Snowballs are quasiballs

Abstract

Abstract. We introduce snowballs, which are compact sets in R 3 homeomorphic to the unit ball. They are 3-dimensional analogs of domains in the plane bounded by snowflake curves. For each snowball B a quasiconformal map f : R 3 → R 3 is constructed that maps B to the unit ball.

Combinatorial modulus, the combinatorial Loewner property, and Coxeter groups

“…The assumptions on Z allow one to check that the proofs of 2) Higher dimensional snowspheres, cf. [Mey10]. Fix n ≥ 2.…”

Lower bounds for codimension-1 measure in metric manifolds

“…Consider, for instance, a metric on S 2 that contains subsets at various small scales that have wildly different Hausdorff dimensions but are bounded by finite-length curves. Such metrics could be constructed as Hausdorff limits of certain polyhedral complexes in R 3 , defined iteratively by cubical sub-division and replacement rules, much like the "snow-sphere" constructions of D. Meyer [10]. By altering the subdivision rules in different regions that are very far from each other in relative distance, it is possible to make the resulting metric spheres highly non-homogeneous for Hausdorff dimension.…”

“…At the same time, these types of non-smooth metric spaces can arise as geometrically controlled deformations of spaces that do support traditional isoperimetric inequalities. For example, the metric spheres described above are quasisymmetrically equivalent to the Euclidean sphere (this equivalence can even be realized by a continuous family of quasiconformal deformations of R 3 , as shown in [10]). This observation prompts the natural question: Can one establish isoperimetric-type inequalities for a broad, quasisymmetrically-invariant class of metric spaces that still gives meaningful geometric information?…”

Quasisymmetric spheres over Jordan domains

“…In higher dimensions the only known characterization is due to Gehring [8] and Väisälä [21]: a topological n-sphere Σ in R n+1 is a quasisphere if and only if the bounded component and the unbounded component of R n+1 \ Σ are quasiconformally equivalent to B n+1 and R n+1 \ B n+1 , respectively. Intriguing examples of quasispheres have been constructed drawing ideas from harmonic analysis, conformal dynamics and classical geometric topology ( [3], [5], [14], [15], [16]). The basic question of a geometric characterization of quasispheres remains.…”