Moduli of families of locally quasisymmetric surfaces
The author introduced [Sov. Math., Dokl. 31, 365-367 (1985); translation from Dokl. Akad. Nauk SSSR 281, 1033-1035 (1985; Zbl 0604.30024)], the k- dimensional modulus M in \(R^ n\), \(2\leq k\leq n\), by means of k- dimensional LQS-surfaces. The definition is now extended to \(\bar R^ n\). This modulus is quasiconformally quasiinvariant. Let B be the open unit cube in \(R^ k\). A k-dimensional chart on the set \(A\subset \bar R^ n\) is a LQS-embedding \(\sigma: B\to A\). If \(a\in R^ n\) and \(\Gamma\) is the family of all subsets (in particular, of all k- dimensional LQS-surfaces) \(A\subset \bar R^ n\) such that there exists a k-dimensional chart \(\sigma: B\to A\) with \(a\in \sigma (B)\) and \(\Lambda^ k(\sigma (B))<\infty\), then \(M(\Gamma)=0\). Similarly, if \(\Gamma\) is the family of all k-dimensional LQS-surfaces in \(R^ n\) through a point \(a\in R^ n\) and \(M_ F\) the classical Fuglede modulus, then \(M_ F(\Gamma)=0\).
scientific article; zbMATH DE number 4144357