Local rigidity of conformally Euclidean metrics for the anisotropic Calder\'on problem

Abstract

We prove local rigidity of every smooth conformally Euclidean metric for the anisotropic Calder\'on problem on smooth compact domains $M\subset\mathbb R^n$, $n\ge3$. Any smooth Riemannian metric $g$ sufficiently close to a fixed background $g_0=e^{2c}\mathsf e$ in $H^s(M)$, with integer $s>n/2+1$, having the same Dirichlet-to-Neumann map satisfies $g=\Phi^*g_0$ for a smooth diffeomorphism $\Phi$ fixing the boundary pointwise.