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137
Analysis of geometric operators on open manifolds: a groupoid approach
, 2008
"... The first five sections of this paper are a survey of algebras of pseudodifferential operators on groupoids. We thus review differentiable groupoids, the definition of pseudodifferential operators on groupoids, and some of their properties. We use then this background material to establish a few n ..."
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Cited by 34 (21 self)
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The first five sections of this paper are a survey of algebras of pseudodifferential operators on groupoids. We thus review differentiable groupoids, the definition of pseudodifferential operators on groupoids, and some of their properties. We use then this background material to establish a few new results on these algebras that are useful for the analysis of geometric operators on noncompact manifolds and singular spaces. The first step is to establish that the geometric operators on groupoids are in our algebras. This then leads to criteria for Fredholmness for geometric operators on suitable noncompact manifolds, as well as to an inductive procedure to study their essential spectrum. As an application, we answer a question of Melrose on the essential spectrum of the Laplace operator on manifolds with multicylindrical ends.
Propagation of singularities for the wave equation on manifolds with corners
 In Séminaire: Équations aux Dérivées Partielles, 2004–2005, Sémin. Équ. Dériv. Partielles
"... Abstract. In this paper we describe the propagation of C ∞ and Sobolev singularities for the wave equation on C ∞ manifolds with corners M equipped with a Riemannian metric g. That is, for X = M ×Rt, P = D2 t −∆M, and u ∈ H1 loc (X) solving Pu = 0 with homogeneous Dirichlet or Neumann boundary condi ..."
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Cited by 29 (14 self)
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Abstract. In this paper we describe the propagation of C ∞ and Sobolev singularities for the wave equation on C ∞ manifolds with corners M equipped with a Riemannian metric g. That is, for X = M ×Rt, P = D2 t −∆M, and u ∈ H1 loc (X) solving Pu = 0 with homogeneous Dirichlet or Neumann boundary conditions, we show that WFb(u) is a union of maximally extended generalized broken bicharacteristics. This result is a C ∞ counterpart of Lebeau’s results for the propagation of analytic singularities on real analytic manifolds with appropriately stratified boundary, [11]. Our methods rely on bmicrolocal positive commutator estimates, thus providing a new proof for the propagation of singularities at hyperbolic points even if M has a smooth boundary (and no corners). 1.
Renormalizing curvature integrals on PoincaréEinstein manifolds
, 2005
"... After analyzing renormalization schemes on a PoincaréEinstein manifold, we study the renormalized integrals of scalar Riemannian invariants. The behavior of the renormalized volume is wellknown, and we show any scalar Riemannian invariant renormalizes similarly. We consider characteristic forms ..."
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Cited by 27 (5 self)
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After analyzing renormalization schemes on a PoincaréEinstein manifold, we study the renormalized integrals of scalar Riemannian invariants. The behavior of the renormalized volume is wellknown, and we show any scalar Riemannian invariant renormalizes similarly. We consider characteristic forms and their behavior under a variation of the PoincaréEinstein structure, and obtain, from the renormalized integral of the Pfaffian, an extension of the GaussBonnet theorem.
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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Cited by 24 (7 self)
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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.
Groupoids and the integration of Lie algebroids
 J. Math. Soc. Japan
"... Abstract. We show that a Lie algebroid on a stratified manifold is integrable if, and only if, its restriction to each strata is integrable. These results allow us to construct a large class of algebras of pseudodifferential operators. They are also relevant for the definition of the graph of certai ..."
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Cited by 23 (8 self)
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Abstract. We show that a Lie algebroid on a stratified manifold is integrable if, and only if, its restriction to each strata is integrable. These results allow us to construct a large class of algebras of pseudodifferential operators. They are also relevant for the definition of the graph of certain singular foliations of manifolds with corners and the construction of natural algebras of pseudodifferential operators on a given complex algebraic variety. Contents
Spectral Analysis of Magnetic Laplacians on Conformally Cusp Manifolds
 ANNALES HENRI POINCARÉ
"... We consider an open manifold which is the interior of a compact manifold with boundary. Assuming gauge invariance, we classify magnetic fields with compact support into being trapping or nontrapping. We study spectral properties of the associated magnetic Laplacian for a class of Riemannian metric ..."
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Cited by 22 (5 self)
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We consider an open manifold which is the interior of a compact manifold with boundary. Assuming gauge invariance, we classify magnetic fields with compact support into being trapping or nontrapping. We study spectral properties of the associated magnetic Laplacian for a class of Riemannian metrics which includes complete hyperbolic metrics of finite volume. When B is nontrapping, the magnetic Laplacian has nonempty essential spectrum. Using Mourre theory, we show the absence of singular continuous spectrum and the local finiteness of the point spectrum. When B is trapping, the spectrum is discrete and obeys the Weyl law. The existence of trapping magnetic fields with compact support depends on cohomological conditions, indicating a new and very strong longrange effect. In the nongauge invariant case, we exhibit a strong AharonovBohm effect. On hyperbolic surfaces with at least two cusps, we show that the magnetic Laplacian associated to every magnetic field with compact support has purely discrete spectrum for some choices of the vector potential, while other choices lead to a situation of limiting absorption principle. We also study perturbations of the metric. We show that in the Mourre theory it is not necessary to require a decay of the derivatives of the perturbation. This very singular perturbation is then brought closer to the perturbation of a potential.
