% Smooth anisotropic uniqueness in the Calderon problem from one boundary patch.
% Self-contained rewrite. Every result of the original paper is proved here.
% Structure: (1) the problem and the theorem, (2) how the proof works,
% (3) boundary data, (4) local metric extension, (5) global isometry,
% (6) the scalar conductivity corollary.
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\title{Smooth anisotropic uniqueness in the Calder\'on problem\\[2pt]
from a single boundary patch}
\author{OpenAI}
\date{September 2026}
\begin{document}
\maketitle
\begin{abstract}
Fix a compact connected smooth manifold with boundary, of dimension at
least three, and a nonempty open piece of its boundary. We measure the
Dirichlet energy of harmonic functions whose boundary values are
supported in that piece. The theorem says that such measurements
determine the Riemannian metric completely, up to a change of
coordinates that fixes the measured boundary points. This holds for
every smooth metric, with input and observation on the same arbitrary
open patch. A scalar-conductivity corollary follows: on a fixed
Euclidean domain, a positive smooth conductivity is determined by its
local Dirichlet-to-Neumann map on any nonempty open boundary patch.
\end{abstract}
\tableofcontents
\section{The problem and the main theorem}
\subsection{What is being measured}
Let $M$ be a compact connected smooth manifold with smooth, possibly
empty-complement, boundary $\partial M$. Write $n=\dim M\ge3$. Let $g$
be a smooth Riemannian metric on $M$, including its boundary. All
integrals below use Riemannian densities, so no orientation is ever
needed.
For a boundary function $f$ in $H^{1/2}(\partial M)$, let $u_f^g\in H^1(M)$
be the unique weak solution of the Dirichlet problem
\[
-\Delta_g u_f^g = 0 \quad\text{in } M, \qquad
u_f^g|_{\partial M} = f ,
\]
where $\Delta_g=\operatorname{div}_g\nabla_g$. Uniqueness and existence
follow from the coercivity of the Dirichlet energy on $H^1_0(M)$.
Fix a nonempty relatively open set $\Gamma\subset\partial M$. Only
boundary values supported in $\Gamma$ are used, and only the resulting
energy flux on $\Gamma$ is observed. Define
\[
H^{1/2}_{co}(\Gamma)
=\bigl\{ f\in H^{1/2}(\partial M): \supp f\subset\Gamma \bigr\},
\]
with the norm of $H^{1/2}(\partial M)$. The \emph{local energy
Dirichlet-to-Neumann map} is the bounded operator
\[
\Lambda_{g,\Gamma}:H^{1/2}_{co}(\Gamma)\to
\bigl(H^{1/2}_{co}(\Gamma)\bigr)^*
\]
given on $f,h$ by
\begin{equation}\label{eq:energy}
\langle \Lambda_{g,\Gamma} f,h\rangle
=\int_M \langle \dd u_f^g,\dd u_h^g\rangle_g \,\dd V_g .
\end{equation}
For smooth supported data, Green's formula rewrites this as
\begin{equation}\label{eq:density}
\langle \Lambda_{g,\Gamma} f,h\rangle
=\int_\Gamma h\,\partial_{\nu_g}u_f^g\,\dd S_g ,
\end{equation}
with $\nu_g$ the outward unit normal and $\dd S_g$ the boundary density.
The output is therefore a \emph{density} on $\Gamma$, not an ordinary
function. Equality of \eqref{eq:energy} on smooth supported pairs is
equivalent to equality of the operators: every element of
$H^{1/2}_{co}(\Gamma)$ has compact support inside $\Gamma$ and is
approximated in $H^{1/2}(\partial M)$ by smooth functions supported in a
fixed larger compact subset of $\Gamma$.
\subsection{The theorem}
\begin{theorem}[Uniqueness from one patch]\label{thm:main}
Let $n\ge3$. Let $M$ be a compact connected smooth $n$-manifold with
nonempty smooth boundary, and let $\Gamma\subset\partial M$ be nonempty,
proper, and relatively open. If two smooth Riemannian metrics $g_1,g_2$
on $M$ satisfy
\[
\Lambda_{g_1,\Gamma}=\Lambda_{g_2,\Gamma},
\]
then there is a smooth diffeomorphism $\Phi:M\to M$, smooth through the
boundary, with
\[
\Phi|_\Gamma=\Id, \qquad g_2=\Phi^*g_1 .
\]
\end{theorem}
The operator hypothesis means equality of the graphs of the two maps.
The conclusion fixes every measured boundary point. It imposes no
condition at points of $\partial M\setminus\Gamma$. Several boundary
components are allowed, and $M$ need not be orientable.
\paragraph{The ambiguity is unavoidable.}
If $\Phi$ fixes $\Gamma$ and $g_2=\Phi^*g_1$, then for $f$ supported in
$\Gamma$ the function $u_f^{g_1}\circ\Phi$ is $g_2$-harmonic with
boundary value $f$ (a boundary diffeomorphism fixing $\Gamma$ preserves
its complement as well). The change of variables $x\mapsto\Phi(x)$ in
\eqref{eq:energy} then gives $\Lambda_{g_1,\Gamma}=\Lambda_{g_2,\Gamma}$.
