A Parametrix Construction for the Laplacian on Q-rank 1 Locally Symmetric Space

This paper presents the construction of parametrices for the Gauss-Bonnet and Hodge Laplace operators on noncompact manifolds modelled on Q-rank 1 locally symmetric spaces. These operators are, up to a scalar factor, $\phi$-differential operators, that is, they live in the generalised $\phi$-calculus studied by the authors in a previous paper, which extends work of Melrose and Mazzeo. However, because they are not totally elliptic elements in this calculus, it is not possible to construct parametrices for these operators within the $\phi$-calculus. We construct parametrices for them in this paper using a combination of the $b$-pseudodifferential operator calculus of R. Melrose and the $\phi$-pseudodifferential operator calculus. The construction simplifies and generalizes the construction done by Vaillant in his thesis for the Dirac operator. In addition, we study the mapping properties of these operators and determine the appropriate Hlibert spaces between which the Gauss-Bonnet and Hodge Laplace operators are Fredholm. Finally, we establish regularity results for elements of the kernels of these operators.Comment: 29 pages; in revision: improved exposition, unified use of rescaled bundles and half densities, corrected misprint

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 $(X,g)$ 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 $X$, 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 $g$ is incomplete, this requires a consideration of the various choices of ideal boundary conditions at the singular set. We also calculate the space of $L^2$ harmonic forms for any complete edge metric on the regular part of $X$

Hodge and signature theorems for a family of manifolds with fibre bundle boundary

Over the past fifty years, Hodge and signature theorems have been proved for various classes of noncompact and incomplete Riemannian manifolds. Two of these classes are manifolds with incomplete cylindrical ends and manifolds with cone bundle ends, that is, whose ends have the structure of a fibre bundle over a compact oriented manifold, where the fibres are cones on a second fixed compact oriented manifold. In this paper, we prove Hodge and signature theorems for a family of metrics on a manifold M with fibre bundle boundary that interpolates between the incomplete cylindrical metric and the cone bundle metric on M . We show that the Hodge and signature theorems for this family of metrics interpolate naturally between the known Hodge and signature theorems for the extremal metrics. The Hodge theorem involves intersection cohomology groups of varying perversities on the conical pseudomanifold X that completes the cone bundle metric on M . The signature theorem involves the summands τ i of Dai’s τ invariant [J Amer Math Soc 4 (1991) 265–321] that are defined as signatures on the pages of the Leray–Serre spectral sequence of the boundary fibre bundle of M . The two theorems together allow us to interpret the τ i in terms of perverse signatures, which are signatures defined on the intersection cohomology groups of varying perversities on X

L2 harmonic forms for a class of complete Kahler metrics

The Hodge theorem for compact manifolds states that every real cohomology class of a compact manifold M is represented by a unique harmonic form. That is, the space of solutions to the differential equation .d Cd / D 0 on L2 forms over M; a space that depends on the metric on M; is canonically isomorphic to the purely topological real cohomology space of M: This isomorphism is enormously useful because it provides a way to transform theorems from geometry into theorems in topology and vice versa. No such result holds in general for complete noncompact manifolds, but in many specific cases there are Hodge-type theorems. One of the oldest is the description, due to Atiyah, Patodi, and Singer [1], of the space of L2 harmonic forms on a manifold with complete cylindrical ends. By calculating the solutions to the equation for harmonic forms on the cylindrical ends, they showed that the space of L2 harmonic forms is isomorphic to the image of the relative cohomology of the manifold in the absolute cohomology. Another Hodge-type result was found by Zucker [14] for a natural class of metrics called Poincaré metrics. These metrics, first constructed by Cornalba and Griffiths [4], are complete Kähler metrics with hyperbolic cusp-type singularities at isolated points on a Riemann surface. Zucker showed that the space of L2 forms on a Riemann surface that are harmonic with respect one of these metrics is isomorphic to the standard cohomology of the surface. This result was extended by Cattani, Kaplan, and Schmid [3] to analogous metrics on bundles over projective varieties with singularities along a divisor. These metrics can be thought of as complete Kähler metrics on the noncompact manifold given by removing the divisor

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 $V$ 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 $Z/\rho^2$, with $\rho(x) = |x - p|$ for $x$ close to $p$ and $Z$ continuous on \RR^3 with $Z(p) > -1/4$ for p \in \maS. Also assume that $\rho$ and $Z$ are smooth outside \maS and $Z$ is smooth in polar coordinates around each singular point. We either assume that $V$ is periodic or that the set \maS is finite and $V$ 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 $\Lambda$ 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 $H = -\Delta + V$ acting on L^2(\TT), as well as for the induced \vt k--Hamiltonians \Hk obtained by restricting the action of $H$ 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