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

Electric Impedance Tomography problem for surfaces with internal holes