Local rigidity of conformally Euclidean metrics for the anisotropic Calderón problem

We prove local rigidity of every smooth conformally Euclidean metric for the anisotropic Calderón 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 $s>n/2+1$, hav