1,232 research outputs found
Pseudodifferential operator calculus for generalized Q-rank 1 locally symmetric spaces, I
This paper is the first of two papers constructing a calculus of
pseudodifferential operators suitable for doing analysis on Q-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 Q-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 \bfa-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.Comment: 44 pages, 2 figures -- Some explanations, references added; changed
normalization of index sets in full calculus to make it more natural; made
full calculus composition result more complet
Harmonic forms on manifolds with edges
Let be a compact Riemannian stratified space with simple edge
singularity. Thus a neighbourhood of the singular stratum is a bundle of
truncated cones over a lower dimensional compact smooth manifold. We calculate
the various polynomially weighted de Rham cohomology spaces of , as well as
the associated spaces of harmonic forms. In the unweighted case, this is
closely related to recent work of Cheeger and Dai \cite{CD}. Because the metric
is incomplete, this requires a consideration of the various choices of
ideal boundary conditions at the singular set. We also calculate the space of
harmonic forms for any complete edge metric on the regular part of
Hodge cohomology of gravitational instantons
We study the space of L^2 harmonic forms on complete manifolds with metrics
of fibred boundary or fibred cusp type. These metrics generalize the geometric
structures at infinity of several different well-known classes of metrics,
including asymptotically locally Euclidean manifolds, the (known types of)
gravitational instantons, and also Poincar\'e metrics on Q-rank 1 ends of
locally symmetric spaces and on the complements of smooth divisors in K\"ahler
manifolds. The answer in all cases is given in terms of intersection cohomology
of a stratified compactification of the manifold. The L^2 signature formula
implied by our result is closely related to the one proved by Dai [dai] and
more generally by Vaillant [Va], and identifies Dai's tau invariant directly in
terms of intersection cohomology of differing perversities. This work is also
closely related to a recent paper of Carron [Car] and the forthcoming paper of
Cheeger and Dai [CD]. We apply our results to a number of examples,
gravitational instantons among them, arising in predictions about L^2 harmonic
forms in duality theories in string theory.Comment: 45 pages; corrected final version. To appear in Duke Math. Journa
Analysis of Schr\"odinger operators with inverse square potentials I: regularity results in 3D
Let be a potential on \RR^3 that is smooth everywhere except at a
discrete set \maS of points, where it has singularities of the form
, with for close to and continuous on
\RR^3 with for p \in \maS. Also assume that and
are smooth outside \maS and is smooth in polar coordinates around each
singular point. We either assume that is periodic or that the set \maS is
finite and extends to a smooth function on the radial compactification of
\RR^3 that is bounded outside a compact set containing \maS. In the
periodic case, we let be the periodicity lattice and define \TT :=
\RR^3/ \Lambda. We obtain regularity results in weighted Sobolev space for the
eigenfunctions of the Schr\"odinger-type operator acting on
L^2(\TT), as well as for the induced \vt k--Hamiltonians \Hk obtained by
restricting the action of to Bloch waves. Under some additional
assumptions, we extend these regularity and solvability results to the
non-periodic case. We sketch some applications to approximation of
eigenfunctions and eigenvalues that will be studied in more detail in a second
paper.Comment: 15 pages, to appear in Bull. Math. Soc. Sci. Math. Roumanie, vol. 55
(103), no. 2/201
Justice Is Not Just a Word
Every civilized society, from the earliest dawn of history, has had some men set apart from the other members of the clan, tribe, province, state or nation, to decide controversies and issues of fact according to the best wisdom they possessed. They were (and are) the wise men of their time and age. They were and are the law men
- …