Abstract:
Consider the relative metric $\rho$ in a domain $D$ of the plane. Denote by $\widetilde{D}$ the completion of $D$ with respect to the metric and denote the generalized boundary of $D$, the set $\widetilde{D}\setminus{D}$, by $\widetilde{\partial}D$. The restriction of $\rho$ onto $\widetilde{\partial}D$ is called the relative metric of the boundary.
The Main Theorem. {\it Let $D\subset\mathbb{R}^2$ be a bounded domain and $D^*\subset\mathbb{R}^2$ be an arbitrary domain. Suppose that there exists a surjective mapping $f\colon\widetilde{\partial}D\to\widetilde{\partial}D^*$ isometric in the relative metrics of the generalized boundaries $\widetilde{\partial}D$ and $\widetilde{\partial}D^*$. Then the domains $D$ and $D^*$ are isometric in Euclidean metrics and, consequently, the mapping is extandible to a Euclidean isometry of the planes.}