Electric impedance tomography problem for surfaces with internal holes

Let (M, g) be a smooth compact Riemann surface with the multicomponent boundary ${\Gamma}={{\Gamma}}_{0}\cup {{\Gamma}}_{1}\cup \cdots \cup {{\Gamma}}_{m}{=:}{{\Gamma}}_{0}\cup ~{{\Gamma}}$. Let u = u^f obey Δu = 0 in M, $u{\vert }_{{{\Gamma}}_{0}}=f,\enspace \enspace u{\vert }_{~{{\Gamma}}}=0$ (the grounded holes) and v = v^h obey Δv = 0 in M, $v{\vert }_{{{\Gamma}}_{0}}=h,\enspace \enspace {\partial }_{\nu }v{\vert }_{~{{\Gamma}}}=0$ (the isolated holes). Let ${{\Lambda}}_{g}^{\mathrm{gr}}:f{\mapsto}{\partial }_{\nu }{u}^{f}{\vert }_{{{\Gamma}}_{0}}$ and ${{\Lambda}}_{g}^{\mathrm{is}}:h{\mapsto}{\partial }_{\nu }{v}^{h}{\vert }_{{{\Gamma}}_{0}}$ be the corresponding Dirichlet-to-Neumann map. The electric impedance tomography problem is to determine M from ${{\Lambda}}_{g}^{\mathrm{gr}}$ or ${{\Lambda}}_{g}^{\mathrm{is}}$. To solve it, an algebraic variant of the boundary control method is applied. The central role is played by the algebra $\mathbb{A}$ of functions holomorphic on the manifold obtained by gluing two examples of M along $~{{\Gamma}}$. We show that $\mathbb{A}$ is determined by ${{\Lambda}}_{g}^{\mathrm{gr}}$ (or ${{\Lambda}}_{g}^{\mathrm{is}}$) up to isometric isomorphism. A relevant copy (M', g', Γ₀') of (M, g, Γ₀) is constructed from the Gelfand spectrum of $\mathbb{A}$. By construction, this copy turns out to be conformally equivalent to (M, g, Γ₀), obeys ${{\Gamma}}_{0}^{\prime }={{\Gamma}}_{0},\enspace \enspace {{\Lambda}}_{{g}^{\prime }}^{\mathrm{gr}}={{\Lambda}}_{g}^{\mathrm{gr}},\enspace \enspace {{\Lambda}}_{{g}^{\prime }}^{\mathrm{is}}={{\Lambda}}_{g}^{\mathrm{is}}$ and provides a solution of the problem. *Supported by RFBR Grant 20-01 627A.