Abstract.
Abstract.
In this dissertation we proved various results for pseudohermitian manifolds, and more
generally for Heisenberg manifolds, as applications of the construction of a non-commutative
residue within the Heisenberg calculus. In chapter 1 we give a thorough overview of this
YVDO calculus. In chapter 2 we build an algebra of YVDO with parameter allowing a
pseudodifferential construction of the resolvent of a subelliptic sublaplacian. In chapter 3
we introduce the notion of holomorphic family of YVDO and we use it to construct the
complex powers of a subelliptic sublaplacian. In chapter 4 we show the existence of an
analytic continuation of the trace on YVDO with non-integral order and on YVDO with
integral order we obtain a residual which is indeed a non-commutative residue. Then we show
that this non-commutative residue extends the Dixmier trace on the whole algebra of YVDO
with integral order and induces the unique trace, up to
a constant mutiple, on this algebra quotiented by the smoothing operators.
The chapter 5 is devoted to geometric applications. We define the zeta function of a
subelliptic sublplacian whose residues are related to the coefficient of the heat trace expansion.
From variational formulae for zeta functions we produce conformal invriants for pseuohermitian
manifolds. Then we study the non-commutative geometry of pseudohermitian manifolds. We define
the area of such a manifold and we show that in dimension 3 this area is computable by a local
formula involving the Tanaka-Webster scalar curvature.
Finally we prove local formulae for the index of a square root of a subelliptic sublaplacian.
First in terms of the heat trace expansion. Then, using cyclic cohomology and the local index formulae
of Connes-Moscovici, we
showed the existence of an even homology class of the manifold whose pairing with the Chern
character gives the index with coefficient in the K0-theory.
Key-words: Heisenberg calculus, pseudohermitian geometry, non-commutative geometry,
non-commutative residue, index theory, spectral invariants.
MSC classification (MSC 2000): 35J06, 58J40, 53C56, 58B34, 58J42, 58J20, 58J50.
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On 8 Jan 2001, 13:45.