\H, g) and the one-dimensional foliation F introduced inSection 1.3 (see [41]). We assume that g corresponds to the identity matrix,and the one-dimensional foliation F is defined by the vector field X(1, 0, 0).

The corresponding Riemannian metric gh on \H defined by (1) is givenby the matrix

1 0 00 h2 00 0 h2

, h > 0.

By Theorem 1.5, it follows that the spectrum of the Laplacian h on 0-and 3- forms on M (with multiplicities) is described as

spec0h = spec3h = 1,h 2,h,

where

1,h = {h(a, b) = 42(a2 + h2b2) : a, b Z},2,h = {h(c, k) = 2c(2k + 1)h + 42c2h2 with mult. 2c,

c Z+, k Z+ {0}}.

First, note that, for any a Z \ {0} and b Z \ {0},

h(a, b) > 42h2, h > 0,

and for any c Z+ and k Z+ {0}}

h(c, k) > 42h2, h > 0.

Therefore, for any h > 0,

00(h) = 30(h) = 0,

01(h) =

31(h) > 4

2h2.

Next, we see that, for any b Z \ {0},

h(0, b) = 42b2h2 = O(h2), h 0.

Since we have infinitely many different branches of eigenvalues h with h =O(h2) as h 0, we conclude that, for any i > 0,

0i (h) = 3i (h) = O(h

2), h 0.

By Theorem 1.6, the spectrum of the Laplace operator h on one andtwo forms on M (with multiplicities) has the form

spec1h = spec2h = 1,h 2,h 3,h,


where

1,h ={h,(a, b) = 42(a2 + h2b2

)+

1

1 + 162(a2 + h2b2)

2with mult. 2 : a, b Z},

2,h ={h(c, k) = 42c2h2 + 2c(2k + 1)hwith mult. 2c : c Z+, k Z+ {0}},

3,h ={h,(c, k) = 42c2h2 + 2c(2k + 1)h +1

(4ch + 1)2 + 16kch

2with mult. 2c : c Z+, k Z+ {0}}.

Observe that, for any b Z \ {0},

h,(0, b) = 42b2h2 +

11 + 162b2h2

2= 164b4h4 +O(h4), h 0,

and, for any spech \ {0}, > Ch4, h > 0,

with some constant C > 0. Therefore, we have

10(h) = 20(h) =

11(h) =

21(h) = 0, h > 0,

and, by the above argument, we obtain, for any i > 1

1i (h) = 2i (h) = O(h

4), h 0.We now turn to the differentiable spectral sequence. By a straightforward

computation, one can show that E3 = E, all the terms Er1 are infinite-

dimensional and, for the basic cohomology, we have

E0,02 = R, E1,02 = R, E

2,02 = C

(S1).

So we get that, in this case, for r = 0 and r = 3,

dim Er1 = {i ri (h) = O

(h2

)as h 0

}= ,

dimErk = {i ri (h) = O

(h2k

)as h 0

}= 1 , k 2 .

and, for r = 1 and r = 2

dim Er1 = {i ri (h) = O

(h2

)as h 0

}= ,

dimEr2 = {i ri (h) = O

(h4

)as h 0

}= ,

dimErk = {i ri (h) = O

(h2k

)as h 0

}= 2 , k 3 .

A more precise information can be obtained from the consideration of thecorresponding eigenspaces that will be discussed elsewhere.

Remark 5. It is quite possible that both the asymptotic formula of The-orem 1.7 and the investigation of small eigenvalues given in this Sectioncan be extended to the differential form Laplace operator on an arbitrary


Riemannian Heisenberg manifold. Nevertheless, we believe that many essen-tially new features of adiabatic limits on Riemannian Heisenberg manifoldscan be already seen in the particular cases, which were considered in thispaper, and we dont expect anything rather different in the general case.

References

[1] J. Alvarez Lopez, Yu. A. Kordyukov. Adiabatic limits and spectral sequences forRiemannian foliations. Geom. Funct. Anal. 10 (2000), 9771027

[2] J. Alvarez Lopez, Yu. A. Kordyukov. Long time behavior of leafwise heat flow forRiemannian foliations. Compositio Math. 125 (2001), 129153

[3] J. Alvarez Lopez, X. M. Masa. Morphisms of pseudogroups and foliation maps. In:Foliations 2005: Proceedings of the International Conference, Lodz, Poland, 13 -24 June 2005, pp. 1 19, World Sci. Publ., Singapore, 2006.

[4] B. Ammann, Ch. Bar. The Dirac operator on nilmanifolds and collapsing circle bun-dles. Ann. Global Anal. Geom. 16 (1998), 221253.

[5] J. M. Bismut, J. Cheeger. -invariants and their adiabatic limits. J. Amer. Math.Soc. 2 (1989), 3370.

[6] J. M. Bismut, D. S. Freed. The analysis of elliptic families, I. Metrics and connectionson determinant bundles. Comm. Math. Phys. 106 (1986), 159176.

[7] J. M. Bismut, D. S. Freed. The analysis of elliptic families, II. Dirac operators, etainvariants and the holonomy theorem. Comm. Math. Phys. 107 (1986), 103163.

[8] J. M. Bismut. Local index theory and higher analytic torsion. In: Proceedings of theInternational Congress of Mathematicians, Vol. I (Berlin, 1998). Doc. Math. 1998,Extra Vol. I, 143162

[9] J. Cheeger, M. Gromov. Collapsing Riemannian manifolds while keeping their curva-ture bounded. I. J. Differential Geom. 23 (1986), 309346.

[10] J. Cheeger. -invariants, the adiabatic approximation and conical singularities. I. Theadiabatic approximation. J. Differential Geom. 26 (1987), 175221.

