On the electric impedance tomography problem for nonorientable surfaces with internal holes

Let $(M,g)$ be a compact smooth (generally speaking, not necessarily orientable) surface, and let $\Gamma_{0},\dots,\Gamma_{m-1}$ be the components of the boundary of $M$. Let $u=u^{f}(x)$ be the solution of te following problem: $\Delta_{g}u=0$ in $M$, $u|_{\Gamma_{0}}=f$, $u|_{\Gamma_{j}}=0$, $j=1,\dots,m'$, $\partial_{\nu}u|_{\Gamma_{j}}=0$, $j=m'+1,\dots,m-1$, where $\nu$ is the outward normal. With this problem, we associate the DN-operator $\Lambda\colon f\mapsto \partial_{\nu}u^{f}|_{\Gamma_{0}}$. The task is to recover $M$ if $\Lambda$ is given.
For solution, a version of the boundary comtrol method is applied. The principal role is played by the algebra $\mathfrak{A}$ of functions holomorphic on the orientable cover of $M$. We show that $\mathfrak{A}$ is determined by $\Lambda$ up to isometric isomorphism. The spectrum of $\mathfrak{A}$ makes it possible to construct a copy $M'$ of $M$. This copy is conformally equivalent to $M$, and its DN-operator coincides with $\Lambda$.