So a diffeomorphism fixing $\Gamma$ cannot be detected, and Theorem
\ref{thm:main} says this is the only thing that cannot be detected.
\subsection{Consequences}
\begin{corollary}[Full-boundary uniqueness]\label{cor:full}
Under the geometric hypotheses of Theorem~\ref{thm:main}, if the
Dirichlet energies of $g_1$ and $g_2$ agree for \emph{every} pair of
smooth boundary values, then there is a smooth diffeomorphism $F:M\to M$
fixing $\partial M$ pointwise with $g_1=F^*g_2$.
\end{corollary}
\begin{proof}
Fix a boundary point $p$ and apply Theorem~\ref{thm:main} to
$\Gamma=\partial M\setminus\{p\}$. This set is nonempty, proper, open and
dense in $\partial M$, so equality of the local maps holds by continuity.
The resulting $\Phi$ is continuous on $M$ and equals the identity on
$\Gamma$, hence on all of $\partial M$. Take $F=\Phi^{-1}$.
\end{proof}
There is also a version with no coordinate ambiguity. Let
$\Omega\subset\R^n$ be bounded and connected with smooth boundary,
$\Gamma\subset\partial\Omega$ nonempty and relatively open, and
$\gamma\in C^\infty(\overline\Omega)$ strictly positive. For
$f\in C_c^\infty(\Gamma)$, extended by zero, let $u_f^\gamma$ solve
\[
\operatorname{div}(\gamma\nabla u_f^\gamma)=0 \text{ in }\Omega,
\qquad u_f^\gamma|_{\partial\Omega}=f .
\]
With $\nu_e$ the outward Euclidean unit normal, define
$\Lambda_\gamma^\Gamma f=(\gamma\partial_{\nu_e}u_f^\gamma)|_\Gamma$.
\begin{corollary}[Scalar uniqueness from one patch]\label{cor:scalar}
Let $n\ge3$ and let $\Omega,\Gamma$ be as above. If two strictly positive
smooth conductivities $\gamma_1,\gamma_2$ satisfy
$\Lambda_{\gamma_1}^\Gamma=\Lambda_{\gamma_2}^\Gamma$, then
$\gamma_1=\gamma_2$ on $\overline\Omega$.
\end{corollary}
The proof is in Section~\ref{sec:scalar}. It uses the exact dictionary
between a conductivity and a conformal metric, and a rigidity lemma
(Lemma~\ref{lem:conformal}) saying a conformal self-map of a Euclidean
domain fixing a boundary patch must be the identity.
\subsection{What was known, and why this is new}
Calder\'on \cite{Calderon1980} posed the inverse conductivity problem.
Kohn and Vogelius \cite{KohnVogelius1984} recovered boundary values of a
scalar conductivity. Sylvester and Uhlmann \cite{SylvesterUhlmann1987}
proved full-data uniqueness for smooth scalar conductivities in
dimension at least three, using complex geometrical optics. For
anisotropic (metric) conductivities, Lee and Uhlmann
\cite{LeeUhlmann1989} determined the boundary jets and recovered
real-analytic metrics; Lassas and Uhlmann \cite{LassasUhlmann2001}
recovered real-analytic manifolds and metrics from any real-analytic
boundary patch; Lassas, Taylor and Uhlmann
\cite{LassasTaylorUhlmann2003} extended to complete manifolds.
Guillarmou and S\'a Barreto \cite{GuillarmouSaBarreto2009} proved
same-patch uniqueness for smooth Einstein metrics.
For partial data, Bukhgeim and Uhlmann \cite{BukhgeimUhlmann2002}
introduced Carleman methods; Kenig, Sj\"ostrand and Uhlmann
\cite{KenigSjostrandUhlmann2007} used disjoint front/back faces; Isakov
\cite{Isakov2007} handled same-patch data by reflection in special
geometries; Dos Santos Ferreira, Kenig, Sj\"ostrand and Uhlmann
\cite{DosSantosFerreiraEtAl2009} proved the linearized density-of-products
theorem; Alessandrini and Kim \cite{AlessandriniKim2012} proved
same-patch stability when the conductivity is known near the whole
boundary. Dos Santos Ferreira, Kenig, Salo and Uhlmann
\cite{DSFKSU2009} and Kenig and Salo \cite{KenigSalo2013} treated
admissible product geometries.
Recent full-data rigidity results hold near prescribed backgrounds: Lin
\cite{Lin2026} near the Euclidean metric, Mu\~noz-Thon and Stefanov
\cite{Stefanov2026} near conformally Euclidean metrics, and Chen,
Jiang, Liu and Tao \cite{ChenJiangLiuTao2026} under quasianalyticity or
extra foliation and ordering assumptions. Uhlmann and Wang
\cite{UhlmannWangNear2026,UhlmannWangPartial2026} recover potentials and
near-boundary conformal factors in dimension three. Daud\'e, Enciso,
Helffer, Kamran and Nicoleau \cite{DaudeEtAlPropagation2026} propagate a
local Dirichlet-to-Neumann equality along a boundary component when the
metrics already agree in a collar.
