The conformal Poisson embedding method for the anisotropic Calderón problem with a potential

We prove that the local Dirichlet-to-Neumann map of ${\mathrm{\Delta }}_g+q$ determines a compact connected real-analytic Riemannian manifold of dimension at least three and a real-analytic real-valued potential, up to a diffeomorphism fixing the observed boundary and the conformal gauge of the equation. The result is new even when $q$ is constant, i.e. for the fixed-frequency (Helmholtz) equation ${\mathrm{\Delta }}_{g}u+\lambda u=0$. The measurements are restricted to an arbitrary nonempty open boundary set $\mathrm{\Gamma }$. The proof develops conformal Poisson embeddings method and uses coordinates given by quotients of solutions.