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125
A survey of foliations and operator algebras
 Proc. Sympos. Pure
, 1982
"... 1 Transverse measure for flows 4 2 Transverse measure for foliations 6 ..."
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Cited by 74 (6 self)
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1 Transverse measure for flows 4 2 Transverse measure for foliations 6
Scalar curvature and geometrization conjectures for 3manifolds
 in Comparison Geometry (Berkeley 1993–94), MSRI Publications
, 1997
"... Abstract. We first summarize very briefly the topology of 3manifolds and the approach of Thurston towards their geometrization. After discussing some general properties of curvature functionals on the space of metrics, we formulate and discuss three conjectures that imply Thurston’s Geometrization ..."
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Cited by 34 (8 self)
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Abstract. We first summarize very briefly the topology of 3manifolds and the approach of Thurston towards their geometrization. After discussing some general properties of curvature functionals on the space of metrics, we formulate and discuss three conjectures that imply Thurston’s Geometrization Conjecture for closed oriented 3manifolds. The final two sections present evidence for the validity of these conjectures and outline an approach toward their proof.
Invariants of conformal Laplacians
 Journal of Differential Geometry
, 1987
"... tions on a Riemannian manifold Mn with scalar curvature s, is a conformally invariant operator. In this paper we will use D to construct new conformal invariants: one of these is a pointwise invariant, one is the integral of a local expression, and one is a nonlocal spectral invariant derived from f ..."
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Cited by 33 (4 self)
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tions on a Riemannian manifold Mn with scalar curvature s, is a conformally invariant operator. In this paper we will use D to construct new conformal invariants: one of these is a pointwise invariant, one is the integral of a local expression, and one is a nonlocal spectral invariant derived from functional determinants. We begin in §1 by describing the Laplacian D and its Green function in the context of conformal geometry. We then derive a basic formula giving the variation in the heat kernel of D. This formula is strikingly simpler than the corresponding formula for the ordinary Laplacian given by Ray and Singer [15]. The heat kernel of D has an asymptotic expansion k(t, x, x) ~ (4πt)~n/2Σak(x)tk. In §2 we prove that a(n_2)/2 is a pointwise conformal invariant of weight2, i.e. it satisfies a(n_2)/2(x; λ2g) = λ2a(n_2)/2(x \ g), where g is the metric and λ is any smooth positive function. In particular, this shows the existence of a nontrivial locally computable conformally invariant density naturally associated to the conformal structure of an even dimensional manifold. The key to the proof is to consider the parametrix of the Green's function, which is obtained from the heat kernel by an integral transform. One finds that a(n_2)/2 occurs as the coefficient of the first log term in this parametrix, and its conformal invariance then follows directly from the conformal invariance of the Green's function. In §3 we show that / a n/2 is a global conformal invariant (the calculations in §4 show that it is not a pointwise invariant). The proof is a direct calculation of the invariant of / a n/2 using equation (1.10).
Spectral Asymmetry, Zeta Functions and the Noncommutative Residue
"... Abstract. In this paper, motivated by an approach developed by Wodzicki, we look at the spectral asymmetry of elliptic ΨDO’s in terms of theirs zeta functions. First, using asymmetry formulas of Wodzicki we study the spectral asymmetry of odd elliptic ΨDO’s and of geometric Dirac operators. In parti ..."
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Abstract. In this paper, motivated by an approach developed by Wodzicki, we look at the spectral asymmetry of elliptic ΨDO’s in terms of theirs zeta functions. First, using asymmetry formulas of Wodzicki we study the spectral asymmetry of odd elliptic ΨDO’s and of geometric Dirac operators. In particular, we show that the eta function of a selfadjoint elliptic odd ΨDO is regular at every integer point when the dimension and the order have opposite parities (this generalizes a well known result of BransonGilkey for Dirac operators), and we relate the spectral asymmetry of a Dirac operator on a Clifford bundle to the Riemmanian geometric data, which yields a new spectral interpretation of the Einstein action from gravity. We also obtain a large class of examples of elliptic ΨDO’s for which the regular values at the origin of the (local) zeta functions can easily be seen to be independent of the spectral cut. On the other hand, we simplify the proofs of two wellknown and difficult results of Wodzicki: (i) The independence with respect to the spectral cut of the regular value at the origin of the zeta function of an elliptic ΨDO; (ii) The vanishing of the noncommutative residue of a zero’th order ΨDO projector. These results were proved by Wodzicki using a quite difficult and involved characterization of local invariants of spectral asymmetry, which we can bypass here. Finally, in an appendix we give a new proof of the aforementioned asymmetry formulas of Wodzicki. 1.
AN ANOMALY FORMULA FOR RAY–SINGER METRICS ON MANIFOLDS WITH BOUNDARY
 GEOMETRIC AND FUNCTIONAL ANALYSIS
, 2006
"... Using the heat kernel, we derive first a local Gauss–Bonnet–Chern theorem for manifolds with a nonproduct metric near the boundary. Then we establish an anomaly formula for Ray–Singer metrics defined by a Hermitian metric on a flat vector bundle over a Riemannian manifold with boundary, not assumin ..."
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Cited by 22 (1 self)
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Using the heat kernel, we derive first a local Gauss–Bonnet–Chern theorem for manifolds with a nonproduct metric near the boundary. Then we establish an anomaly formula for Ray–Singer metrics defined by a Hermitian metric on a flat vector bundle over a Riemannian manifold with boundary, not assuming that the Hermitian metric on the flat vector bundle is flat nor that the Riemannian metric has product structure near the boundary.
Full asymptotic expansion of the heat trace for non–self–adjoint . . .
, 2001
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