Adiabatic Limit, Heat Kernel and Analytic Torsion

The adiabatic limit refers to the geometric degeneration in which the metric is been blown up along certain directions. The study of the adiabatic limit of geometric invariants is initiated by E. Witten [39], who relates the adiabatic limit of the η-invariant to the holonomy of determinant line bundle, the so called “global anomaly”. In this case the manifold is fibered over a circle and the metric is been blown up along the circle direction. Witten’s result was given full mathematical treatment in [8], [9] and [13], see also [16]. In [4], J.-M. Bismut and J. Cheeger studied the adiabatic limit of the eta invariant for a general fibration of closed manifolds. Assuming the invertibility of the Dirac family along the fibers, they showed that the adiabatic limit of the η-invariant of a Dirac operator on the total space is expressible in terms of a canonically constructed differential form, η̃, on the base. The Bismut-Cheeger η̃ form is a higher dimensional analogue of the η-invariant and it is exactly the boundary correction term in the families index theorem for manifolds with boundary, [5], [6]. The families index theorem for manifolds with boundary has since been established in full generality by MelrosePiazza in [31], [32]. Around the same time, Mazzeo and Melrose took on the analytic aspect of the adiabatic limit [27] and studied the uniform structure of the Green’s operator of the Laplacian in the adiabatic limit. Their analysis enables the first author to prove the general adiabatic limit formula in [14]. The adiabatic limit formula is used in [7] to prove a generalization of the Hirzebruch conjecture on the signature defect (Cf. [1],[35]). Other applications of adiabatic limit technique can be found in [40], [18] and [36]. The main purpose of this paper is to study the uniform behavior of the heat kernel in the adiabatic limit. The adiabatic limit introduces degeneracy along the base directions and gives rise to new singularity for the heat kernel which interacts in a complicated way with the usual diagonal singularity. We resolve this difficulty by lifting the heat kernel to a larger space obtained by blowing up certain submanifolds of the usual carrier space of the heat kernel (times the adiabatic direction). The new space is a manifold with corner and the uniform structure of the adiabatic heat kernel can be expressed by stating that it gives rise to a polyhomogeneous conormal distribution on the new space. More precisely, if φ : M −→ Y is a fibration with typical fibre F, the adiabatic metric is the one-parameter family of metrics xgx, with gx = φ h + xg, on M, where h is a metric on Y and g a symmetric 2-tensor on M which restricts to Riemannian metrics on the fibers. Note that gx collapses the fibration to the base space in the limit x → 0. Our main object of study is the regularity of the heat