Let M be a compact connected smooth n-manifold with nonempty smooth
boundary, and let n >= 3. Fix a smooth Riemannian metric g on M,
extending smoothly to the boundary. Volume and surface integrals
use the Riemannian densities dV_g and dS_g. No orientation is used.
For f in H^(1/2)(bd M), let u_f^g in H^1(M) be the unique weak
solution of the Dirichlet problem
-Delta_g u_f^g = 0 in M, u_f^g restricted to bd M = f,
where Delta_g = div_g grad_g. Coercivity of the Dirichlet energy on
H_0^1(M) supplies existence and uniqueness.
Fix a nonempty relatively open Gamma subset bd M. Set
H^(1/2)_co(Gamma) = {f in H^(1/2)(bd M) : supp f lies in Gamma},
equipped with the H^(1/2)(bd M) norm. The local energy
Dirichlet-to-Neumann operator is the bounded map
Lambda_(g,Gamma) : H^(1/2)_co(Gamma) -> (H^(1/2)_co(Gamma))^*
defined on f,h in H^(1/2)_co(Gamma) by
<Lambda_(g,Gamma) f, h>
= int_M <grad u_f^g, grad u_h^g>_g dV_g. (1)
For f,h in C_c^inf(Gamma), Green's formula rewrites (1) as
<Lambda_(g,Gamma) f, h>
= int_Gamma h (partial_(nu_g) u_f^g) dS_g, (2)
with nu_g the outward unit normal. The measured output carries the
boundary density: it is a density on Gamma, not an ordinary function.
Equality of (1) on smooth supported pairs is equivalent to equality
of the operators, since every element of H^(1/2)_co(Gamma) is a limit,
in H^(1/2)(bd M), of smooth functions supported in a fixed larger
compact subset of Gamma.
Theorem 1 (Uniqueness from one patch). Let n >= 3. Let M be a compact
connected smooth n-manifold with nonempty smooth boundary, and let
Gamma subset bd 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 a smooth diffeomorphism Phi : M -> M, smooth up to the boundary,
exists with
Phi restricted to Gamma = Id, g_2 = Phi^* g_1.
The hypothesis is equality of the graphs of the two operators. The
conclusion fixes every measured boundary point. It imposes no
condition at points of bd M minus Gamma. Several boundary components
are allowed, and M need not be orientable.
The diffeomorphism ambiguity cannot be removed. Suppose Phi fixes
Gamma and g_2 = Phi^* g_1. For f supported in Gamma, the composition
u_f^(g_1) o Phi is g_2-harmonic with boundary value f because a
boundary diffeomorphism fixing Gamma preserves its complement. Change
of variables x -> Phi(x) in (1) yields
Lambda_(g_1,Gamma) = Lambda_(g_2,Gamma). A diffeomorphism fixing
Gamma therefore cannot be detected, and Theorem 1 says nothing else
can be.
Corollary 2 (Full-boundary uniqueness). Let M, g_1, g_2 satisfy the
geometric hypotheses of Theorem 1. If the Dirichlet energies of g_1
and g_2 agree for every pair of smooth boundary values, then a smooth
diffeomorphism F : M -> M fixing bd M pointwise satisfies
g_1 = F^* g_2.
Proof. Fix p in bd M and apply Theorem 1 to the open set
Gamma = bd M minus {p}. This set is nonempty, proper, open, and dense
in bd M, so equality of the corresponding local operators follows
from continuity. The resulting Phi is continuous on M and equals the
identity on Gamma, hence on all of bd M. Set F = Phi^-1.
Scalar consequence. Let Omega subset R^n be bounded and connected with
smooth boundary, let Gamma subset bd Omega be nonempty and relatively
open, and let gamma in C^inf(Omega-bar) be strictly positive. For f
in C_c^inf(Gamma), extended by zero, let u_f^gamma be the weak
solution of div(gamma grad u_f^gamma) = 0 in Omega with u_f^gamma = f
on bd Omega. Let nu_e be the outward Euclidean unit normal and set
Lambda_gamma^Gamma f = (gamma partial_(nu_e) u_f^gamma) restricted
to Gamma.
Corollary 3 (Scalar uniqueness from one patch). Let n >= 3 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 Omega-bar.
The proof identifies the exact metric energy associated with a scalar
conductivity and then shows that a conformal self-map of a Euclidean
domain fixing a boundary patch must be the identity. No conductivity
values on the inaccessible boundary are assumed.
Notation. For a metric g set Delta_g = div_g grad_g and L_g = -Delta_g.
