1,806 research outputs found
Boundedness of the number of nodal domains for eigenfunctions of generic Kaluza-Klein -folds
This article concerns the number of nodal domains of eigenfunctions of the
Laplacian on special Riemannian -manifolds, namely nontrivial principal
bundles over Riemann surfaces equipped with certain
invariant metrics, the Kaluza-Klein metrics. We prove for generic Kaluza-Klein
metrics that any Laplacian eigenfunction has exactly two nodal domains unless
it is invariant under the action.
We also construct an explicit orthonormal eigenbasis on the flat -torus
for which every non-constant eigenfunction belonging to the
basis has two nodal domains.Comment: 59 pages, will appear at Annales de l'Institut Fourie
Calculi, Hodge operators and Laplacians on a quantum Hopf fibration
We describe Laplacian operators on the quantum group SUq (2) equipped with
the four dimensional bicovariant differential calculus of Woronowicz as well as
on the quantum homogeneous space S2q with the restricted left covariant three
dimensional differential calculus. This is done by giving a family of Hodge
dualities on both the exterior algebras of SUq (2) and S2q . We also study
gauged Laplacian operators acting on sections of line bundles over the quantum
sphere.Comment: v3, one reference corrected, one reference added. 31 page
Complex valued Ray-Singer torsion
In the spirit of Ray and Singer we define a complex valued analytic torsion
using non-selfadjoint Laplacians. We establish an anomaly formula which permits
to turn this into a topological invariant. Conjecturally this analytically
defined invariant computes the complex valued Reidemeister torsion, including
its phase. We establish this conjecture in some non-trivial situations.Comment: Fixed two sign mistakes and added a few more details here and ther
Holomorphic factorization of determinants of Laplacians using quasi-Fuchsian uniformization
For a quasi-Fuchsian group \Ga with ordinary set , and
the Laplacian on \n differentials on \Ga\bk\Omega, we define a notion of a
Bers dual basis for . We prove that
, is, up to an anomaly computed by
Takhtajan and the second author in \cite{TT1}, the modulus squared of a
holomorphic function F(n), where F(n) is a quasi-Fuchsian analogue of the
Selberg zeta Z(n). This generalizes the D'Hoker-Phong formula
, and is a quasi-Fuchsian counterpart of the result
for Schottky groups proved by Takhtajan and the first author in \cite{MT}.Comment: 15 page
- …