We give an estimate for the distance function related to Kobayashi metric on a bounded strictly pseudoconvex domain with C 2 -smooth boundary.Our formula relates Carnot-Carathéodory boundary.The is precise up additive term.As corollary we conclude that equipped this hyperbolic in sense of Gromov.

Suppose G is a Gromov hyperbolic group, and ∂ ∞ quasisymmetrically homeomorphic to an Ahlfors Q-regular metric 2-sphere Z with regular conformal dimension Q.Then acts discretely, cocompactly, isometrically on

We prove that every quasisymmetric self-homeomorphism of the standard 1/3-Sierpiński carpet S3 is a Euclidean isometry.For carpets in more general family, 1/p-Sierpiński Sp, p ≥ 3 odd, we show groups self-maps are finite dihedral.We also establish Sp and Sq quasisymmetrically equivalent only if = q.The main tool proof for these facts new invariant-a certain discrete modulus path family-that preserved under maps carpets.

Let S i , i∈I, be a countable collection of Jordan curves in the extended complex plane $\widehat{\mathbb{C}}$ that bound pairwise disjoint closed regions. If are uniform quasicircles and uniformly relatively separated, then there exists quasiconformal map $f\colon\widehat{\mathbb{C}}\rightarrow\widehat{\mathbb{C}}$ such f(S ) is round circle for all i∈I. This implies every Sierpiński carpet whose peripheral circles separated can mapped to by quasisymmetric map.

If a group acts by uniformly quasi-Möbius homeomorphisms on compact Ahlfors n-regular space of topological dimension n such that the induced action distinct triples is cocompact, then quasisymmetrically conjugate to an standard n-sphere Möbius transformations.

We call the complement of a union at least three disjoint (round) open balls in unit sphere ${\Bbb S}^n$ Schottky set. prove that every quasisymmetric homeomorphism set spherical measure zero to another is restriction M\"obius transformation on S}^n$. In other direction we show S}^2$ positive admits nontrivial maps sets. These results are applied establish rigidity statements for convex subsets hyperbolic space have totally geodesic boundaries.

The lower bound for Bloch’s constant is slightly improved.

We study densities ρ on the unit ball in euclidean space which satisfy a Harnack type inequality and volume growth condition for measure associated with ρ. For these geometric theory can be developed captures many features of quasiconformal mappings. example, we prove generalizations Gehring-Hayman theorem, radial limit theorem find analogues compression expansion phenomena boundary. 1991 Mathematics Subject Classification: 30C65.

The hyperbolic plane $\mathbb {H}^2$ admits a quasi-isometric embedding into every group which is not virtually free.

Abstract Let Z be an Ahlfors Q -regular compact metric measure space, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>Q</m:mi><m:mo>&gt;</m:mo><m:mn>0</m:mn></m:mrow></m:math> {Q&gt;0} . For xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>p</m:mi><m:mo>&gt;</m:mo><m:mn>1</m:mn></m:mrow></m:math> {p&gt;1} we introduce a new (fractional) Sobolev space...

Abstract Every Thurston map $f\colon S^2\rightarrow S^2$ on a $2$ -sphere $S^2$ induces pull-back operation Jordan curves $\alpha \subset S^2\smallsetminus {P_f}$ , where ${P_f}$ is the postcritical set of f . Here isotopy class $[f^{-1}(\alpha )]$ (relative to ) only depends $[\alpha ]$ We study this for maps with four points. In case, obstruction can be seen as fixed point operation. show that if hyperbolic orbifold and points has obstruction, then one ‘blow up’ suitable arcs in underlying...

By using quasiconformal flows, we establish that exponentials of logarithmic potentials measures small mass are comparable to Jacobians homeomorphisms Rn, n≥2. As an application, obtain the fact certain complete conformal deformations even-dimensional Euclidean space Rn with total Paneitz or Q-curvature bi-Lipschitz equivalent standard