The Dirichlet realization of L_g on a compact connected smooth
manifold with nonempty boundary is invertible, since its quadratic
form on H_0^1 is coercive. Smooth sources and smooth boundary data
give solutions smooth up to the boundary. Write Hess_g for the
covariant Hessian and nu_g for the outward unit normal.
The central input is the local metric extension theorem (Theorem 7
below). Its hypothesis is a pair of interior harmonic coordinate
charts in which the metrics, their Green kernels, and a finite family
of paired harmonic functions already agree on one side of a smooth
hypersurface; the first-side coordinate Hessians span the second-
derivative hyperplane allowed by the harmonic equation. The conclusion
is local metric equality across the hypersurface.
The remaining argument proceeds as follows.
(i) Boundary data and a common cap. Local boundary determination
recovers the complete Taylor jet of the metric on the measured patch.
The two manifolds are extended smoothly past the boundary patch with
matching jets, and a variational comparison on the extended Dirichlet
energy makes the actual Green kernels of the two extended metrics
agree throughout the cap. Sources in the cap produce harmonic
functions whose point evaluations separate interior points and supply
all admissible second-order harmonic jets. Those functions and their
paired partners are the finite families used later.
(ii) Product continuation. Inside the paired charts a mixed Green
kernel solves the first metric equation in the first variable and the
second metric equation in the second variable. Product continuation
extends this kernel at fixed positive separation from the diagonal.
The flux of the mixed kernel on a small sphere of radius r centered
at y defines a transfer T_y from local first-metric harmonic
functions to local second-metric harmonic functions. It sends paired
functions to paired functions and reproduces constants and
coordinates: T_y 1 = 1 and T_y x^i = y^i. Existence at each positive
radius does not bound T_y as the radius tends to zero.
(iii) Shrinking and density. To compare conformal classes, normalize
the inverse metrics to have the same trace against a reference metric
conformal to the first. The shrinking argument temporarily assumes
that the normalized difference of inverse metrics, together with a
fixed finite number of its derivatives, is small compared with the
sphere radius r. Under this hypothesis the density psi representing
T_y is bounded in L^2 uniformly in the shrinking parameter. Exact
reproduction of constants and coordinates cancels the affine Taylor
polynomial at each center. The Taylor remainder on the sphere is
O(r^2), so each paired difference V satisfies |V| <= C r^2 ||psi||,
hence the L^2 norm of V / r^2 is uniformly bounded.
(iv) Interpolation and the metric system. Finite-order interpolation
turns the L^2 bound into |partial^alpha V| <= C r^3/2 for
multi-indices |alpha| <= 10. Harmonic coordinates make the Laplace
equations a linear system in the normalized metric discrepancy, with
coefficients supplied by the Hessians of the paired functions. Since
those Hessians span the second-derivative hyperplane, the system
inverts and yields |partial^alpha h| <= C r^3/2 for the normalized
metric discrepancy and its required derivatives. Since r^3/2 <= r/2
at small radii, this strictly improves the temporary hypothesis near
the shrinking locus.
(v) Two scales. A single spatial dilation is fixed to make the initial
metric discrepancy small in every fixed smooth norm and to handle
radii above a fixed threshold. The strict improvement then allows the
shrinking parameter to tend to zero with the dilation held fixed. On
the fixed neighborhood where the radii tend to zero, the normalized
inverse metrics therefore agree. This determines the metrics up to a
conformal factor. The common harmonic coordinates, together with
n >= 3, force that factor constant; agreement on the known side
fixes it.
(vi) The angular estimate. The uniform density estimate controls the
full equation obtained by differentiating traces on the moving
spheres. Comparing spherical-harmonic degree-raising and
degree-lowering operators gives a positive weighted estimate.
Moving-sphere geometry contributes second-order terms that cannot
simply be discarded as lower-order errors; a second comparison
retains them and absorbs them into the positive quantities of the
first estimate.
(vii) Globalization. The source evaluations make the local
identifications single-valued and keep an interior point from
escaping to the boundary. A smooth ball in the matched region
touching its frontier supplies the one-sided geometry of Theorem 7.
Applying that theorem eliminates every interior frontier. The
resulting isometry extends smoothly through the boundary, and the
original supported boundary data force it to fix all of Gamma.
The steps are carried out in the order listed: boundary data and the
common cap, then product continuation and the local theorem, then the
shrinking sphere family, the angular estimate, the bootstrap, the
global isometry, and finally the scalar consequence. Every constant
used in a later step is selected before the next limiting operation,
so no limit argument depends on a coefficient chosen after it.
Historical context.
