Pseudodifferential operators on manifolds with fibred boundaries

Abstract

Respectfully dedicate to Professor M. Sato on the occasion of his 70th birthday Abstract. Let X be a compact manifold with boundary. Suppose that the boundary is fibred, φ : ∂X −→ Y, and let x ∈ C ∞ (X) be a boundary defining function. This data fixes the space of 'fibred cusp' vector fields, consisting of those vector fields V on X satisfying V x = O(x 2 ) and which are tangent to the fibres of φ; it is a Lie algebra and C ∞ (X) module. This Lie algebra is quantized to the 'small calculus' of pseudodifferential operators Ψ * Φ (X). Mapping properties including boundedness, regularity, Fredholm condition and symbolic maps are discussed for this calculus. The spectrum of the Laplacian of an 'exact fibred cusp' metric is analyzed as is the wavefront set associated to the calculus.

Finiteness of theL2-index of the Dirac operator of generalized Euclidean Taub–NUT metrics

“…We first note that the essential spectrum of the associated Dirac operator is R, and we describe its domain. This is a direct application of the work done in [13] and of the theory of Φ-pseudodifferential calculus developed in [14]. Next we show that the dimension of the kernel is finite.…”

“…We have noticed in [13] that ds K 2 belongs to the class of fibered cusp metrics from [14]. Moreover, its Dirac operator D is elliptic but not fully elliptic in this calculus.…”

“…From [14], D is Fredholm from its domain H 1 Φ to L 2 if and only if it is fully elliptic. Thus let us compute its normal operator.…”

“…The normal operator is obtained in two steps (see [14]). We first formally replace x 2 ∂ x with iξ, ξ ∈ R, and xJ/2, xK/2 with τ 2 , τ 3 where τ 2 , τ 3 ∈ R are global coordinates on the vector bundle φ * T S 2 over S 3 (note that T S 2 is not trivial, but its pull-back to S 3 through the Hopf fibration is).…”

Bott periodicity for fibred cusp operators

“…In this thesis, we will describe how Bott periodicity arises when one considers instead fibred cusp operators Ψ * Φ (X; E) on a compact manifold with boundary X acting on some complex vector bundle E. These operators were introduced by Mazzeo and Melrose in [12]. The definition involves a defining function for the boundary ∂X and a fibration Φ : ∂X → Y of the boundary.…”