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1.
We consider a compact manifold X whose boundary is a locally trivial fiber bundle, and an associated pseudodifferential algebra
that models fibered cusps at infinity. Using tracelike functionals that generate the 0-dimensional Hochschild cohomology groups
we first express the index of a fully elliptic fibered cusp operator as the sum of a local contribution from the interior
of X and a term that comes from the boundary. This leads to an abstract answer to the index problem formulated in [11]. We
give a more precise answer for firstorder differential operators when the base of the boundary fiber bundle is S1. In particular, for Dirac operators associated to a metric of the form
near ∂X = {x = 0} with twisting bundle T we obtain
in terms of the integral of the Atiyah-Singer form in the interior of X, and the adiabatic limit of the η-invariant of the
restriction of the operator to the boundary. 相似文献
2.
Robert Lauter Sergiu Moroianu 《Transactions of the American Mathematical Society》2003,355(8):3009-3046
The Hochschild homology of the algebra of pseudodifferential operators on a manifold with fibered cusps, introduced by Mazzeo and Melrose, is studied and computed using the approach of Brylinski and Getzler. One of the main technical tools is a new convergence criterion for tri-filtered half-plane spectral sequences. Using trace-like functionals that generate the -dimensional Hochschild cohomology groups, the index of a fully elliptic fibered cusp operator is expressed as the sum of a local contribution of Atiyah-Singer type and a global term on the boundary. We announce a result relating this boundary term to the adiabatic limit of the eta invariant in a particular case.
3.
G. Ya. Areshkin 《Journal of Mathematical Sciences》1989,47(6):2903-2907
Let X be the F-space of the functions x(t) defined on the measurable space (T, Σ, μ) with values in B-space Y. We consider
the operators f mapping X to the B-space Z. X, Y, and Z are considered over the scalar field R. To each operator f is associated
the family Φf of vector-valued functions
. The characteristics of these families are given for various classes of operators. The relationship of convergence and continuation
of the operators f with convergence and continuation of the corresponding families Φf is considered. Riesz' theorem on integral representation of linear functionals is generalized.
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR,
Vol. 159, pp. 113–118, 1987. 相似文献
4.
《K-Theory》2006,37(1-2):25-104
A families index theorem in K-theory is given for the setting of Atiyah, Patodi, and Singer of a family of Dirac operators with spectral boundary condition.
This result is deduced from such a K-theory index theorem for the calculus of cusp, or more generally fibred-cusp, pseudodifferential operators on the fibres
(with boundary) of a fibration; a version of Poincaré duality is also shown in this setting, identifying the stable Fredholm
families with elements of a bivariant K-group.
(Received: February 2006) 相似文献
5.
We consider the heat operator of a Bismut superconnection for a family of generalized Dirac operators defined along the leaves of a foliation with Hausdorff graph. We assume that the strong Novikov–Shubin invariants of the Dirac operators are greater than three times the codimension of the foliation. We compute the t asymptotics associated to a rescaling of the metric by 1/t and show that the heat operator converges to the Chern character of the index bundle of the operator. Combined with previous results, this gives a general families index theorem for such operators. 相似文献
6.
J?rn M��ller 《Geometric And Functional Analysis》2011,21(2):443-482
A manifold with fibered cusp metrics X can be considered as a geometrical generalization of locally symmetric spaces of
\mathbbQ{\mathbb{Q}}-rank one at infinity. We prove a Hodge-type theorem for this class of Riemannian manifolds, i.e. we find harmonic representatives
of the de Rham cohomology H
p
(X). Similar to the situation of locally symmetric spaces, these representatives are computed by special values or residues
of generalized eigenforms of the Hodge–Laplace operator on Ω
p
(X). 相似文献
7.
V. Nistor 《Acta Mathematica Hungarica》2003,99(1-2):155-183
We define the gauge-equivariant index of a family of elliptic operators invariant with respect to the free action of a family
of Lie groups (these families are called ``gauge-invariant families' in what follows). If the fibers of
are simply-connected and solvable, we compute the Chern character of the gauge-equivariant index, the result being given
by an Atiyah–Singer type formula that incorporates also topological information on the bundle
. The algebras of invariant pseudodifferential operators that we study,
and
, are generalizations of ``parameter dependent' algebras of pseudodifferential operators (with parameter in R
q), so our results provide also an index theorem for elliptic, parameter dependent pseudodifferential operators. We apply these
results to study Fredholm boundary conditions on a simplex.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
8.
In this paper for a positive real number α we consider two partial differential operators D and Dα on the half–plane
We define a generalized Fourier transform
associated with the operators D and Dα. We establish an analogue of Beurling–H?rmander’s Theorem for this transform
and we give some applications of this theorem. 相似文献
9.
10.
The ellipticity of operators on a manifold with edge is defined as the bijectivity of the components of a principal symbolic
hierarchy , where the second component takes values in operators on the infinite model cone of the local wedges. In the general understanding
of edge problems there are two basic aspects: Quantisation of edge-degenerate operators in weighted Sobolev spaces, and verifying
the ellipticity of the principal edge symbol which includes the (in general not explicitly known) number of additional conditions of trace and potential type on the edge.
We focus here on these questions and give explicit answers for a wide class of elliptic operators that are connected with
the ellipticity of edge boundary value problems and reductions to the boundary. In particular, we study the edge quantisation
and ellipticity for Dirichlet–Neumann operators with respect to interfaces of some codimension on a boundary. We show analogues
of the Agranovich–Dynin formula for edge boundary value problems.