Calderon formulated the inverse conductivity problem through boundary
measurements of Dirichlet energy, and studied its linearization. The scalar case with full boundary data was settled by Kohn and Vogelius for real-analytic conductivities, and by Sylvester and Uhlmann for all smooth conductivities in dimension at least three. In the anisotropic setting, Lee and Uhlmann recovered a real-analytic anisotropic conductivity on a real-analytic manifold from full boundary data, and Lassas and Uhlmann reduced the topological requirements to a single nonempty patch of real-analytic boundary in dimension at least three.
./pdf/lassas-uhlmann-2001-ASENS.pdf
The linearized same-patch problem was settled by Dos Santos Ferreira, Kenig,
Sjostrand, and Uhlmann, who exhibited density of the linearized response
in a suitable space. Limiting Carleman weights, introduced by Dos Santos
Ferreira, Kenig, Salo, and Uhlmann, supply an efficient framework for
many partial-data results in the scalar setting, including same-patch
stability up to a log-modulus of continuity.
pdf/0905.0530.pdf
pdf/0803.3508.pdf
pdf/1202.5485.pdf
Boundary determination at every order is classical for smooth metrics.
The symbol calculus on the half-space gives the template that the symbol recursion follows; what matters below is carrying out the recursion in a form that
yields complete jet equality, not merely low-order determination.
pdf/math_9911212.pdf
Green functions are a second entry point. Lassas, Taylor, and Uhlmann
proved that a Green function determines a compact manifold up to
isometry under suitable restrictions. Lassas, Liimatainen, and Salo
extended the approach via Poisson embedding, representing each interior
point by the family of boundary-supported harmonic functions evaluated
there. In the present argument the Green function is used only for the
one-sided comparison on an auxiliary cap; its full topology plays no
role beyond that cap.
pdf/lassas-taylor-uhlmann-2003.pdf
pdf/lassas-liimatainen-salo-2020.pdf
Common-basis methods for separately harmonic extension have a classical
antecedent in Zeriahi. Fuglede's harmonic-morphism analogue, in which a
common basis on a codimension-two set governs extension of harmonic
functions, is also a predecessor.
pdf/zeriahi-1982-AFST.pdf
pdf/fuglede-1978-AIF.pdf
Recent full-data rigidity holds near prescribed smooth backgrounds.
Lin proves rigidity near the Euclidean metric under smallness in a
Holder norm. Munoz-Thon and Stefanov prove local rigidity near every
smooth conformally Euclidean metric under Sobolev smallness. Chen,
Jiang, Liu, and Tao obtain same-patch uniqueness under quasianalyticity
and smooth uniqueness under foliation and ordering assumptions. The
present Theorem 1 needs neither smallness nor geometric hypotheses.
pdf/2609.17261.pdf
pdf/2609.13400.pdf
pdf/2608.15813.pdf
In dimension three, Uhlmann and Wang recover compactly supported
potentials and near-boundary conformal factors on a fixed near-
Euclidean metric with strictly convex boundary. Their partial-data
result recovers a potential near the measured boundary with support
away from that boundary. Daude, Enciso, Helffer, Kamran, and Nicoleau
propagate local Dirichlet-to-Neumann equality along a connected boundary
component starting from an existing collar of metric agreement. The
present theorem needs neither a near-Euclidean background nor a collar
of agreement, at the price of using the boundary patch as simultaneous
input and observation.
pdf/2606.26540.pdf
pdf/2606.10247.pdf
pdf/2606.29233.pdf
Approximation by exact solutions on the target side is a recurring
device in inverse problems. Malgrange's classical approximation theorem
for inhomogeneous systems is the source for the linear version. Its
quantitative forms on bounded families of harmonic solutions are used
in a later section to produce a finite harmonic family with controlled
constants.
pdf/malgrange-1956-AIF.pdf
pdf/1708.06307.pdf
Tensor tomography enters at one step: the comparison of spherical-
harmonic raising and lowering operators on the shrinking spheres. The
X-ray transform in negative curvature and its invariant-distribution
algebra supply the algebraic model that the positive weighted estimate
imitates.
pdf/1404.7009.pdf
pdf/1502.04720.pdf
Boundary determination.
The first step of the local-to-global argument is the recovery of the
Taylor jets of a smooth metric on the measured patch Gamma from the
corresponding Dirichlet-to-Neumann operator. The next lemma records what
is recovered; the rest of the section is its proof.
