An Introduction to the Theory of Higher-Dimensional Quasiconformal Mappings

Triangular ratio metric in the unit disk

“…Clearly, x 2 = y 2 since both j, c ∈ R. Let x 2 = u + ri and y 2 = u − ri. Now, h 2 = r 2 + (u − j) 2 and d 2 = r 2 + (c − u) 2 . Thus,…”

“…From the equality h 2 = r 2 + (u − j) 2 , it follows that Proof: Suppose without loss of generality that 0 < q < 1, x 2 = u + ri and y 2 = x 2 . From Lemma 5.2, it follows that…”

“…In geometric function theory, metrics are often used to define new types of geometries of subdomains of the Euclidean, Hilbert, Banach and other metric spaces [1][2][3][4]. One can introduce a metric topology and build new types of geometries of a domain G ⊂ R n , n ≥ 2, based on metrics.…”

Homeomorphic extension of quasi-isometries for convex domains in $${\mathbb {C}}^d$$ and iteration theory

“…Hence, Example 8.3 also shows the failure of the Carathéodory extension theorem for (non quasiconformal) univalent maps in complex dimension strictly greater than one. In fact, Carathéodory extension theorem holds for quasiconformal maps in R n (see, for instance, [28, Cor. 6.5.15]).…”

Space quasiconformal composition operators with applications to Neumann eigenvalues

“…Let R(E, F )) be a ring with a, b ∈ E and c, ∞ ∈ F . Then for the Teichmüller ring R T (n, t) we have the estimate [11].…”