Reconstructing anisotropic conductivities on two-dimensional Riemannian manifolds from power densities

We consider an electrically conductive compact two-dimensional Riemannian manifold with smooth boundary. This setting defines a natural conductive Laplacian on the manifold and hence also voltage potentials, current fields and corresponding power densities arising from suitable boundary conditions. Motivated by Acousto-Electric Tomography we show that if the manifold has genus zero and the metric is known, then the anisotropic conductivity can be recovered uniquely and constructively from knowledge of a few power densities. We illustrate the procedure numerically by reconstructing an anisotropic conductivity on the catenoid, i.e. the classical genus zero minimal surface in three-space.