Spectral asymptotics of foliated manifolds
James L. Heitsch, Connor Lazarov · Illinois Journal of Mathematics · 1994
LAZAROV transversals forms a r-ring and a transverse measure v is a measure on this g-ring.The transverse measure is called invariant if it is invariant under the holonomy pseudogroup acting on smooth transversals.A Riemannian metric g on M gives rise to a volume form h on each leaf L of F. The family h {h L} and an invariant transverse measure v give rise to a measure tz h du on M ([C], [M-S] chapterIV).Let E be a smooth vector bundle on M with smooth metric, ElL the bundle restricted to the leaf L, and L x the leaf through the point x.Let H x LE(Lx, EILx), the Hilbert space of L 2 sections of ElL x over L x with inner product ( )x.A section of the family {H x} is a function s: M -13 H x with s(x) H x. A measurable structure on {H x} is a sequence {s n} of sections such that for each x, {Sn(X)} generates H x as a Hilbert space.See [Dix], p. 161 for a complete discussion.The measurable structure we use is given in the appendix of [H-L2].A section s of {H x} is called measurable if (s(x), Sn(X)) x is a Borel function on M for all n.An inner product on the measurable sections is given by (s,t) fM(S(X),t(X))xdl(x).A measurable section s is square integrable if Ilsl[ 2 (s, s) < oo.Let be the collection of square integrable sections with the inner product ( ), and identify two sections if t.hey are equal almost everywhere on M (with respect to the measure/).H is then the direct integral of the family {Hx}.Let U and V be foliation charts on M and y a leafwise path from U to V. Let U yV denote the subset of U V consisting of those points (x, y) such that x domain of h and y h(Px).Here h is the holonomy map corresponding to y, Px is the placque of x in U and h(Px) is the placque of hv(x) in V. Let C(U, V, y) denote the set of leafwise smooth, uniformly bounded, measurable sections of the bundle E (R) E* over M M which are compactly supported in U y V.An element of C(U, V, 3') gives rise to a measurable family of bounded operators on {H x} and hence a bounded operator on .W(F; E) is the von Neumann algebra of operators on generated by the C6(U, V, y) where U and V are in a fixed cover of M by foliation charts and 3' ranges over the holonomy classes of leafwise paths from U to V. (See [H-L1, 2], [C], [M-S] and [Dix] p. 181.)An invariant transverse measure u determines a trace, try, on W(F; E) ([C], [M-S], [H-L1].For an element k C(U, V, y), k(x, y) E x (R) E, so tr k(x, x) is well defined and tr k is given by tr.k fMtr k(x, x)h dr.We can replace the given family of metrics {gL} on the leaves by any other family {w L} of leafwise smooth transversely measurable leafwise metrics