Lectures on n-Dimensional Quasiconformal Mappings

A Density Problem for Sobolev Spaces on Planar Domains

“…See [12] for more details. Actually, [12] states these results for "modulus", but "modulus" is equivalent to conformal capacity (see e.g.…”

Uniformization of Sierpiński carpets in the plane

“…In this section we summarize basic facts on quasiconformal and related maps (see [20,23,33] for general background). Let f : C → C be a homeomorphism, and for x ∈ C and small r > 0 define…”

“…Note that points are "removable singularities" for quasiconformal maps [33,Theorem 17.3]. Moreover, the dilatation of F does not change by the passage from the Euclidean metric on C to the chordal metric on C ⊆ C, because these metrics are "asymptotically" conformal, i.e., the identity map from C equipped with the Euclidean metric to C equipped with the chordal metric is 1-quasiconformal.…”

“…We make the preliminary choice m = m 1 . Then there exists a rectifiable path α in connecting E and F satisfying (33). If α stays inside (with the possible exception of its endpoints), we can take γ = α.…”

“…Since quasiconformal maps on C preserve such sets (see [33,Definition 24.6 and Theorem 33.2]), the round carpet T is also a set of spherical measure zero. Let g : T → T be another quasisymmetry onto a round carpet T ⊆ C. Then g • f −1 is a quasisymmetry of T onto T .…”