We investigate spectral properties of the Laplacian in <i>L</i>² (<i>Q</i>), where <i>Q</i> is a tubular region in ³ of a fixed cross section, and the boundary conditions combined a Dirichlet and a Neumann part. We analyze two complementary situations, when the tube is bent but not twisted, and secondly, it is twisted but not bent. In the first case we derive sufficient conditions for the presence and absence of the discrete spectrum showing, roughly speaking, that they depend on the direction in which the tube is bent. In the second case we show that a constant twist raises the threshold of the essential spectrum and a local slowndown of it gives rise to isolated eigenvalues. Furthermore, we prove that the spectral threshold moves up also under a sufficiently gentle periodic twist.

A family of coordinates h for the Teichmller space of a compact surface with boundary was introduced by the second author. Mondello showed that the coordinate 0 can be used to produce a natural cell decomposition of the Teichmller space, invariant under the action of the mapping class group. In this paper, we show that, for any h 0, the coordinate h produces a natural cell decomposition of the Teichmller space.

A spherical polyhedral surface is a triangulated surface obtained by isometric gluing of spherical triangles. For instance, the boundary of a generic convex polytope in the 3-sphere is a spherical polyhedral surface. This paper investigates these surfaces from the point of view of inner angles. A rigidity result is obtained. A characterization of spherical polyhedral surfaces in terms of the triangulation and the angle assignment is established.

It is shown that every non-compact hyperbolic manifold of finite volume has a finite cover admitting a geodesic ideal triangulation. Also, every hyperbolic manifold of finite volume with non-empty, totally geodesic boundary has a finite regular cover which has a geodesic partially truncated triangulation. The proofs use an extension of a result due to Long and Niblo concerning the separability of peripheral subgroups.

Dunfield-Garoufalidis and Boyer-Zhang proved that the A-polynomial of a nontrivial knot in S3 is nontrivial. In this paper, we use holonomy perturbations to prove the non-triviality of the A-polynomial for a nontrivial, null-homotopic knot in an irreducible 3-manifold. Also, we give a strong constraint on the A-polynomial of a knot in the 3-sphere.