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.

References

This has remained an open problem for more than 35 years if $\dim(M) \geq 3$ .

— Semiglobal uniqueness for the Lorentzian Calderón problem  (2608.13116 - Oksanen et al., 13 Aug 2026) in Section 1, Introduction

Again, when λ = 0 and n ≥ 3, the problem remains largely open in the smooth category.

— The anisotropic Calderón problem for Riemannian metrics at high fixed frequency  (2610.01280 - Cekić et al., 1 Oct 2026) in Section 1.3, “State of the art and previous work”

However, for the stationary Laplace--Beltrami operator, the corresponding problem remains open.

— The anisotropic Calderón problem for Riemannian metrics at high fixed frequency  (2610.01280 - Cekić et al., 1 Oct 2026) in Section 1.3, “State of the art and previous work”

To our knowledge, no theorem establishes global uniqueness from the full Dirichlet-to-Neumann map for all uniformly elliptic scalar conductivities under a weaker Sobolev assumption in dimensions n\geq3.

— Determining $W^{1,n}$ conductivities in Calderón's problem for $n\geq4$  (2610.01529 - Chen et al., 1 Oct 2026) in Section 1, subsection “Motivation and statement” (Introduction)

The latter problem remains open for general smooth metrics in dimensions three and higher.

— Fractional Calderón problem with a potential  (2610.06766 - Feizmohammadi et al., 5 Oct 2026) in Section 1, Introduction