2

1. INTRODUCTION

[Wo2]) and Guillemin [Gul] and to study the zeta and eta functions of hypoelliptic

operators. In turn this has several geometric consequences. In particular, this allows

us to make use of the framework of Connes' noncommutative geometry, including

the local index formula of [CM].

Third, we make use of the Heisenberg calculus and of the results of this mono-

graph to derive spectral asymptotics for hypoelliptic operators on Heisenberg mani-

folds. The advantage of using the Heisenberg calculus is illustrated by reformulating

in a geometric fashion these asymptotics for the main geometric operators on CR

and contact manifolds, namely, the Kohn Laplacian, the horizontal sublaplacian

and the Gover-Graham operators in the CR setting and the horizontal sublapla-

cian and the contact Laplacian in the contact setting.

On the other hand, although the setting of this monograph is the hypoelliptic

calculus on Heisenberg calculus, it is believed that the results herein can be extended

to more general settings such as the hypoelliptic calculus on Carnot-Caratheodory

manifolds which are equiregular in the sense of [Gro].

Following is a more detailed description of the contents of this memoir.

1.1. Heisenberg manifolds and their main differential operators

A Heisenberg manifold (M, H) consists of a manifold M together with a dis-

tinguished hyperplane bundle H C TM. This definition covers many examples:

Heisenberg group and its quotients by cocompact lattices, (codimension 1) folia-

tions, CR and contact manifolds and the confolations of Elyahsberg and Thurston.

In this setting the relevant tangent structure for a Heisenberg manifold (M, H)

is rather that of a bundle GM of two-step nilpotent Lie groups (see [BG], [Be],

[EM], [EMM], [FS1], [Gro], [Po6], [Ro2]).

The main examples of differential operators on Heisenberg manifolds are the

following.

(a) Hormander's sum of squares on a Heisenberg manifold (M, H) of the form,

(1-1-1) A = V^

l

Vx

1

+ ... + V^

m

V

X m

,

where the (real) vector fields X\,..., X

m

span H and V is a connection on a vector

bundle £ over M and the adjoint is taken with respect to a smooth positive measure

on M and a Hermitian metric on £.

(b) Kohn Laplacian \^b;p,q acting on (p, g)-forms on a CR manifold M 2 n + 1

endowed with a CR compatible Hermitian metric (not necessarily a Levi metric).

(c) Horizontal sublaplacian A ^ acting on horizontal differential forms of degree

A : on a Heisenberg manifold (M, H). When M 2 n + 1 is a CR manifold the horizontal

sublaplacian preserves the bidegree and so we can consider its restriction &b;p,q to

foms of bidegree (p, q).

(d) Gover-Graham operators Q^ \ k = l , . . . , n + l,n + 2,n + 4,.. . on a strictly

pseudoconvex CR manifold M 2 n + 1 endowed with a CR compatible contact form 6

(so that 9 defines a pseudohermitian structure on M). These operators have been

constructed by Gover-Graham [GG] as the CR analogues of the conformal GJMS

operators of [GJMS]. In particular, they are differential operators which tranforms

conformally under a conformal change of contact form and for k = 1 we recover the

conformal sublaplacian of Jerison-Lee [JL1].

(e) Contact Laplacian on a contat manifold M 2 n + 1 associated to the contact

complex of Rumin [Ru]. This complex acts between sections of a graded subbunbdle