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
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