Boundary values of mappings close to an isometric mapping (Q1076999)

A mapping f of a domain \(\Delta\) of \(R^ n\) into \(R^ n\) is said to be K-quasi-isometric if it is continuous and injective and at each point we have \[ K^{-1}\leq \lim \frac{| f(y)-f(x)|}{| y-x|}\leq \lim \frac{| f(y)-f(x)|}{| y-x|}\leq K, \] where K is a real number. Here we note that, if \(K=1\), f is an isometry with respect to the usual Euclidean metrics. This paper presents two results as follows. Let \(G(\epsilon,\omega_ 1)\) be the class of mappings \((\epsilon,\omega_ 1)\) close to the class G of isometries of domains of \(R^ n\) with values in \(R^{n+1}\), and \(\omega_ 1\) be the class of all injective mappings of \(R^ n\) into \(R^{n+1}\). Then, the family \(\{W(\epsilon)\}\) of class \(W(\epsilon)\) of boundary values of \((1+\epsilon)\)-quasi-isometric mappings \(f: R_+^{n+1}\to R^{n+1}\) coincides asymptotically with the family \(\{G(\epsilon,\omega_ 1)\cap G\}\) as \(\epsilon\) \(\to 0\). The second part is devoted to the study of boundary values of isometric and quasi-isometric mappings, whose main result may be summarized in the following theorem: let n be a natural number such that it is greater than or equal to 2. There exists a real number \(C=C(n)>1\) and for each number \(\epsilon >0\) a domain \(\Delta_{\epsilon}\in R^ n\) and a mapping \(f_{\epsilon}\) of the bounding sphere \(\partial B(0,1)\) of the n-dimensional ball \(B(0,1)<R^ n\) onto the boundaries of the ball B(0,1) and the domain \(\Delta_{\epsilon}\), and at the same time, the domain \(\Delta_{\epsilon}\) cannot be mapped C-quasi-isometrically onto B(0,1).