ISSN:
1420-8970
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Let (M, h) be an odd-dimensional compact spin manifold in whichH is an embedded hypersurface with quadratic defining functionx 2∈C ∞ (M). Let be the Dirac operator associated to the metric jhen where ∈〉0 is a parameter. The limiting metric,g 0, is an exact b-metric on the compact manifold with boundary $$\overline M $$ obtained by cuttingM alongH and compactifying as a manifold with boundary, i.e. it givesM/H asymptotically cylindrical ends with cross-section $$\partial \overline M $$ , a double cover ofH. Under the assumption that the induced Dirac operator on this double cover is invertible we show that where is the ‘b’ version of the eta invariant introduced in [Me1],r 1(∈) andr 2(∈) are smooth, vanish at ∈=0 and are integrals of local geometric data, and where $$\tilde \eta ( \in )$$ is the finite dimensional eta invariant, or signature, for the small eigenvalues of . If is invertible then $$\tilde \eta ( \in ) \equiv 0$$ and Even if is not invertible this holds in ℝ/ℤ. In fact the discussion takes place more naturally in the context of the generalized Dirac operators associated to Hermitian Clifford modules. These results are proved by analyzing the resolvent family of , uniformly away from the spectrum and near zero. This leads to a precise description of the behaviour of the small eigenvalues. The corresponding ‘heat calculus’ is also constructed. It contains, and hence describes rather precisely, the heat kernel for uniformly as ∈→0. This calculus is related to, but different from, the surgery pseudodifferential operator calculus of McDonald ([Mc]).
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01928215
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