The authors consider infinitely connected domains on the complex plane with one or two marked components of the boundaries and functions regular and univalent in such domains. Extremal properties of such functions mapping a domain \(D\) onto a circular domain with radial cuts are discussed. Let \(\Gamma\) be a family of locally rectifiable curves in \(D\), \(\text{adm}\;\Gamma\) be the set of Borel functions for which \(\int_{\gamma} \rho(z)\,| dz| \geq 1\) for all \(\gamma \in \Gamma\). The value \(m_2(\Gamma) = \inf_{\rho \in \text{adm}\;\Gamma} \int\int \rho^2(z)\, dx\,dy\) is called the 2-modulus of \(\Gamma\). The function \(\rho_0(z)\) for which the infimum is attached is called the generalized extremal metrics. A domain is called the normal annulus with radial cuts, if \(\rho_0(z)=(| z| \log \frac Rr)^{-1}\) and some other natural conditions are fulfilled. The authors study properties of the normal annulus and other similar domains.