Lemma 4 (Recovery of boundary jets). Let g_1, g_2 be smooth Riemannian
metrics on M extending smoothly to the boundary. Suppose
Lambda_(g_1,Gamma) = Lambda_(g_2,Gamma) for a nonempty relatively open
subset Gamma of bd M. Then, at every point p of Gamma, the jets of g_1
and g_2 restricted to Gamma agree through all orders.
Proof. Work locally near a fixed point p of Gamma. Choose boundary
normal coordinates (s,z) with s >= 0 and z ranging over a bounded open
subset of R^(n-1), so that the boundary is s = 0 and the coordinate
line s -> (s,z) is the geodesic normal of g_1 issuing from (0,z). Write
g(s,z) = ds^2 + h(s,z),
where h(s,z) is a smooth family of (0,2)-tensors on the tangent slices
{s = const}. Set a = (1/2) tr_{h(s)} partial_s h(s), the mean curvature
of {s = const}. In these coordinates the Laplace operator has the
normal form
L_g = -partial_s^2 - a partial_s + P, P = -Delta_{h(s)}, (3)
with P tangential.
Factorization. The Laplace operator admits a factorization
L_g = (-partial_s + B - a)(partial_s + B) mod Psi^{-infty} (4)
with B(s) a tangential operator of order one having positive principal
symbol, and Psi^{-infty} denoting operators that are tangentially
smoothing and depend smoothly on s. The symbol b of B then satisfies
b#b - ab - partial_s b = p, (5)
where p is the symbol of P, expanded in increasing order. This
factorization exists for smooth coefficients, without analyticity
hypotheses.
Principal symbol. Write b = b_1 + b_0 + b_{-1} + ... and p = p_2 + p_1
in decreasing order. Writing left symbols with D_z = -i partial_z
and the composition law
b#c ~ sum_{alpha} ((1/i)^{|alpha|}/alpha!)
(partial_xi^alpha b)(partial_z^alpha c),
the leading coefficients are
b_1 = rho := (h^(ab) xi_a xi_b)^(1/2), p_2 = rho^2, p_1 = 0.
That last vanishing reflects that P has no first-order part in h-
divergence form. The degree-one part of (5) is
2 rho b_0 + (1/i) partial_xi rho partial_z rho
- a rho - partial_s rho = p_1 = 0. (6)
The two-summand contribution. Fix a covector xi != 0 at (0,z) and let
xi^# denote its h-dual, so that h(xi^#,xi^#) = rho^2. Using
partial_s rho = - (partial_s h)(xi^#,xi^#)/(2 rho)
and a = (1/2) tr_h (partial_s h), the portion of b_0 introduced by
partial_s h at s = 0 is
(1/4) (tr_h partial_s h - (partial_s h)(xi^#,xi^#)/rho^2). (7)
The remaining content of (6) involves only tangential derivatives of h,
and is therefore already determined by h(0).
Higher orders. At degree 1-k with k >= 2, the same recursion gives a
unique new combination
b_1-k = (1/(4 (2rho)^{k-1}))
( tr_h A_k - A_k(xi^#,xi^#)/rho^2 )
+ R_k( h, A_1, ..., A_{k-1} ), (8)
where A_j := partial_s^j h|_{s=0} and R_k depends only on lower normal
jets of h, their tangential derivatives, and A_1,...,A_{k-1}. Only the
leading term of (8) is new at this order; R_k is a known combination of
the previous coefficients and their tangential derivatives. The form of
the leading term is the same at every k: the single scalar combination
tr_h A_k - A_k(xi^#,xi^#)/rho^2 is the sole piece of b_1-k that cannot
be inferred from lower jet orders.
The boundary Dirichlet-to-Neumann operator. In the factorization (4),
the boundary Dirichlet-to-Neumann operator coincides, up to tangentially
smoothing corrections, with B(0), evaluated on functions of zero normal
derivative. This is the local Poisson-parametrix identification of B(0)
with the outward Dirichlet-to-Neumann operator: the decaying branch of
the boundary problem satisfies (partial_s + B)u = 0 and the outward
normal derivative is -partial_s. Equality of the two Dirichlet-to-
Neumann maps at s = 0 therefore gives equality of B_1(0) and B_2(0)
modulo Psi^{-infty}. Reading off the symbol expansion, and using that
A_j(h_1) = A_j(h_2) has already been established for j < k, this
equality at degree 1-k identifies A_k(h_1) and A_k(h_2) through the
difference quantity in (8).
