The author considers the following sewing problem. We are given metric spaces \(X_1\) and \(X_2\), two closed subspaces \(Y_1\subset X_1\) and \(Y_2\subset X_2\), and a homeomorphism \(f:Y_1\rightarrow Y_2\). Does there exist a metric \(\hat d\) on \(\hat X =X_1 \cup X_2 /(f)\) such that \(\text{Id} :X_j\rightarrow \hat X\) is quasisymmetric and \(\text{Id}: X_j\setminus Y_j\rightarrow \hat X\) is locally quasisimilar for \(j=1,2\)?
In this article, a positive answer is given in the case \(f\) is assumed to be quasisymmetric, \(X_1\), \(X_2\) are proper, and \(Y_1\), \(Y_2\) are uniformly perfect. Although there are in principle many possible metrics answering the problem, the method used here defines a unique conformal gauge which depends only on the gauges of \(X_1\) and \(X_2\). Some properties of this gauge are also studied.