Cheeger–Müller theorem on manifolds with cusps

Abstract

We prove equality between the renormalized Ray-Singer analytic torsion and the intersection R-torsion on a Witt-manifold with cusps, up to an error term determined explicitly by the Betti numbers of the cross section of the cusp and the intersection R-torsion of a model cone. In the first step of the proof we compute explicitly the renormalized Ray-Singer analytic torsion of a model cusp in general dimension and without the Witt-condition. In the second step we establish a gluing formula for renormalized Ray-Singer analytic torsion on a general class of non-compact manifolds in any dimension that includes Witt-manifolds with cusps, but also scattering manifolds with asymptotically conical ends. In the final step, a Cheeger-Müller theorem on cusps follows by a combination of the previous explicit computation and the gluing formula.

Zeta-determinants of Sturm–Liouville operators with quadratic potentials at infinity

“…Spectral geometry of such manifolds has been initiated by Müller in his paper [Mül83]. A recent work by the third named author [Ver14] discusses analytic torsion on such spaces and in particular strongly relies on computations of zeta-determinants of such operators.…”

“…The statement now follows from the fact that integrals of the Schwartz kernels for (D µ,1 + z 2 ) −1 and (D µ,2 + z 2 ) −1 along the diagonal in [1, δ] admit an asymptotic expansion of the form ∞ k=0 a k (µ)z −1−k , where the leading order term a 0 is independent of the boundary conditions. The next proposition is proved in [Ver14] without specifying the leading coefficients in the asymptotic expansion. By keeping track of the coefficients in the asymptotic expansions of Sec.…”