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1 Transverse measure for flows 4 2 Transverse measure for foliations 6

Abstract. We first summarize very briefly the topology of 3-manifolds 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 3-manifolds. The final two sections present evidence for the validity of these conjectures and outline an approach toward their proof.

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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 Branson-Gilkey 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 well-known 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.

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. Continuing our study of global conformal invariants for Riemannian manifolds, we find new classes of such invariants corresponding to boundary value problems and to Lefschetz fixed point theorems. We also study the eta invariant, functional determinants, and analytic continuation to metrics of indefinite signature. In particular, we find global invariants of CR manifolds coming from bundles which support Lorentz metrics. 0. Introduction Given a geometric structure on a manifold M , one may attempt to study the topology and geometry of M via the global behavior of solutions to naturally associated partial differential equations. Often the functional analytic constructs in this situation will reveal deep facts about M , locally as well as globally. A standard example is that of the Laplace operator \Delta on differential forms in a compact, Riemannian, m-dimensional, smooth manifold (M; g) without boundary: here asymptotically, (0.1) Tr L 2 exp(\Gammat\Delta) ¸ t \Gammam=2 1 X i=...

Abstract. In this paper, we first prove a local family version of the Atiyah-Bott-Segal-Singer Lefschetz fixed point formula, then we extend the famous Witten’s rigidity Theorems to the family case. Several family vanishing theorems for elliptic genera are also proved. 0 Introduction. Let M, B be two compact smooth manifolds, and π: M → B be a submersion with compact fibre X. Assume that a compact Lie group G acts fiberwise on M, that is the action preserves each fiber of π. Let P be a family of elliptic operators

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