Stability in the anisotropic Calderón problem for Painlevé-Liouville Riemannian manifolds

We study the question of stability of the global and partial anisotropic Calderón inverse problems for the class of Painlevé-Liouville Riemannian manifolds, that is compact $n$-dimensional manifolds with boundary $(M,g)$, where $M=[0,1]\times K\,$, $K$ is any smooth closed connected orientable manifold of dimension $n-1$ endowed with a Riemannian metric $g_K$, and $g=α^4 g_{0}$ is any conformal deformation of the product metric $g_{0}=dx^2+g_{K}$ on $M$ which is compatible with the Painlevé block-separability of the Laplace-Beltrami operator $Δ_{g_0}$. Given a pair of Painlevé-Liouville Riemannian manifolds $(M,g)$ and $(M,\tilde{g})$ satisfying some technical hypothesis, denoting the corresponding Dirichlet-to-Neumann maps by $Λ_{g}$ and $Λ_{\tilde{g}}$, and assuming that $\lVert Λ_{g}-Λ_{\tilde{g}}\rVert_{\mathcal{B}(H^{1/2}(\partial M), H^{-1/2}(\partial M))}\ = ε$, we show a logarithmic stability result for the global anisotropic Calderón problem which says that there exists constants $C$ and $0