arXiv subjects
Publications and source records attributed to Mario Bonk.
Let $Z$ be an Ahlfors $Q$-regular compact metric measure space, where $Q>0$. For $p>1$ we introduce a new (fractional) Sobolev space $A^p(Z)$ consisting of functions whose extensions to the hyperbolic filling of $Z$ satisfies a weak-type gradient condition. If $Z$ supports a $Q$-Poincaré inequality with $Q>1$, then $A^{Q}(Z)$ coincides with the familiar (homogeneous) Hajłasz-Sobolev space.
We prove that if $ξ$ is a quasisymmetric homeomorphism between Sierpiński carpets that are the Julia sets of postcritically-finite rational maps, then $ξ$ is the restriction of a Möbius transformation to the Julia set. This implies that the group of quasisymmetric homeomorphisms of a Sierpiński carpet Julia set of a postcritically-finite rational map is finite.
Let $S_i$, $i\in I$, be a countable collection of Jordan curves in the extended complex plane $\Sph$ that bound pairwise disjoint closed Jordan regions. If the Jordan curves are uniform quasicircles and are uniformly relatively separated, then there exists a quasiconformal map $f\: \Sph\ra \Sph$ such that $f(S_i)$ is a round circle for all $i\in I$. This implies that every Sierpiński carpet in $\oC$ whose peripheral circles are uniformly relatively separated uniform quasicircles can be mapped to a round Sierpiński carpet by a quasisymmetric map.
We call a complement of a union of at least three disjoint (round) open balls in the unit sphere S^n a Schottky set. We prove that every quasisymmetric homeomorphism of a Schottky set of spherical measure zero to another Schottky set is the restriction of a Mobius transformation on S^n. In the other direction we show that every Schottky set in S^2 of positive measure admits non-trivial quasisymmetric maps to other Schottky sets. These results are applied to establish rigidity statements for convex subsets of hyperbolic space that have totally geodesic boundaries.
We prove that every quasisymmetric self-homeomorphism of the standard 1/3-Sierpiński carpet $S_3$ is a Euclidean isometry. For carpets in a more general family, the standard $1/p$-Sierpiński carpets $S_p$, $p\ge 3$ odd, we show that the groups of quasisymmetric self-maps are finite dihedral. We also establish that $S_p$ and $S_q$ are quasisymmetrically equivalent only if $p=q$. The main tool in the proof for these facts is a new invariant---a certain discrete modulus of a path family---that is preserved under quasisymmetric maps of carpets.
Suppose G is a Gromov hyperbolic group, and the boundary at infinity of G is quasisymmetrically homeomorphic to an Ahlfors Q-regular metric 2-sphere Z with Ahlfors regular conformal dimension Q. Then G acts discretely, cocompactly, and isometrically on hyperbolic 3-space.
The hyperbolic plane admits a quasi-isometric embedding into a hyperbolic group if and only if the group is not virtually free.
Suppose G is a hyperbolic group whose boundary has topological dimension k. If the boundary is quasisymmetrically homeomorphic to an Ahlfors k-regular metric space, then, modulo a finite normal subgroup, G is isomorphic to a uniform lattice in the isometry group of hyperbolic (k+1)-space.
We study metric spaces homeomorphic to the 2-sphere, and find conditions under which they are quasisymmetrically homeomorphic to the standard 2-sphere. As an application of our main theorem we show that an Ahlfors 2-regular, linearly locally contractible metric 2-sphere is quasisymmetrically homeomorphic to the standard 2-sphere, answering a question of Heinonen and Semmes.
Every nonconstant meromorphic function in the plane univalently covers spherical discs of radii arbitrarily close to arctan(sqrt 8) ~ 70^\circ 32'. If in addition all critical points of the function are multiple, then a similar statement holds with pi/2. These constants are the best possible. The proof is based on the consideration of negatively curved singular surfaces associated with meromorphic functions.