Luiz Hartmann

The analytic torsion of a cone over a sphere

Abstract We compute the analytic torsion of a cone over a sphere of dimensions 1, 2, and 3, and we conjecture a general formula for the cone over an odd dimensional sphere.

The analytic torsion of a cone over an odd dimensional manifold

Abstract We study the analytic torsion of a cone over an orientable odd dimensional compact connected Riemannian manifold W . We prove that the logarithm of the analytic torsion of the cone decomposes as the sum of the logarithm of the root of the analytic torsion of the boundary of the cone, plus a topological term, plus a further term that is a rational linear combination of local Riemannian invariants of the boundary. We show that this last term coincides with the anomaly boundary term appearing in the Cheeger Muller theorem [3] , [2] for a manifold with boundary, according to Bruning and Ma (2006) [5] . We also prove Poincare duality for the analytic torsion of a cone.

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

Abstract We consider Sturm–Liouville operators on a half line [ a , ∞ ) , a > 0 , with potentials that are growing at most quadratically at infinity. Such operators arise naturally in the analysis of hyperbolic manifolds, or more generally manifolds with cusps. We establish existence and a formula for the associated zeta-determinant in terms of the Wronski-determinant of a fundamental system of solutions adapted to the boundary conditions. Despite being the natural objects in the context of hyperbolic geometry, spectral geometry of such operators has only recently been studied in the context of analytic torsion.