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Smooth quasiregular mappings with branching

Mario Bonk;  Juha Heinonen — 2004

Publications Mathématiques de l'IHÉS

We give an example of a 𝒞 3 - ϵ -smooth quasiregular mapping in 3-space with nonempty branch set. Moreover, we show that the branch set of an arbitrary quasiregular mapping in-space has Hausdorff dimension quantitatively bounded away from . By using the second result, we establish a new, qualitatively sharp relation between smoothness and branching.

Quasiconformal mappings with Sobolev boundary values

Kari Astala;  Mario Bonk;  Juha Heinonen — 2002

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider quasiconformal mappings in the upper half space ℝ + n + 1 of ℝ n + 1 , n ≥ 2 , whose almost everywhere defined trace in ℝ n has distributional differential in L n ( ℝ n ) . We give both geometric and analytic characterizations for this possibility, resembling the situation in the classical Hardy space H 1 . More generally, we consider certain positive functions defined on ℝ + n + 1 , called conformal densities. These densities mimic the averaged derivatives of quasiconformal mappings, and we prove analogous trace theorems for them....

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