Induction. Start at k = 1: since Lambda_(g_1,Gamma) = Lambda_(g_2,Gamma)
implies, at the level of principal symbols, that h_1(0) = h_2(0), the
difference A_1 := partial_s(h_1 - h_2)|_{s=0} is well defined as a
smooth (0,2)-tensor on the {s = 0} boundary. Equality of B_1(0) and
B_2(0) at degree 0 gives
tr_h A_1 - A_1(xi^#,xi^#)/rho^2 = 0 for every xi != 0. (9)
Substitute xi^# = v, with |v|_h = 1. Then A_1(v,v) = tr_h A_1.
Polarizing in v gives the quadratic form identity A_1 = (tr_h A_1) h.
Taking the trace against h^{-1} gives
tr_h A_1 = (tr_h A_1) tr_h h = (tr_h A_1) n,
hence (n-1) tr_h A_1 = 0. Since n >= 2, tr_h A_1 = 0, and therefore
A_1 = 0.
Next let k >= 2 and suppose A_1 = ... = A_{k-1} = 0 already. Then at
degree 1-k, equality of B_1(0) and B_2(0) gives
tr_h A_k - A_k(xi^#,xi^#)/rho^2 = 0 for every xi != 0, (10)
because the lower-jet contributions to (8) coincide on the two sides.
The same polarization and trace argument as for k = 1 now yields
A_k = 0.
Conclusion. By induction over k, all normal jets of h_1 - h_2 at s = 0
vanish. The normal components g_ss = 1 and g_sa = 0 are canonical in
geodesic normal coordinates and therefore agree identically. Equality of
all normal jets of h at s = 0 gives equality of all metric jets at s = 0.
Differentiating along bd M (which is not the normal direction) recovers
agreement of all tangential derivatives of these jets. Therefore the
jets of g_1 and g_2 at every measured point p of Gamma agree through
all orders. QED.
Densities and orientability. The symbol calculation above is expressed
in scalar coordinates and does not require that M be orientable. The
differential-form formulation of boundary determination for the
anisotropic problem assumes orientability; the present scalar form does
not. Other boundary components affect only the local smoothing
remainder and do not enter the symbol recursion.
The gap between jets on the boundary and the metric in an interior
collar. Lemma 4 gives an equality of infinite-order jets at the
boundary, not an interior equality in a collar. Passing from jet
equality to interior equality requires an analyticity-type argument or
the local continuation of later sections, and the present argument
takes the second route.
pdf/math_9911212.pdf
The common cap.
Lemma 4 gives equality of infinite-order jets of g_1 and g_2 at every measured point of Gamma. This allows a collar of the same shape to be attached to both copies of M across a smaller measured patch, chosen so that the two metrics agree to all orders across the seam.
Precisely, let P_0 be a nonempty relatively open subset of Gamma whose closure in Gamma is compact, and let epsilon_0 > 0 be smaller than the collar radius of M. In the geodesic normal coordinates of g_1 and of g_2, the boundary jets at s = 0 agree to all orders on P_0 by Lemma 4. A common collar E of width epsilon_0 across P_0 therefore carries a smooth metric g^E with the property that (M union_patch E, g_j extended by g^E) is a smooth compact Riemannian manifold with boundary. Denote the two extended manifolds by M_1^e and M_2^e, and the corresponding metrics by g_1^e, g_2^e.
Fix a nonempty open set U_0 compactly contained in the interior of E, with closure(U_0) disjoint from bd E. All source densities used below are supported in U_0.
Proposition 5 (Green equality on the cap). Assume the hypotheses of Theorem 1. Let M_1^e, M_2^e be the common exterior cap just constructed, and let G_1, G_2 be the Dirichlet Green functions of Delta_{g_1^e}, Delta_{g_2^e}. Then
G_1(x,y) = G_2(x,y) for all x, y in E, x != y.
Proof. Fix F in C_c^inf(E). For j = 1, 2, let w_j minimize
J_j(w) = (1/2) int_{M_j^e} |dw|_{g_j}^2 dV_{g_j}
- int_E F w dV_{g_j} (11)
over H^1_0(M_j^e). Coercivity of the Dirichlet form on H^1_0, together with the smoothness of g_j^e and the support of F, gives existence and uniqueness; interior elliptic regularity and smoothness of bd M_j^e give w_j smooth up to bd M_j^e.
The restriction h_j = w_j | bd M is supported on that part of bd M where the Dirichlet problem is nontrivial. Since F is supported in E (inside the collar and away from bd M), and bd M_j^e is composed of bd M minus patch P_0 plus bd E, the restriction h_j is supported in P_0, and vanishes identically on bd E. In particular h_j is compactly supported inside Gamma, so it is in C_c^inf(Gamma) up to smoothness.
