Electric impedance tomography problem for surfaces with internal holes ^*

Let $(M,g)$ be a smooth compact Riemann surface with the multicomponent boundary $\Gamma=\Gamma_0\cup\Gamma_1\cup\dots\cup\Gamma_m=:\Gamma_0\cup\tilde\Gamma$. $u=u^f$ obey $\Delta u=0$ in $M$, $u|_{\Gamma_0}=f,\,\,u|_{\tilde\Gamma}=0$ (the grounded holes) and $v=v^h$ v=0$ $v|_{\Gamma_0}=h,\,\,\partial_\nu v|_{\tilde\Gamma}=0$ isolated holes). $\Lambda_{g}^{\rm gr}: f\mapsto\partial_\nu u^f|_{\Gamma_{0}}$ is}: h\mapsto\partial_\nu v^h|_{\Gamma_{0}}$ corresponding DN-maps. The EIT problem is to determine $M$ from gr}$ or is}$. To solve it, an algebraic version of BC-method applied. main instrument algebra holomorphic functions on ma\-ni\-fold ${\mathbb M}$, which obtained by gluing two examples along $\tilde{\Gamma}$. We show that this determined (or is}$) up isometric isomorphism. Its Gelfand spectrum set characters) plays role material for constructing relevant copy $(M',g',\Gamma_{0}')$ $(M,g,\Gamma_{0})$. This conformally equivalent original, provides $\Gamma_{0}'=\Gamma_{0},\,\,\Lambda_{g'}^{\rm gr}=\Lambda_{g}^{\rm gr},\,\,\Lambda_{g'}^{\rm is}=\Lambda_{g}^{\rm is}$, thus solves problem.