The spectral projections and the resolvent for scattering metrics
 J. Anal. Math
, 1999
"... Abstract. In this paper we consider a compact manifold with boundary X equipped with a scattering metric g as defined by Melrose [9]. That is, g is a Riemannian metric in the interior of X that can be brought to the form g = x−4 dx2 + x−2h ′ near the boundary, where x is a boundary defining function ..."
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Cited by 22 (14 self)
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Abstract. In this paper we consider a compact manifold with boundary X equipped with a scattering metric g as defined by Melrose [9]. That is, g is a Riemannian metric in the interior of X that can be brought to the form g = x−4 dx2 + x−2h ′ near the boundary, where x is a boundary defining function and h ′ is a smooth symmetric 2cotensor which restricts to a metric h on ∂X. Let H = ∆ + V where V ∈ x2C ∞ (X) is real, so V is a ‘shortrange’ perturbation of ∆. Melrose and Zworski started a detailed analysis of various operators associated to H in [11] and showed that the scattering matrix of H is a Fourier integral operator associated to the geodesic flow of h on ∂X at distance π and that the kernel of the Poisson operator is a Legendre distribution on X ×∂X associated to an intersecting pair with conic points. In this paper we describe the kernel of the spectral projections and the resolvent, R(σ ±i0), on the positive real axis. We define a class of Legendre distributions on certain types of manifolds with corners, and show that the kernel of the spectral projection is a Legendre distribution associated to a conic pair on the bstretched product X2 b (the blowup of X2 about the corner, (∂X) 2). The structure of the resolvent is only slightly more complicated. As applications of our results we show that there are ‘distorted Fourier transforms ’ for H, ie, unitary operators which intertwine H with a multiplication operator and determine the scattering matrix; and give a scattering wavefront set estimate for the resolvent R(σ ± i0) applied to a distribution f. 1.
On boundary value problems for Einstein metrics
 Geom. & Topology
"... Abstract. On any given compact manifold M n+1 with boundary ∂M, it is proved that the moduli space E of Einstein metrics on M is a smooth, infinite dimensional Banach manifold. The Dirichlet and Neumann boundary maps to data on ∂M are smooth Fredholm maps of index 0. These results also hold for mani ..."
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Cited by 20 (11 self)
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Abstract. On any given compact manifold M n+1 with boundary ∂M, it is proved that the moduli space E of Einstein metrics on M is a smooth, infinite dimensional Banach manifold. The Dirichlet and Neumann boundary maps to data on ∂M are smooth Fredholm maps of index 0. These results also hold for manifolds with compact boundary which have a finite number of locally asymtotically flat ends, as well as for the Einstein equations coupled to many other fields. 1. Introduction. Let M = M n+1 be a compact (n + 1)dimensional manifold with boundary ∂M, n ≥ 2. In this paper, we consider the structure of the space of Einstein metrics on (M,∂M), i.e. metrics g on ¯M = M ∪ ∂M satisfying the Einstein equations
Ricci flow and the determinant of the Laplacian on noncompact surfaces
 Comm. Par. Diff. Eq
, 2013
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A resolvent approach to traces and zeta Laurent expansions
 AMS Contemp. Math. Proc
, 2005
"... Abstract. Classical pseudodifferential operators A on closed manifolds are considered. It is shown that the basic properties of the canonical trace TR A introduced by Kontsevich and Vishik are easily proved by identifying it with the leading nonlocal coefficient C0(A, P) in the trace expansion of A( ..."
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Cited by 19 (2 self)
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Abstract. Classical pseudodifferential operators A on closed manifolds are considered. It is shown that the basic properties of the canonical trace TR A introduced by Kontsevich and Vishik are easily proved by identifying it with the leading nonlocal coefficient C0(A, P) in the trace expansion of A(P −λ) −N (with an auxiliary elliptic operator P), as determined in a joint work with Seeley 1995. The definition of TR A is extended from the cases of noninteger order, or integer order and eveneven parity on odddimensional manifolds, to the case of evenodd parity on evendimensional manifolds. For the generalized zeta function ζ(A, P, s) = Tr(AP −s), extended meromorphically to C, C0(A, P) equals the coefficient of s 0 in the Laurent expansion at s = 0 when P is invertible. In the mentioned parity cases, ζ(A, P, s) is regular at all integer points. The higher Laurent coefficients Cj(A, P) at s = 0 are described as leading nonlocal coefficients C0(B, P) in trace expansions of resolvent expressions B(P −λ) −N, with B logpolyhomogeneous as defined by Lesch (here −C1(I, P) = C0(log P, P) gives the zetadeterminant). C0(B, P) is shown to be a quasitrace in general, a canonical trace TR B in restricted cases, and the formula of Lesch for TR B in terms of a finite part integral of the symbol is extended to the parity cases.