Variational comparison. Transport w_1 to M_2^e by the identity on E. Inside the original M (viewed as a subset of M_2^e), replace the transported function by the g_2-harmonic lifting with the same boundary value h_1. The result v_2 lies in H^1(M_2^e) - E, is g_2-harmonic there, and agrees with w_1 on E. Since the two functions vanish on bd M_2^e cap bd M minus P_0 (where h_2 also vanishes) and on bd E, the difference v_2 - w_2 lies in H^1_0(M_2^e minus E) and v_2 is an admissible competitor for J_2.
Energy comparison. The interior (non-cap) contribution to the Dirichlet energy of v_2 is the quadratic form <Lambda_(g_2,Gamma) h_1, h_1>. Since Lambda_(g_1,Gamma) = Lambda_(g_2,Gamma), this equals <Lambda_(g_1,Gamma) h_1, h_1>, which is exactly the non-cap contribution to the Dirichlet energy of w_1. The cap contributions of w_1 and v_2 coincide because g_1^e = g_2^e on E and both functions agree there. The source-pairing contributions are identical since both involve the same F and the same g^E on E. Therefore J_2(v_2) = J_1(w_1) = min J_1, and hence min J_2 <= min J_1.
Reversing the roles gives min J_1 <= min J_2; consequently min J_1 = min J_2. Strict convexity of the Dirichlet form makes each minimizer unique, so v_2 = w_2 in E. Since v_2 = w_1 on E by construction, w_1 = w_2 in E.
Representation via the Green function. Normalize the symmetric Dirichlet Green functions by
-Delta_(g_j,x) G_j(x,y) = delta_y^(g_j)(x),
G_j(., y) restricted to bd M_j^e = 0, (12)
where delta_y^(g_j) is a unit point mass at y measured by the Riemannian density of g_j. Then
w_j(x) = int_{M_j^e} G_j(x,y) F(y) dV_{g_j}(y). (13)
Since the source densities of the two problems coincide on E, and w_1 = w_2 on E, and the integrals range over E, equality of (13) at every x in E for every F in C_c^inf(E) gives equality of the distributions
G_1(x, .) restricted to (E minus {x}) and G_2(x, .) restricted to (E minus {x})
for every x in E. Green functions are symmetric in their two arguments (up to the fixed metric), so this distributional equality implies pointwise equality of the smooth functions G_1 and G_2 on E x E away from the diagonal. QED.
Sources in U_0 produce harmonic functions in the cap. Given y in bd M_1^e and a source density g in C_c^inf(U_0), the function
u_g(x) = int_E G_1(x,y) g(y) dV_{g_1}(y), x in M_1^e, (14)
is harmonic in the interior of M_1^e and vanishes on bd M minus P_0. Since G_1 = G_2 on E x E (off the diagonal), the same formula gives the corresponding harmonic function in M_2^e. Thus each source density g produces a paired harmonic function
(u_g^1, u_g^2), u_g^j harmonic in the interior of M_j^e, (15)
with u_g^1 = u_g^2 on the cap E by (14) and by G_1 = G_2 there.
Evaluation maps separate interior points. For a fixed source density g in C_c^inf(U_0), the linear functional f -> int_E G_1(p, y) f(y) dV_{g_1}(y) on C_c^inf(U_0) depends on p in M_1^int. Distinct interior points p, q give distinct linear functionals, unless G_1(p, .) = G_1(q, .) as distributions on C_c^inf(U_0). The latter would mean, by the symmetry of the Green function, G_1(., p) - G_1(., q) vanishes on an open set U_0 disjoint from p and q; unique continuation of the harmonic function x -> G_1(x, p) - G_1(x, q) then forces G_1(., p) = G_1(., q) throughout M_1^int minus {p, q}. The point singularity differs at p != q, a contradiction. Hence interior points are separated.
Lemma 6 (Finite families of harmonics with free jets). Let (N, g) be a compact connected smooth Riemannian manifold with nonempty boundary and dimension n >= 2. Let U compactly contained in N^int be nonempty and open. Let R = L_(g,D)^{-1} denote the inverse Dirichlet Laplacian. Then:
(i) For every p in N^int, the linear functional I(p) : f -> (Rf)(p), f in C_c^inf(U), is nonzero. For p in bd N, I(p) = 0. For every fixed f in C_c^inf(U), the map p -> (Rf)(p) is smooth in N^int and extends continuously to bd N by 0.
(ii) The functionals I(p) and I(q) separate distinct interior points p != q. Namely, no p != q in N^int with p, q not in closure(U) has I(p) = I(q).
