OP-00717 Solution claimed

Smooth anisotropic Calderón uniqueness

Let $M$ be a compact connected smooth manifold of dimension $n \ge 3$ with nonempty smooth boundary. For a smooth Riemannian metric $g$ on $M$ define $\Lambda_g f = \partial_{\nu_g} u|_{\partial M}$, where $\Delta_g u = 0$ in $M$ and $u|_{\partial M} = f$. If $F : M \to M$ is a smooth diffeomorphism with $F|_{\partial M} = \mathrm{Id}$, then $\Lambda_{F^* g} = \Lambda_g$. Lee and Uhlmann proved that this is the only obstruction for real-analytic metrics on strongly convex, simply connected manifolds, and Lassas and Uhlmann for real-analytic manifolds and metrics with no topological assumption, from measurements restricted to a real-analytic part of the boundary.

Problem. Let $g_1$ and $g_2$ be smooth Riemannian metrics on $M$ that induce the same metric on $\partial M$, so that their boundary operators have the same interpretation. Does $\Lambda_{g_1} = \Lambda_{g_2}$ imply that there is a smooth diffeomorphism $F : M \to M$ with $F|_{\partial M} = \mathrm{Id}$ and $g_1 = F^* g_2$? The data are this single zero-frequency boundary operator for each metric.

Source: The anisotropic Calderón problem in the geometric form of Lee and Uhlmann (1989), who proved the real-analytic case; stated in this form, for smooth metrics and full-boundary data, by AIM — Open Applied Problems.

Imported from github.com/MColbrook/AIM.

Classification

Primary: AP — Analysis of PDEs
Secondary: DG, OC
Keywords: AIM Open Applied Problems, Calderón problem, Dirichlet-to-Neumann map, Riemannian manifolds

References

@article{lee1989,
  author  = {Lee, John M. and Uhlmann, Gunther},
  title   = {Determining anisotropic real-analytic conductivities by boundary measurements},
  journal = {Comm. Pure Appl. Math.},
  volume  = {42},
  number  = {8},
  year    = {1989},
  pages   = {1097--1112},
  doi     = {10.1002/cpa.3160420804},
}
@article{lassas2001,
  author  = {Lassas, Matti and Uhlmann, Gunther},
  title   = {On determining a {R}iemannian manifold from the {D}irichlet-to-{N}eumann map},
  journal = {Ann. Sci. {\'E}cole Norm. Sup. (4)},
  volume  = {34},
  number  = {5},
  year    = {2001},
  pages   = {771--787},
  doi     = {10.1016/S0012-9593(01)01076-X},
}
@article{uhlmann2013,
  author  = {Uhlmann, Gunther},
  title   = {30 years of {C}alder{\'o}n's problem},
  journal = {S{\'e}minaire Laurent Schwartz --- EDP et applications (2012--2013)},
  year    = {2013},
  pages   = {Expos{\'e} no. XIII, 1--25},
  doi     = {10.5802/slsedp.40},
}
@misc{aim-smooth-anisotropic-calderon,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{Smooth anisotropic Calderón uniqueness}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/047-smooth-anisotropic-calderon.md},
  note         = {Entry 047 at commit 1b7c7fd, 2026-10-13},
}

Resolution claims

Smooth anisotropic uniqueness in the Calderón problem from one boundary patch v1 GitHub ↗

OpenAI

We resolve the smooth anisotropic Calderón uniqueness problem with arbitrary same-patch measurements. A smooth Riemannian metric on a compact connected manifold of dimension at least three is determined, up to a diffeomorphism fixing the measured patch, by its zero-frequency Dirichlet-to-Neumann energy form with both input and observation on any nonempty open boundary patch.

Full resolution · not verified · fidelity not established · posted on GitHub 6 Oct 2026

Status history

When	Change	By	Reason
2026-10-18	Needs verification → Solution claimed	—	Recorded here: math @ 1581d4c, build/paper.tex.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00717, "Smooth anisotropic Calderón uniqueness", clopen.solutions.