[11] B. Colbois, G. Courtois. Petites valeurs propres et classe dEuler des S1-fibres. Ann.Sci. Ecole Norm. Sup. (4) 33 (2000), 611645.

[12] A. Connes. Sur la theorie non commutative de lintegration. In Algebres doperateurs(Sem., Les Plans-sur-Bex, 1978), Lecture Notes in Math. Vol. 725, pp. 19143.Springer, Berlin, Heidelberg, New York, 1979.

[13] X. Dai. Adiabatic limits, non-multiplicity of signature and the Leray spectral se-quence. J. Amer. Math. Soc. 4 (1991), 265231.

[14] X. Dai. APS boundary conditions, eta invariants and adiabatic limits. Trans. Amer.Math. Soc. 354 (2002), 107122.

[15] A. El Kacimi-Alaoui, M. Nicolau. On the topological invariance of the basic coho-mology. Math. Ann., 295 (1993), 627634.

[16] R. Forman. Spectral sequences and adiabatic limits. Comm. Math. Phys. 168 (1995),57116.

[17] K. Fukaya. Collapsing of Riemannian manifolds and eigenvalues of Laplace operator.Invent. Math. 87 (1987), 517547.

[18] Z. Ge. Adiabatic limits and Rumins complex. C. R. Acad. Sci., Paris. 320 (1995),699 702.

[19] C. S. Gordon, E. N.Wilson. The spectrum of the Laplacian on Riemannian Heisenbergmanifolds. Michigan Math. J. 33 (1986), 253271.

[20] A. Haefliger. Some remarks on foliations with minimal leaves. J. Differential Geom.15 (1980), 269 284.

[21] J. L. Heitsch. A cohomology for foliated manifolds. Comment. Math. Helv. 50 (1975),197218.


[22] P. Jammes. Sur le spectre des fibres en tore qui seffondrent. Manuscripta Math. 110(2003), 13 31.

[23] P. Jammes. Effondrement, spectre et prorietes diophantiennes des flots riemanniens.Preprint math.DG/0505417, 2005.

[24] Yu. A. Kordyukov. Functional calculus for tangentially elliptic operators on foliatedmanifolds. In Analysis and Geometry in Foliated Manifolds, Proceedings of the VIIInternational Colloquium on Differential Geometry, Santiago de Compostela, 1994,113136, Singapore, 1995. World Scientific.

[25] Yu. A. Kordyukov. Adiabatic limits and spectral geometry of foliations. Math. Ann.313 (1999), 763783.

[26] Yu. A. Kordyukov. Noncommutative geometry of foliations, Preprintmath.DG/0504095, 2005; to appear in K-theory.

[27] J. Lott. Collapsing and the differential form Laplacian: the case of a smooth limitspace. Duke Math. J. 114 (2002), 267306.

[28] J. McCleary. Users Guide to Spectral Sequences, volume 12 of Mathematics LectureSeries. Publish or Perish Inc., Wilmington, Del., 1985.

[29] X. Masa. Duality and minimality in Riemannian foliations. Comment. Math. Helv.67 (1992), 17 27.

[30] R. R. Mazzeo, R. B. Melrose. The adiabatic limit, Hodge cohomology and Leraysspectral sequence for a fibration. J. Differential Geom. 31 (1990), 185213.

[31] P. Molino. Geometrie globale des feuilletages riemanniens. Nederl. Akad. Wetensch.Indag. Math. 44 (1982), 4576.

[32] P. Molino. Riemannian foliations, Progress in Mathematics. Vol. 73. BirkhauserBoston Inc., Boston, MA, 1988.

[33] B. L. Reinhart. Foliated manifolds with bundle-like metrics. Ann. of Math. (2) 69(1959), 119132.

[34] B. L. Reinhart. Differential geometry of foliations, Ergebnisse der Mathematik undihrer Grenzgebiete, Vol. 99. Springer-Verlag, Berlin, 1983.

[35] C. Roger. Methodes Homotopiques et Cohomologiques en Theorie de Feuilletages.Universite de Paris XI, Paris, 1976.

[36] M. Rumin. Sub-Riemannian limit of the differential form spectrum of contact mani-folds. Geom. Func. Anal. 10 (2000), 407 452.

[37] K.S. Sarkaria. A finiteness theorem for foliated manifolds. J. Math. Soc. Japan 30(1978), 687 696.

[38] E. Witten. Global gravitational anomalies. Comm. Math. Phys. 100 (1985), 197229.[39] A. A. Yakovlev. Adiabatic limits on Riemannian Heisenberg manifolds. Preprint,

2006, submitted to Mat. sb.[40] A. A. Yakovlev. The spectrum of the Laplace-Beltrami operator on the two-dimensi-

onal torus in adiabatic limit. Preprint math.DG/0612695, 2006.[41] A. A. Yakovlev. Adiabatic limits and spectral sequences for one-dimensional foliations

on Riemannian Heisenberg manifolds, in preparation.

Institute of Mathematics, Russian Academy of Sciences, 112 Chernyshevsky

str., 450077 Ufa, Russia

E-mail address: [email protected]

Department of Mathematics, Ufa State Aviation Technical University, 12

K. Marx str., 450000 Ufa, Russia

E-mail address: [email protected]

http://arXiv.org/abs/math/0505417http://arXiv.org/abs/math/0504095http://arXiv.org/abs/math/0612695

Introduction1. Adiabatic limits and eigenvalue distribution1.1. Riemannian foliations1.2. A linear foliation on the 2-torus1.3. Riemannian Heisenberg manifolds

2. Adiabatic limits and differentiable spectral sequence2.1. Preliminaries on the differentiable spectral sequence2.2. Riemannian foliations2.3. A linear foliation on the 2-torus2.4. Riemannian Heisenberg manifolds

References

kor yak adiabaticlimandspec

Documents