Index and Spectral Theory for Manifolds with Generalized Fibred Cusps

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.