In this paper we investigate the geometry of the quasihyperbolic metric of domains in R". This metric arises from the conformally flat generalized Riemannian metric d(x, aD)' ldx I. Due to the fact that the density cl(x, aD) -t is not necessarily differentiable, the classical theories of Riemannian geometry do not apply to this metric. The quasihyperbolic metric has been found to have many inte...

We will review work with Tatiana Toro yielding a characterization of those domains for which the harmonic measure has a density whose logarithm has vanishing mean oscillation. 2000 Mathematics Subject Classification: 31B25, 35R35, 42B35, 51M25.

In this paper we consider flat metrics (semi-translation structures) on surfaces of finite type. There are two main results. The first is a complete description of when a set of simple closed curves is spectrally rigid, that is, when the length vector determines a metric among the class of flat metrics. Secondly, we give an embedding into the space of geodesic currents and use this to get a bou...

Let M and N be doubly connected Riemann surfaces with $${\mathscr {C}}^{1,\alpha }$$ boundaries nonvanishing conformal metrics $$\sigma $$ $$\wp respectively, assume that is a smooth metric bounded Gauss curvature $${\mathcal {K}}$$ finite area. Assume {H}}^\wp (M, N)$$ the class of all {W}}^{1,2}$$ homeomorphisms between {E}}^\wp : {\mathcal N)\rightarrow {\mathbf {R}}$$ Dirichlet-energy funct...

A new, more widely applicable, constructive definition of locally compact metric space is given, and a metric one-point compactification is constructed. Classically, this provides a simpler, more direct, construction of a metric on the one-point compactification of a separable locally compact metric

1. A principal bundle 33(X, G) with base space X and group G is called "locally flat" if the structural group G can be reduced to a totally disconnected subgroup GiCG, or in other words: if there exists a totally disconnected subgroup GiQG, a principal bundle 33i(X, Gi) and an injection j: Soi—>33 [l ]. For instance, if a differentiable bundle 33 admits an infinitesimal connection whose curvatu...

We analyse in a systematic way the four dimensionnal Einstein-Weyl spaces equipped with a diagonal Kähler Bianchi IX metric. In particular, we show that the subclass of EinsteinWeyl structures with a constant conformal curvature is the one with a conformally flat but not necessarily flat metric ; we exhibit its 3-parameter distance and Weyl one-form. This extends previous analysis of Pedersen e...

We analyse in a systematic way the four dimensionnal Einstein-Weyl spaces equipped with a diagonal Kähler Bianchi IX metric. In particular, we show that the subclass of EinsteinWeyl structures with a constant conformal scalar curvature is the one with a conformally scalar flat but not necessarily scalar flat metric ; we exhibit its 3-parameter distance and Weyl one-form. This extends previous a...

We introduce the notion of an ls-π-Ponomarev-system to give necessary and sufficient conditions for f : (M,M0) → X to be a strong wcmapping (wc-mapping, wk-mapping) where M is a locally separable metric space. Then, we systematically get characterizations of weakly continuous strong wc-images (wc-images, wk-images) of locally separable metric spaces by means of certain networks. Also, we give c...