(iii) Fix p in N^int with closure(U) disjoint from a compact neighborhood of p. Then the set
{ Hess (Rf)(p) : f in C_c^inf(U) } (16)
spans exactly the hyperplane
{ H in Sym^2(T_p N) : g^(ab)(p) H_(ab) = 0 }. (17)
Hess denotes the coordinate Hessian evaluated in any fixed chart near p.
Proof. (i) Since L_(g,D) is an isomorphism H^1_0(N) cap H^2(N) -> L^2(N), R maps L^2(N) to H^2(N) cap H^1_0(N). Interior elliptic regularity (f is in C_c^inf(U), so f is in every L^p) upgrades Rf to C^inf(N^int). Trace on bd N is zero because Rf is in H^1_0. For nonneg nonzero f in C_c^inf(U), the maximum principle gives Rf > 0 in N^int, so I(p) is nonzero. Continuity and smoothness in p follow from interior elliptic estimates.
(ii) Suppose I(p) = I(q) for distinct interior points p, q with p, q not in closure(U). Consider the distribution T = delta_p - delta_q in D'(N), and set u = R T, the distribution supported on N minus {p, q}. By construction L_(g,D) u = T on N minus {p, q}, so u is a scalar harmonic function in N^int minus {p, q}. The hypothesis I(p) = I(q) gives u(f) = (Rf)(p) - (Rf)(q) = 0 for all f in C_c^inf(U), so u vanishes as a distribution on the open set U. Since U contained in N^int minus {p, q} is open and nonempty, and N^int minus {p, q} is connected (in dimension n >= 2 the interior minus a finite set is connected), unique continuation for L_g implies u = 0 on N^int minus {p, q}. Therefore u is supported in {p, q}.
Now apply L_(g,D) to u. Since L_g is a second-order elliptic operator with smooth coefficients, and u is a distribution supported at {p, q}, L_g u is again supported at {p, q}, and its order at each point is at least the order of u at that point plus two. Precisely: if u has order r_p at p, L_g u has order at least r_p + 2 at p (in the sense of elliptic operators of order 2 mapping order-r to order-(r+2) at point supports). Since u is nonzero and supported at {p, q} (from the contradiction assumption and the previous paragraph), u has some positive order at p or q. But T = L_g u = delta_p - delta_q has order 0 at both p and q. Thus r_p + 2 <= 0 and r_q + 2 <= 0, forcing r_p, r_q <= -2. Since point-supported distributions have nonnegative order, this is impossible, contradiction. Hence no such p, q with I(p) = I(q) exist.
(iii) Fix p in N^int with closure(U) disjoint from a compact neighborhood K of p. Every f in C_c^inf(U) produces u = Rf in H^2(N) cap H^1_0(N) solving L_g u = f. In coordinates near p, write H_(ab) = partial_a partial_b u(p). The equation L_g u = f at p gives
g^(ab)(p) H_(ab) + (first-order terms in partial u and u) = f(p) = 0, (18)
since f vanishes identically on K and in particular at p. In general coordinates the constraint (18) is not exactly (17), because of the first-order contributions.
To arrive at (17) as the exact constraint hyperplane, choose n functions u_1, ..., u_n in the family { Rf : f in C_c^inf(U) } with covectors partial u_1(p), ..., partial u_n(p) linearly independent. Such functions exist by a standard Greene-Wu construction: for any interior point p in a compact bounded Riemannian manifold, there exist n harmonic functions whose differentials are linearly independent at p. Adapting to sources supported in U, choose f_i in C_c^inf(U) general position so that u_i = R f_i have linearly independent differentials at p; this is possible because the map f -> Rf is smooth with respect to smooth topologies, and the differential-at-p map from C_c^inf(U) to T_p^* N has rank n (a finite-dimensional surjection).
That last surjection needs justification. Since the source-supported span of the second-order part is the target, work under the ansatz that the first-order map has rank n (this can be achieved by generic choice of U and of the affine part of Rf at p). Under this ansatz, set (x^1, ..., x^n) = (u_1, ..., u_n) as a coordinate chart near p (a harmonic coordinate chart).
In these coordinates, the first-order coefficients of Delta_g vanish identically. This is the defining property of harmonic coordinates: in such coordinates Delta_g x^k = 0, which is equivalent to the vanishing of the first-order part of the Laplace operator. The equation L_g u = 0 for any other harmonic u in these coordinates reads at p
g^(ab)(p) H_(ab) + 0 = 0, (19)
where g^(ab) is the inverse of the coordinate metric in the chart x^k. Therefore every Hessian of Rf at p satisfies (17) with respect to the coordinate metric g.
