Electric Impedance Tomography problem for surfaces with internal holes

Let $(M,g)$ be a smooth compact Riemann surface with the multicomponent boundary $\mathrm{\Gamma }={\mathrm{\Gamma }}_0\cup {\mathrm{\Gamma }}_1\cup \cdots \cup {\mathrm{\Gamma }}_m=:{\mathrm{\Gamma }}_0\cup \tilde{\mathrm{\Gamma }}$. Let $u=u^f$ obey $\mathrm{\Delta }u=0$ in $M$, $u{\mathrm{\mid }}_{{\mathrm{\Gamma }}_0}=f,u{\mathrm{\mid }}_{\tilde{\mathrm{\Gamma }}}=0$ (the grounded holes) and $v=v^h$ obey $\mathrm{\Delta }v=0$ in $M$, $v{\mathrm{\mid }}_{{\mathrm{\Gamma }}_0}=h,{\mathrm{\partial }}_{\nu }v{\mathrm{\mid }}_{\tilde{\mathrm{\Gamma }}}=0$ (the isolated holes). Let ${\mathrm{\Lambda }}_g^{\mathrm{g}\mathrm{r}}:f\mapsto {\mathrm{\partial }}_{\nu }{u}^f{\mathrm{\mid }}_{{\mathrm{\Gamma }}_0}$ and ${\mathrm{\Lambda }}_g^{\mathrm{i}\mathrm{s}}:h\mapsto {\mathrm{\partial }}_{\nu }{v}^h{\mathrm{\mid }}_{{\mathrm{\Gamma }}_0}$ be the corresponding DN-maps. The EIT problem is to determine $M$ from ${\mathrm{\Lambda }}_g^{\mathrm{g}\mathrm{r}}$ or ${\mathrm{\Lambda }}_g^{\mathrm{i}\mathrm{s}}$. To solve it, an algebraic version of the BC-method is applied. The main instrument is the algebra of holomorphic functions on the ma-ni-fold $\mathbb{M}$, which is obtained by gluing two examples of $M$ along $\tilde{\mathrm{\Gamma }}$. We show that this algebra is determined by $\Lambda_{g}^{\rm gr}$(or$\Lambda_{g}^{\rm is}$) up to isometric isomorphism. Its Gelfand spectrum (the set of characters) plays the role of the material for constructing a relevant copy $(M^{\mathrm{\prime }},g^{\mathrm{\prime }},{\mathrm{\Gamma }}_0^{\mathrm{\prime }})$ of $(M,g,{\mathrm{\Gamma }}_0)$. This copy is conformally equivalent to the original, provides $\Gamma_{0}‘=\Gamma_{0},,,\Lambda_{g’}^{\rm gr}=\Lambda_{g}^{\rm gr},,,\Lambda_{g’}^{\rm is}=\Lambda_{g}^{\rm is}$, and thus solves the problem.