Related papers: Conic degeneration and the determinant of the Lapl…

Let M be a surface with conical singularities, and consider a degenerating family of surfaces obtained from M by removing disks of smaller and smaller radius around a subset of the conical singularities. Such families arise naturally in the…

By a discrete torus we mean the Cayley graph associated to a finite product of finite cycle groups with generating set given by choosing a generator for each cyclic factor. In this article we study the spectral theory of the combinatorial…

Compact polyhedral surfaces (or, equivalently, compact Riemann surfaces with conformal flat conical metrics) of an arbitrary genus are considered. After giving a short self-contained survey of their basic spectral properties, we study the…

We study the regularized determinant of the Laplacian as a functional on the space of Mandelstam diagrams (noncompact translation surfaces glued from finite and semi-infinite cylinders). A Mandelstam diagram can be considered as a compact…

In this paper, we investigate the long-time structure of the heat kernel on a Riemannian manifold M which is asymptotically conic near infinity. Using geometric microlocal analysis and building on results of Guillarmou and Hassell on the…

In this paper we define a new convergence called "asymptotically conic convergence" in which a smooth family of Riemannian metrics on a fixed compact manifold degenerate to a metric with isolated conic singularity. Our results are:…

Let $X$ be a Riemann surface of genus $g\ge 1$ endowed with a flat conical metric $m$ and let ${\rm det}\,\Delta$ be the $\zeta$-regularized determinant of the Friedrichs Laplacian on $(X,m)$. We derive variational formulas for ${\rm…

In this paper we study the behaviour of the continuous spectrum of the Laplacian on a complete Riemannian manifold of bounded curvature under perturbations of the metric. The perturbations that we consider are such that its covariant…

The goal of this paper is to compute the zeta function determinant for the massive Laplacian on Riemann caps (or spherical suspensions). These manifolds are defined as compact and boundaryless $D-$dimensional manifolds deformed by a…

We study two types of regularizations of the determinant of Laplacian on Riemann manifold from the viewpoint of resurgence theory. One is the formal logarithmic derivative of the determinant, and the other is its exponential deformation.…

For a one-parameter degeneration of compact Riemann surfaces endowed with the K\"ahler metric induced from the K\"ahler metric on the total space of the family, we determine the exact magnitude of the small eigenvalues of the Laplacian as a…

We find an explicit expression for the zeta-regularized determinant of (the Friedrichs extension) of the Laplacian on a compact Riemann surface of genus one with conformal metric of curvature $1$ having a single conical singularity of angle…

We construct a determinant of the Laplacian for infinite-area surfaces which are hyperbolic near infinity and without cusps. In the case of a convex co-compact hyperbolic metric, the determinant can be related to the Selberg zeta function…

We investigate the functional determinant of the laplacian on piece-wise flat two-dimensional surfaces, with conical singularities in the interior and/or corners on the boundary. Our results extend earlier investigations of the determinants…

We consider a class of singular Riemannian manifolds, the deformed spheres $S^N_k$, defined as the classical spheres with a one parameter family $g[k]$ of singular Riemannian structures, that reduces for $k=1$ to the classical metric. After…

Manifolds with fibered cusps are a class of complete noncompact Riemannian manifolds including all locally symmetric spaces of rank one. We study the spectrum of the Hodge Laplacian with coefficients in a flat bundle on a closed manifold…

We consider families of degenerating hyperbolic surfaces. The surfaces are geometrically finite of fixed topological type. Let Z(s) be the Selberg Zeta function of a surface, and let Z_d(s) be the contribution of the pinched geodesics to…

We study the asymptotics of the determinant of Laplacian on a translation surface (a compact Riemann surface equipped with a conformal flat conical metric with trivial holonomy) of genus g with 2g-2 conical points of angle 4\pi as two…

We give an integral representaion of the zeta-reguralized determinant of Laplacians on three dimensional Heisenberg manifolds, and study a behaivior of the values when we deform the uniform discrete subgroups. Heiseberg manifolds are the…

Let $(M,g)$ be a compact smoothly stratified pseudomanifold with an iterated cone-edge metric satisfying a spectral Witt condition. Under these assumptions the Hodge-Laplacian $\Delta$ is essentially self-adjoint. We establish the…