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Index and Spectral Theory for Manifolds with Generalized Fibred Cusps | | Boris Vaillant
; | | Date: |
8 Feb 2001 | | Subject: | Differential Geometry; Spectral Theory MSC-class: 58G25; 58Gxx | math.DG math.SP | | 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 $$ C(M)=M imes [A,infty[_r, g= d r^2+ phi^*g_Y+ e^{-2cr}g_Z, $$ at infinity. Here $phi:M^{h+v} o Y^h$ 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. | | Source: | arXiv, math.DG/0102072 | | Services: | Forum | Review | PDF | Favorites | |
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