Pseudodifferential operators on manifolds with foliated boundaries

Abstract

Let X be a smooth compact manifold with boundary. For smooth foliations on the boundary of X admitting a 'resolution' in terms of a fibration, we construct a pseudodifferential calculus generalizing the fibred cusp calculus of Mazzeo and Melrose. In particular, we introduce certain symbols leading to a simple description of the Fredholm operators inside the calculus. When the leaves of the fibration 'resolving' the foliation are compact, we also obtain an index formula for Fredholm perturbations of Dirac-type operators. Along the way, we obtain a formula for the adiabatic limit of the eta invariant for invertible perturbations of Dirac-type operators, a result of independent interest generalizing the well-known formula of Bismut and Cheeger.

Geometrically Finite Poincaré–Einstein Metrics

“…Near H 0 , but away from the other boundary hypersurfaces, a (0 − F)-vector field behaves like a 0-vector field of [MM87]. Near H α , but away from H 0 , it looks instead like a foliated cusp vector field (or F i -vector field) of [Roc12], the foliated version of fibered cusp vector fields of [MM98]. Near H i , i ≥ n ir + 1, it looks like a cusp vector field (a fibred cusp vector field associated to a trivial fibration over a point).…”

Singular spaces, groupoids and metrics of positive scalar curvature

“…The vector fields in V F (M, H), together with C ∞ (M ), generate an algebra of differential operators on M that we denote Diff * F (M, H). In the case of M being a manifold with foliated boundary we use the notation Diff * F (M ) as in [Roc12]. If D ∈ Diff * F (M, H) then D lifts uniquely to a differential operator on the groupoid (67) G F (M, H) := Hol(M, H) |M ∪ r * N H F × R with N H F := T H |∂M /T F and r denoting the range map for the holonomy groupoid Hol(∂M, F).…”

On a Hitchin–Thorpe inequality for manifolds with foliated boundaries

“…1) The meaning of the term "product-type" metric differs from one author to another. In this paper, we are following the convention used in [Vai01] and [DW07], but in [MM98] and [Roc12], "product-type" is what we call "asymptotic" here.…”

“…where h is a Γ-invariant metric on Σ, and κ ∈ C ∞ ( W , S 2 T * W ) is a Γ-invariant tensor restricting to a metric on the fibres of W → Σ. With the same notations, a metric g F c on M is a (product-type) foliated cusp metric ( [Roc12], (1.6) p.1314) if it is such that…”

“…Manifolds admitting such metrics are the main topic of [Roc12], where a pseudodifferential calculus adapted to this setting is constructed, before addressing the index theory of the associated Dirac-type operators. More recently, a study of their Hodge cohomology has been carried-out in [GR15].…”