1,645 research outputs found
Spectral Asymmetry, Zeta Functions and the Noncommutative Residue
In this paper we study the spectral asymmetry of (possibly nonselfadjoint)
elliptic PsiDO's in terms of the difference of zeta functions coming from
different cuttings. Refining previous formulas of Wodzicki in the case of odd
class elliptic PsiDO's, our main results have several consequence concerning
the local independence with respect to the cutting, the regularity at integer
points of eta functions and a geometric expression for the spectral asymmetry
of Dirac operators which, in particular, yields a new spectral interpretation
of the Einstein-Hilbert action in gravity.Comment: v8: final version. To appear in Int. Math. J., 22 page
The Logarithmic Singularities of the Green Functions of the Conformal Powers of the Laplacian
Green functions play an important role in conformal geometry. In this paper,
we explain how to compute explicitly the logarithmic singularities of the Green
functions of the conformal powers of the Laplacian. These operators include the
Yamabe and Paneitz operators, as well as the conformal fractional powers of the
Laplacian arising from scattering theory for Poincar\'e-Einstein metrics. The
results are formulated in terms of Weyl conformal invariants arising from the
ambient metric of Fefferman-Graham. As applications we obtain "Green function"
characterizations of locally conformally flat manifolds and a spectral
theoretic characterization of the conformal class of the round sphere.Comment: 22 pages. v3: final version. To appear in Contemp. Mat
Noncommutative residue for Heisenberg manifolds. Applications in CR and contact geometry
This paper has four main parts. In the first part, we construct a
noncommutative residue for the hypoelliptic calculus on Heisenberg manifolds,
that is, for the class of Heisenberg PsiDOs introduced by Beals-Greiner and
Taylor. This noncommutative residue appears as the residual trace on integer
order Heisenberg PsiDOs induced by the analytic extension of the usual trace to
non-integer order Heisenberg PsiDOs. Moreover, it agrees with the integral of
the density defined by the logarithmic singularity of the Schwartz kernel of
the corresponding Heisenberg PsiDO. In addition, we show that this
noncommutative residue provides us with the unique trace up to constant
multiple on the algebra of integer order Heisenberg PsiDOs. In the second part,
we give some analytic applications of this construction concerning zeta
functions of hypoelliptic operators, logarithmic metric estimates for Green
kernels of hypoelliptic operators, and the extension of the Dixmier trace to
the whole algebra of integer order Heisenberg PsiDOs. In the third part, we
present examples of computations of noncommutative residues of some powers of
the horizontal sublaplacian and the contact Laplacian on contact manifolds. In
the fourth part, we present two applications in CR geometry. First, we give
some examples of geometric computations of noncommutative residues of some
powers of the horizontal sublaplacian and of the Kohn Laplacian. Second, we
make use of the framework of noncommutative geometry and of our noncommutative
residue to define lower dimensional volumes in pseudohermitian geometry, e.g.,
we can give sense to the area of any 3-dimensional CR manifold. On the way we
obtain a spectral interpretation of the Einstein-Hilbert action in
pseudohermitian geometry.Comment: v5: major improvement of the presentation, final version. To appear
in Journal of Functional Analysis. 56 page
- …