On the hidden mechanism behind non-uniqueness for the anisotropic Calder{\'o}n problem with data on disjoint sets

We show that there is generically non-uniqueness for the anisotropic Calder\'on problem at fixed frequency when the Dirichlet and Neumann data are measured on disjoint sets of the boundary of a given domain. More precisely, we first show that given a smooth compact connected Riemannian manifold with boundary $(M,g)$ of dimension $n\geq 3$, there exist in the conformal class of $g$ an infinite number of Riemannian metrics $\tilde{g}$ such that their corresponding DN maps at a fixed frequency coincide when the Dirichlet data $\Gamma_D$ and Neumann data $\Gamma_N$ are measured on disjoint sets and satisfy $\overline{\Gamma_D \cup \Gamma_N} \ne \partial M$. The conformal factors that lead to these non-uniqueness results for the anisotropic Calder\'on problem satisfy a nonlinear elliptic PDE of Yamabe type on the original manifold $(M,g)$ and are associated to a natural but subtle gauge invariance of the anisotropic Calder\'on problem with data on disjoint sets. We then construct a large class of counterexamples to uniqueness in dimension $n\geq 3$ to the anisotropic Calder\'on problem at fixed frequency with data on disjoint sets and \emph{modulo this gauge invariance}. This class consists in cylindrical Riemannian manifolds with boundary having two ends (meaning that the boundary has two connected components), equipped with a suitably chosen warped product metric.