Harmonic functions describe points and coordinates intrinsically; this
goes back to Greene and Wu \cite{GreeneWu1975}, and the Poisson embedding
of Lassas, Liimatainen and Salo \cite{LLS20} represents an interior
point by its evaluations on boundary-supported harmonic functions. The
local approximation and finite-jet tools come from Lax
\cite{Lax1956}, Malgrange \cite{Malgrange1956}, and the quantitative
supported-boundary form of R\"uland and Salo \cite{RulandSalo2019}.
Theorem~\ref{thm:main} needs no background metric, no real analyticity,
no convexity, and no condition on the inaccessible boundary. The price
is a technical local continuation theorem (Theorem~\ref{thm:local}) whose
proof is the analytical heart of the paper.
\section{How the proof works}\label{sec:strategy}
The proof has five movements. We describe them in words first, so the
technical sections have a destination.
\paragraph{Movement 1: read the boundary.}
The measured operator is an energy, i.e.\ a density. From its principal
symbol one first recovers the boundary metric and the boundary density;
dividing by the density gives the ordinary Dirichlet-to-Neumann operator.
Factoring the Laplace operator in normal coordinates then gives a
symbol recursion that recovers the full Taylor jet of the metric on
$\Gamma$ (Lemma~\ref{lem:jets}). This is classical boundary
determination \cite{LeeUhlmann1989,JoshiLionheart2005}; we prove it in
full so the paper is self-contained.
\paragraph{Movement 2: build a common exterior cap.}
Because the boundary jets agree on an open piece $P_0\Subset\Gamma$, we
can extend both manifolds past the boundary by a thin cap in such a way
that (a) the cap is common and (b) after identifying the caps the two
metrics agree near the seam. A variational (energy minimizing)
comparison using the measured map shows the Dirichlet Green kernels of
the two extended metrics agree on the whole cap
(Proposition~\ref{prop:cap}). Sources inside the cap then produce
harmonic functions, and evaluation at interior points gives
functionals that separate points and realize every admissible second
jet (Lemma~\ref{lem:jet-family}). These are our probes.
\paragraph{Movement 3: the local extension theorem.}
This is the heart. We are given two metrics in the same harmonic
coordinates that agree on one side $W$ of a smooth hypersurface through
$0$, whose Green kernels agree on $W\times W$, and a finite list of
paired harmonic functions $u_1^a,u_2^a$ agreeing on $W$, whose
Hessians span the hyperplane of second derivatives allowed by the
harmonic equation. Claim: the two coordinate metrics agree near $0$
(Theorem~\ref{thm:local}, proved in Section~\ref{sec:bootstrap}).
The mechanism is a \emph{transfer functional}. Inside the region where
a mixed Green kernel has been continued, its flux on a small sphere
centered at $y$ defines $T_y$, mapping first-metric harmonic functions
to second-metric harmonic functions. It reproduces constants,
coordinates, and every paired function:
\[
T_y 1=1,\qquad T_y x^i=y^i,\qquad T_y u_1^a=u_2^a .
\]
Existence of $T_y$ at every positive radius uses a product-continuation
lemma across a cross (Lemma~\ref{lem:cross}) and a strict-weight
criterion for crossing moving surfaces
(Proposition~\ref{prop:criterion}). Existence alone does not give a
bound as the radius shrinks, so we prove one.
The temporary hypothesis is that the normalized difference $h$ of the
two inverse metrics, together with eight derivatives, is small compared
with the sphere radius $r$ (Hypothesis \eqref{eq:bootstrap}). Under
this hypothesis the density $\psi$ representing $T_y$ is bounded in
$L^2$ uniformly in the shrinking parameter
(Lemma~\ref{lem:density}). Then subtracting the affine Taylor
polynomial at $y$ and using $T_y1=1$, $T_yx^i=y^i$ gives, for each
paired difference $V=u_2^a-u_1^a$,
\[
|V(y)|\le C r^2\,\|\psi\|_{L^2(S_y)} ,
\]
hence $\|V/r^2\|_{L^2}\le C$ (Lemma~\ref{lem:moment}). A standard
finite-order interpolation lemma (Lemma~\ref{lem:interp}) turns the
$L^2$ bound into pointwise bounds $\partial^\alpha V\le Cr^{3/2}$ for
$|\alpha|\le10$. The common harmonic coordinates make the Laplace
equations linear in the metric discrepancy, with coefficients given by
these Hessians; solving that finite linear system (the Hessians span
the allowed hyperplane) yields $H_8(y)\le Cr^{3/2}$
(Lemma~\ref{lem:improvement}). Since $r^{3/2}\le \tfrac12 r$ for small
$r$, this strictly improves the temporary hypothesis near the
shrinking locus, and a single spatial dilation handles radii above a
fixed threshold.
\paragraph{The two parameters.}
There are two scales. The \emph{spatial dilation} $\lambda$ is chosen
once, to make the metric difference small in every fixed Smooth norm
and to handle radii above a threshold. The \emph{shrinking parameter}
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