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 in dimension greater than or equal to 3. More precisely, we show that if a smooth Riemannian metric is sufficiently close (in certain Sobolev norm) to a conformally Euclidean one, and such that their Dirichlet-to-Neumann maps are equal, then the metrics are isometric by a diffeomorphism that fixes the boundary pointwise. The proof uses a harmonic embedding, complex geometric optics, and estimates on frequencies which leads to a compactness property. This is a joint work with Plamen Stefanov.