Anisotropic Calderón problem in dimensions at least three

Establish recovery of a smooth Riemannian metric on a compact manifold with boundary, up to the natural boundary-fixing diffeomorphism gauge, from the Cauchy data set of the Laplace–Beltrami equation when the manifold dimension is at least three.

Background

The anisotropic Calderón problem asks whether the coefficients of the Laplace equation on a compact Riemannian manifold can be determined from boundary Cauchy data. Specifically, given the set of boundary Dirichlet and normal-derivative traces of all harmonic functions, one seeks to recover the underlying Riemannian metric up to the natural diffeomorphism invariance fixing the boundary.

The paper identifies this elliptic inverse problem as unresolved in dimensions at least three and contrasts it with the Lorentzian Calderón problem studied in the paper, for which the authors prove uniqueness results under semiglobal and perturbative hypotheses.