Boris Vertman

Analytic Torsion of a Bounded Generalized Cone

We compute the analytic torsion of a bounded generalized cone by generalizing the computational methods of M. Spreafico and using the symmetry in the de Rham complex, as established by M. Lesch. We evaluate our result in lower dimensions and further provide a separate computation of analytic torsion of a bounded generalized cone over S1, since the standard cone over the sphere is simply a flat disc.

Regular singular Sturm–Liouville operators and their zeta-determinants☆

Abstract We consider Sturm–Liouville operators on the line segment [ 0 , 1 ] with general regular singular potentials and separated boundary conditions. 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. This generalizes the earlier work of the first author, treating general regular singular potentials but only the Dirichlet boundary conditions at the singular end, and the recent results by Kirsten–Loya–Park for general separated boundary conditions but only special regular singular potentials.

The Metric Anomaly of Analytic Torsion on Manifolds with Conical Singularities

In this paper we study the analytic torsion of a manifold with isolated conical singularities. First we show that the analytic torsion is invariant under deformations of the metric which are of higher order near the singularities. Then we identify the metric anomaly of analytic torsion for a bounded generalized cone at its regular boundary in terms of spectral information of the cross-section. In view of previous computations of analytic torsion on cones, this leads to a detailed geometric identification of the topological and spectral contributions to analytic torsion, arising from the conical singularity. The contribution exhibits a torsion-like spectral invariant of the cross-section of the cone, which we study under scaling of the metric on the cross-section.

Mapping properties of the heat operator on edge manifolds

We consider the heat operator acting on differential forms on spaces with complete and incomplete edge metrics. In the latter case we study the heat operator of the Hodge Laplacian with algebraic boundary conditions at the edge singularity. We establish the mapping properties of the heat operator, recovering and extending the classical results from smooth manifolds and conical spaces. The estimates, together with strong continuity of the heat operator, yield short-time existence of solutions to certain semilinear parabolic equations. Our discussion reviews and generalizes earlier work by Jeffres and Loya.

HEAT-TRACE ASYMPTOTICS FOR EDGE LAPLACIANS WITH ALGEBRAIC BOUNDARY CONDITIONS

We consider the Hodge-Laplace operator on manifolds with incomplete edge singularities and an intricate elliptic boundary value theory. We single out the class of algebraic self-adjoint extensions for the Hodge Laplacian. Our microlocal heat kernel construction for algebraic boundary conditions is guided by the method of signaling solutions by Mooers, though crucial arguments in the conical case obviously do not carry over to the setting of edges. We establish the heat kernel asymptotics for the algebraic extensions of the Hodge operator on edges, and elaborate on the exotic phenomena in the heat trace asymptotics which appear in the case of a non-Friedrichs extension.

Regularizing infinite sums of zeta-determinants

We present a new multiparameter resolvent trace expansion for elliptic operators, polyhomogeneous in both the resolvent and auxiliary variables. For elliptic operators on closed manifolds the expansion is a simple consequence of the parameter dependent pseudodifferential calculus. As an additional nontrivial toy example we treat here Sturm–Liouville operators with separated boundary conditions. As an application we give a new formula, in terms of regularized sums, for the

Multiparameter resolvent trace expansion for elliptic boundary problems

\zeta

Refined analytic torsion as analytic function on the representation variety and applications

ζ–determinant of an infinite direct sum of Sturm–Liouville operators. The Laplace–Beltrami operator on a surface of revolution decomposes into an infinite direct sum of Sturm–Louville operators, parametrized by the eigenvalues of the Laplacian on the cross-section