Index and Spectral Theory for Manifolds with Generalized Fibred Cusps
| dc.creator | Vaillant, Boris | |
| dc.date | 2001-02-08 | |
| dc.date.accessioned | 2026-07-07T04:40:05Z | |
| dc.date.available | 2026-07-07T04:40:05Z | |
| dc.description | Generalizing work of W. Müller we investigate the spectral theory for the Dirac operator D on a noncompact manifold X with generalized fibred cusps $$ C(M)=M\times [A,\infty[_r, g= d r^2+ ϕ^*g_Y+ e^{-2cr}g_Z, $$ at infinity. Here $ϕ:M^{h+v}\to Y^h$ 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 ``$ϕ$-calculus'' developed by R. Mazzeo and R. Melrose we explicitly construct the meromorphic continuation of the resolvent $G(λ)$ 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 $λ$. 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. | |
| dc.description | 122 pages, 8 figures, doctoral thesis | |
| dc.identifier | https://arxiv.org/abs/math/0102072 | |
| dc.identifier | http://arxiv.org/abs/math/0102072 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/60919 | |
| dc.subject | Differential Geometry | |
| dc.subject | Spectral Theory | |
| dc.subject | 58G25; 58Gxx | |
| dc.title | Index and Spectral Theory for Manifolds with Generalized Fibred Cusps | |
| dc.type | text |