Sobolev spaces and hyperbolic fillings

Anisotropic Hölder and Sobolev Spaces for Hyperbolic Diffeomorphisms

We study spectral properties of transfer operators for diffeomorphisms T : X → X on a Riemannian manifold X: Suppose that Ω is an isolated hyperbolic subset for T , with a compact isolating neighborhood V ⊂ X. We first introduce Banach spaces of distributions supported on V , which are anisotropic versions of the usual space of C functions C(V ) and of the generalized Sobolev spaces W (V ), res...

Weakly Hyperbolic Equations with Time Degeneracy in Sobolev Spaces

The theory of nonlinear weakly hyperbolic equations was developed during the last decade in an astonishing way. Today we have a good overview about assumptions which guarantee local well posedness in spaces of smooth functions (C∞, Gevrey). But the situation is completely unclear in the case of Sobolev spaces. Examples from the linear theory show that in opposite to the strictly hyperbolic case...

Simplicial Volume and Fillings of Hyperbolic Manifolds

Let M be a hyperbolic n–manifold whose cusps have torus crosssections. In [FM10], the authors constructed a variety of nonpositively and negatively curved spaces as “2π–fillings” of M by replacing the cusps of M with compact “partial cones” of their boundaries. These 2π–fillings are closed pseudomanifolds, and so have a fundamental class. We show that the simplicial volume of any such 2π–fillin...

Sobolev Spaces

Configuration Spaces of Points on the Circle and Hyperbolic Dehn Fillings

A purely combinatorial compactification of the configuration space of n (≥ 5) distinct points with equal weights in the real projective line was introduced by M. Yoshida. We geometrize it so that it will be a real hyperbolic cone-manifold of finite volume with dimension n − 3. Then, we vary weights for points. The geometrization still makes sense and yields a deformation. The effectivity of def...