Index and Spectral Theory for Manifolds with Generalized Fibred Cusps

TL;DR

This paper develops a spectral theory for Dirac operators on manifolds with generalized fibred cusps, constructing resolvent and heat kernel expansions, and deriving an index formula involving eta invariants.

Contribution

It extends spectral analysis techniques to manifolds with generalized fibred cusps, explicitly constructing resolvent and heat kernels, and deriving an index formula involving eta invariants.

Findings

Explicit meromorphic continuation of the resolvent G(λ)

Construction of the heat kernel for small times

Index formula involving eta invariants

Abstract

Generalizing work of W. M\"uller we investigate the spectral theory for the Dirac operator D on a noncompact manifold X with generalized fibred cusps $$ C(M)=M\times [A,\infty[_r, g= d r^2+ \phi^*g_Y+ e^{-2cr}g_Z, $$ at infinity. Here $\phi:M^{h+v}\to Y^h$ is a compact fibre bundle with fibre Z and a distinguished horizontal space HM. The metric $g_Z$ is a metric in the fibres and $g_Y$ is a metric on the base of the fibration. We also assume that the kernel of the vertical Dirac operator at infinity forms a vector bundle over $Y$. Using the ``$\phi$-calculus'' developed by R. Mazzeo and R. Melrose we explicitly construct the meromorphic continuation of the resolvent $G(\lambda)$ of D for small spectral parameter as a special ``conormal distribution''. From this we deduce a description of the generalized eigensections and of the spectral measure of D. Complementing this, we perform an explicit construction of the heat kernel $[\exp(-tD^2)]$ for finite and small times t, corresponding to large spectral parameter $\lambda$. Using a generalization of Getzler's technique, due to R. Melrose, we can describe the singular terms in the heat kernel expansion and prove an index formula for D, calculating the extended $L^2$-index of D in terms of the usual local expression, the family eta invariant for the family of vertical Dirac operators at infinity and the eta invariant for the horizontal ``Dirac'' operator at infinity.