Index and Spectral Theory for Manifolds with Generalized Fibred Cusps

Abstract

Generalizing work of W. Müller we investigate the spectral theory for the Dirac operator D on a noncompact manifold X with generalized fibred cusps $$C(M)=M\times [A,\infty {[}_r,g=d{r}^2+{\varphi }^{\ast }{g}_Y+e^{-2cr}{g}_Z,$$ at infinity. Here $\varphi :M^{h+v}\to Y^h$ 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 ``$\varphi$-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 $[\exp (-t{D}^2)]$ 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.

122 pages, 8 figures, doctoral thesis

Cited by in corpus (20)

• Hodge and signature theorems for a family of manifolds with fibration boundary
• Analytic torsion and R-torsion of Witt representations on manifolds with cusps
• Geometry of Spin and Spin^c structures in the M-theory partition function
• Ricci flow and the determinant of the Laplacian on non-compact surfaces
• L^2 harmonics forms on non compact manifolds
• The signature package on Witt spaces, I. Index classes
• On the Homotopy Classification of Elliptic Operators on Manifolds with Edges
• The heat kernel on curvilinear polygonal domains in surfaces
• Asymptotics of relative heat traces and determinants on open surfaces of finite area
• Cheeger-Mueller Theorem on manifolds with cusps
• Pseudodifferential operator calculus for generalized Q-rank 1 locally symmetric spaces, I
• Spectral geometry on manifolds with fibred boundary metrics I: Low energy resolvent
• The relative L^2 index theorem for Galois coverings
• Low energy limit for the resolvent of some fibered boundary operators
• Polyhomogénéité des métriques asymptotiquement hyperboliques complexes le long du flot de Ricci
• Resolvent, heat kernel and torsion under degeneration to fibered cusps
• $L^2$ cohomology of Manifolds with flat ends
• Additivity and non-additivity for perverse signatures
• A Hodge-Type Theorem for Manifolds with Fibered Cusp Metrics
• Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps