The Lee-Uhlmann conjecture: smooth anisotropic Calderon uniqueness in dimension at least three, from one boundary patch

Theorem 1.1: for $n\ge3$, compact connected smooth $M$ with smooth boundary and any nonempty proper relatively open $\Gamma\subset\partial M$, if smooth metrics satisfy $\Lambda_{g_1,\Gamma}=\Lambda_{g_2,\Gamma}$ (energy form with input and output on $\Gamma$), then $g_2=\Phi^*g_1$ for a smooth diffeomorphism with $\Phi|_\Gamma=\mathrm{Id}$. Corollaries: full-boundary uniqueness, and uniqueness of smooth positive scalar conductivities on smooth Euclidean domains from one same-patch measurement. The October 5 companion adds a smooth unitary connection on a trivial Hermitian rank-two bundle, determined up to gauge fixed on the patch. Not shown: dimension two, nonsmooth metrics, stability estimates, or reconstruction.

Theorem 1.1: for $n\ge3$, compact connected smooth $M$ with smooth boundary and any nonempty proper relatively open $\Gamma\subset\partial M$, if smooth metrics satisfy $\Lambda_{g_1,\Gamma}=\Lambda_{g_2,\Gamma}$ (energy form with input and output on $\Gamma$), then $g_2=\Phi^*g_1$ for a smooth diffeomorphism with $\Phi|_\Gamma=\mathrm{Id}$. Corollaries: full-boundary uniqueness, and uniqueness of smooth positive scalar conductivities on smooth Euclidean domains from one same-patch measurement. The October 5 companion adds a smooth unitary connection on a trivial Hermitian rank-two bundle, determined up to gauge fixed on the patch. Not shown: dimension two, nonsmooth metrics, stability estimates, or reconstruction.