Quasiconformal homogeneity of hyperbolic manifolds

Identities on hyperbolic manifolds and quasiconformal homogeneity of hyperbolic surfaces

Mazurkiewicz Manifolds and Homogeneity

It is proved that no region of a homogeneous locally compact, locally connected metric space can be cut by an Fσ-subset of a “smaller” dimension. The result applies to different finite or infinite topological dimensions of metrizable spaces. The classical Hurewicz-Menger-Tumarkin theorem in dimension theory says that connected topological n-manifolds (with or without boundary) are Cantor manifo...

Harmonic Quasiconformal Mappings of Riemannian Manifolds

Quasiconformal Harmonic Maps into Negatively Curved Manifolds

Let F : M → N be a harmonic map between complete Riemannian manifolds. Assume that N is simply connected with sectional curvature bounded between two negative constants. If F is a quasiconformal harmonic diffeomorphism, then M supports an infinite dimensional space of bounded harmonic functions. On the other hand, if M supports no non-constant bounded harmonic functions, then any harmonic map o...

A lower bound for Torelli-K-quasiconformal homogeneity

A closed hyperbolic Riemann surfaceM is said to beK-quasiconformally homogeneous if there exists a transitive family F of K-quasiconformal homeomorphisms. Further, if all [f ] ⊂ F act trivially on H1(M ;Z), we say M is Torelli-K-quasiconformally homogeneous. We prove the existence of a uniform lower bound on K for Torelli-K-quasiconformally homogeneous Riemann surfaces. This is a special case o...