The anisotropic partial Calderón conjecture for singular metrics

Let $M$ be a smooth compact connected manifold with smooth boundary $\partial M$, and let $g$ and $\tilde{g}$ be smooth Riemannian metrics on $M$. Let $\Gamma_D, \Gamma_N$ be open subsets of $\partial M$. Assume that $\lambda \in \mathbb{R}$ does not belong to the Dirichlet spectra $\sigma(-\Delta_g)$ or $\sigma(-\Delta_{\tilde{g}})$, and suppose that

$$\Lambda_{g,\Gamma_D,\Gamma_N}(\lambda)=\Lambda_{\tilde{g},\Gamma_D,\Gamma_N}(\lambda).$$

Anisotropic partial Calderón conjecture. It should follow that $g=\tilde{g}$ up to the diffeomorphism gauge invariance if $\dim M\geq 3$, and up to the diffeomorphism and conformal gauge invariances if $\dim M=2$ and $\lambda=0$.

Here the diffeomorphism gauge consists of pullbacks by diffeomorphisms of $M$ equal to the identity on $\Gamma_D\cup\Gamma_N$; in dimension two at zero frequency, the additional conformal gauge consists of multiplying the metric by a positive smooth function equal to $1$ on $\Gamma_N$. The conjecture asks whether the partial Dirichlet-to-Neumann map at one frequency determines the metric modulo precisely these unavoidable invariances; its status is not resolved in the supplied text.