The anisotropic Calderón conjecture for Riemannian manifolds

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Let (M,g1)(M,g_1) and (M,g2)(M,g_2) be two compact oriented Riemannian manifolds with smooth boundary and dimension n≥3n\geq 3. For a metric gg, let

Cg={(u∣∂M,∂νu∣∂M); Δgu=0 in M, u∈C∞(M)}C_g=\{(u|_{\partial M},\partial_{\nu}u|_{\partial M});\,\Delta_g u=0\text{ in }M,\ u\in C^{\infty}(M)\}

be the Cauchy data set of harmonic functions. The anisotropic Calderón conjecture. If Cg1=Cg2C_{g_1}=C_{g_2}, then

g2=ψ∗g1g_2=\psi^*g_1

for some diffeomorphism ψ:M→M\psi:M\to M satisfying ψ∣∂M=Id⁡\psi|_{\partial M}=\operatorname{Id}. This is the geometric formulation of Calderón's inverse problem: boundary measurements determine the Riemannian metric up to a diffeomorphism fixing the boundary. The supplied text does not indicate whether the conjecture was resolved, so its status is left open.

References

Primary source

Tony Liimatainen and Mikko Salo, “Nowhere conformally homogeneous manifolds and limiting Carleman weights”, arXiv:1011.2507 (2010).

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