Metric recovery from one boundary patch in the anisotropic Calderón problem
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For a compact connected smooth manifold of dimension at least three, do density-valued energy Dirichlet-to-Neumann forms on any nonempty proper relatively open boundary patch determine a smooth metric up to a patch-fixing diffeomorphism?
Why it matters and what counts as progress
Why it matters: It asks whether local boundary measurements suffice for the smooth anisotropic inverse problem without a larger accessible boundary region.
What counts as progress: Prove arbitrary-patch anisotropic uniqueness up to patch-fixing diffeomorphism for smooth metrics, or exhibit a new gauge obstruction.
How the problem developed
- 24 Sept 2026 · Uniqueness theorem [1]
OpenAI Reported uniqueness of smooth metrics in dimension at least three, up to a boundary-fixing diffeomorphism, from Dirichlet-to-Neumann data on one arbitrary open boundary patch.
AI contribution
Smooth anisotropic uniqueness in the Calderón problem from one boundary patch
For compact connected smooth manifolds of dimension at least three, the manuscript claims that equal density-valued energy Dirichlet-to-Neumann forms measured on any nonempty proper relatively open boundary patch determine two smooth metrics up to a smooth diffeomorphism fixing that patch.
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Publication dates and editorial changes.
OpenAI reports smooth anisotropic recovery from one boundary patch