Nicoleta Dines and Bert-Wolfgang Schulze were supported by Chinese-German Cooperation Program “Partial Differential Equations”,
NNSF of China and DFG of Germany. Xiaochun Liu was supported by NNSF of China through Grant No. 10501034, and Chinese-German
Cooperation Program “Partial Differential Equations”, NNSF of China and DFG of Germany. 相似文献
11.
Sergiu Moroianu 《Proceedings of the American Mathematical Society》2006,134(11):3395-3404
We give an analytic proof of the fact that the index of an elliptic operator on the boundary of a compact manifold vanishes when the principal symbol comes from the restriction of a -theory class from the interior. The proof uses non-commutative residues inside the calculus of cusp pseudodifferential operators of Melrose.
12.
Guoyuan Chen 《数学学报(英文版)》2013,29(5):975-992
A version of the "Fredholm index = spectral flow" theorem is proved for general families of elliptic operators {A(t)} t∈R on closed (compact and without boundary) manifolds. Here we do not require that A(t), t∈R or its leading part is self-adjoint. 相似文献
13.
Dan Kucerovsky 《Proceedings of the American Mathematical Society》2001,129(12):3729-3736
We give a -theoretical proof of an index theorem for Dirac-Schrödinger operators on a noncompact manifold.
14.
This paper is the first of two papers constructing a calculus of pseudodifferential operators suitable for doing analysis on -rank 1 locally symmetric spaces and Riemannian manifolds generalizing these. This generalization is the interior of a manifold with boundary, where the boundary has the structure of a tower of fibre bundles. The class of operators we consider on such a space includes those arising naturally from metrics which degenerate to various orders at the boundary, in directions given by the tower of fibrations. As well as -rank 1 locally symmetric spaces, examples include Ricci-flat metrics on the complement of a divisor in a smooth variety constructed by Tian and Yau. In this first part of the calculus construction, parametrices are found for “fully elliptic differential a-operators,” which are uniformly elliptic operators on these manifolds that satisfy an additional invertibility condition at infinity. In the second part we will consider operators that do not satisfy this condition. 相似文献
15.
François Labourie 《Publications Mathématiques de L'IHéS》2007,106(1):139-213
This article relates representations of surface groups to cross ratios. We first identify a connected component of the space
of representations into PSL(n,ℝ) – known as the n-Hitchin component – to a subset of the set of cross ratios on the boundary at infinity of the group. Similarly, we study some representations
into associated to cross ratios and exhibit a “character variety” of these representations. We show that this character variety
contains all n-Hitchin components as well as the set of negatively curved metrics on the surface. 相似文献
16.
Francesco L. Chiera 《Mathematische Zeitschrift》2006,253(3):461-477
We show that spaces of vector–valued singular modular forms for principal congruence subgroups of the symplectic group Sp(n,ℤ) of integral weight are generated by suitable finite dimensional families of Siegel theta series. This is obtained as an application of some
results concerning the action of trace operators on non–homogeneous theta series. 相似文献
17.
Jean-Pierre Magnot 《Acta Appl Math》2006,91(1):67-95
We state a Chern–Weil type theorem which is a generalization of a Chern–Weil type theorem for Fredholm structures stated by Freed in [4]. Using this result, we investigate Chern forms on based manifold of maps following two approaches, the first one using the Wodzicki residue, and the second one using renormalized traces of pseudo-differential operators along the lines of [1, 19, 20]. We specialize to the case to study current groups. Finally, we apply these results to a class of holomorphic connections on the loop group . In this last example, we precise Freed's construction [5] on the loop group: The cohomology of the first Chern form of any holomorphic connection in the class considered is given by the Kähler form. 相似文献
18.
Julius Borcea 《Arkiv f?r Matematik》2006,44(2):211-240
The spectral order on R
n
induces a natural partial ordering on the manifold of monic hyperbolic polynomials of degree n. We show that all differential operators of Laguerre–Pólya type preserve the spectral order. We also establish a global monotony
property for infinite families of deformations of these operators parametrized by the space ℓ∞ of real bounded sequences. As a consequence, we deduce that the monoid of linear operators that preserve averages of zero sets and hyperbolicity consists only of differential operators of Laguerre–Pólya
type which are both extensive and isotonic. In particular, these results imply that any hyperbolic polynomial is the global
minimum of its -orbit and that Appell polynomials are characterized by a global minimum property with respect to the spectral order. 相似文献
19.
Xinhui Jiang 《K-Theory》1997,12(4):319-359
In this paper we shall study the indices of some leafwise elliptic operators on foliated bundles, by using some concretely constructed cyclic cocycles on the smooth foliation algebras. The key is a decomposition of the KK-classes associated to the operators. The group action analogue of this approach provides a bridge from the classical Atiyah-Singer index theorem to the higher -index theorem of Connes and Moscovici. 相似文献
20.
Ian Short 《Geometriae Dedicata》2006,119(1):91-104
Every open ball within has an associated hyperbolic metric and Möbius transformations act as hyperbolic isometries from one ball to another. The Hillam–Thron Theorem is concerned with images of balls under Möbius transformation, yet existing proofs of the theorem do not make use of hyperbolic geometry. We exploit hyperbolic geometry in proving a generalisation of the Hillam–Thron Theorem and examine the precise configurations of points and balls that arise in that theorem.This work was supported by Science Foundation Ireland grant 05/RFP/MAT0003 相似文献