Year: 1981
Type: article
Abstract: We consider quasi-symmetric embeddings f : G → R n f:G \to {R^n} , G G open in R p {R^p} , p ⩽ n p \leqslant n . If p = n p = n , quasi-symmetry implies quasi-conformality. The converse is true if G G has a sufficiently smooth boundary. If p > n p > n , the Hausdorff dimension of f G fG is less than n n . If f G fG has a finite p p -measure, f f preserves the property of being of p p -measure zero. If p > n p > n and n ⩾ 3 n \geqslant 3 , R n {R^n} contains a quasi-symmetric p p -cell which is topologically wild. We also prove auxiliary results on the relations between Hausdorff measure and Čech cohomology.
Author: Jussi Väisälä
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W2086909375