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Index and spectral theory for manifolds with generalized fibred cusps. (English) Zbl 1059.58018

Bonner Mathematische Schriften 344. Bonn: Univ. Bonn, Mathematisches Institut (Diss.). ii, 124 p. (2001).
Summary: Generalizing work of W. Müller [‘Manifolds with cusps of rank one. Spectral theory and \(L^2\)-index theorem’. Lect. Notes Math. (1987; Zbl 0632.58001)] we investigate the spectral theory for the Dirac operator D on a noncompact manifold \(X^n\) with generalized fibred cusps \[ C(M)=M\times[A,\infty [_r,\quad g=dr^2+\varphi^*g_Y+ e^{-2cr}g_X, \] at infinity. Here \(\varphi:M^{h+v} \to Y^h\) is a compact fibre bundle with fibre \(Z\) and a distinguished horizontal spare \(HM\). 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 [Asian J. Math. 2, No. 4, 833–866 (1998; Zbl 1125.58304)] 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 \([\exp(-t\text{D}^2)]\) 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 for small times in the interior of the manifold as well as at spatial infinity. This then allows us to prove an index formula for D, \[ \text{ind}_-(\text{D})= \frac{1}{(2\pi i)^{n/2}}\int_X\widehat A(R)\text{Ch}(F^{E/S})+ \frac{1}{(2\pi i)^{(h+1)/2}}\int_Y\widehat A(R^Y)\widehat\eta(\text{D}^V)+\frac 12\eta (\text{D}_Y), \] which calculates 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.

MSC:

58J20 Index theory and related fixed-point theorems on manifolds
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58-02 Research exposition (monographs, survey articles) pertaining to global analysis

Keywords:

Zbl 1125.58304
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