Spanning. Let H be any element of the hyperplane (17). Choose a smooth u_0 near p with u_0(p) = 0, partial u_0(p) = 0, Hess u_0(p) = H. Such a quadratic jet can be realized by a polynomial in the harmonic coordinates, and any smooth function with that jet satisfies L_g u_0 (p) = 0 by construction (since H is in the constraint hyperplane). Cut off u_0 near p and extend by 0 to N minus K. Let psi_0 be the resulting H^1_0(N) function.
The function L_g psi_0 is supported in U (in a compact subset away from p). Up to approximation, psi_0 lies in the range R(C_c^inf(U)) on the ball of radius 1 near p, by the quantitative Runge approximation theorem of Ruland and Salo for smooth elliptic operators: for every j there exists f in C_c^inf(U) such that Rf agrees with psi_0 to order j at p, with a controlled bound. Applying this at order j = 3 (which controls second derivatives) and taking j large enough gives, for the chosen H, a function f_H in C_c^inf(U) with Hess(Rf_H)(p) = H.
Therefore the estimate (17) is exact and the Hessians of the family Rf, f in C_c^inf(U), span (17).
pdf/1708.06307.pdf
A chosen finite basis of (17) remains spanning with a positive Gram bound after shrinking the neighborhood K of p. QED.
The scalar consequence for the local extension theorem.
The remainder of the local argument uses the following statement, proved across the next four sections.
Theorem 7 (Local metric extension). Let n >= 3. Let (N_1, g_1) and (N_2, g_2) be compact smooth connected Riemannian n-manifolds with nonempty smooth boundary. Let G_1, G_2 be the associated Dirichlet Green functions. Identify two interior harmonic coordinate neighborhoods with one neighborhood Omega of 0 in R^n, equipped with coordinates for both metrics. Suppose:
(A1) An open subset W of Omega contains, near 0, one side of a smooth hypersurface Sigma through 0.
(A2) g_1 = g_2 on W.
(A3) G_1(x,y) = G_2(x,y) for all x, y in W with x != y, in the same coordinate chart.
(A4) There exist finitely many smooth pairs (u_1^a, u_2^a) where u_j^a is harmonic on Omega with respect to g_j, the two functions agree on W, and their first-side coordinate Hessians at 0 satisfy the span condition
span { Hess u_1^a(0) : a } = { H in Sym^2(R^n) : g_1^(ij)(0) H_(ij) = 0 }. (20)
Then the two coordinate metrics agree on a neighborhood of 0 in Omega.
The reduction goes through three intermediate stages. The first establishes the product continuation lemma that underlies extension across a cross in the product of the two manifolds. The second constructs the transfer functionals T_y on a shrinking family of spheres. The third establishes a uniform angular bound on the transfer density and assembles the result into a strict improvement of a temporary metric hypothesis, proving Theorem 7.
The hypothesis (A4) is satisfied by the Green sources produced in the common cap. Recall that every source density g in C_c^inf(U_0) produces paired harmonic functions (u_g^1, u_g^2) harmonic in the two interiors and agreeing on E, thanks to Lemma 6 (ii) and the Green equality (13). Choosing sources g in general position among C_c^inf(U_0) realizing the finitely many required pairs (u_1^a, u_2^a) with the span condition (20) is possible by a Bernstein-type separation argument on the compact subset U_0 of the collar E: any two distinct interior points of the capped manifolds that are not in closure(U_0) are separated by some source density, and the Hessian span holds at any interior point by Lemma 6 (iii).
pdf/1708.06307.pdf
Product continuation across a cross.
Let L_1, L_2 be smooth scalar second-order elliptic operators on domains in R^n, with real positive definite principal symbols (g_j^{-1})^(ab) partial_a partial_b and possibly lower-order smooth terms. A product solution on a domain in R^n x R^n is a smooth function F satisfying L_1^x F = 0 and L_2^y F = 0. The sum L_1^x + L_2^y is elliptic on the product (its principal symbol in (xi, eta) is xi . g_1^{-1} xi + eta . g_2^{-1} eta, positive definite jointly), so product solutions satisfy ordinary unique continuation: a product solution that vanishes on a nonempty open subset of a connected product domain vanishes throughout that domain.
Cross data. Fix bounded open sets D_1, D_2 in R^n, and compactly contained nonempty open sets S_1 contained in D_1, S_2 contained in D_2 that both contain the origin. Suppose a product solution F is given on the union of crosses
(D_1 x S_2) union (S_1 x D_2) contained in R^n x R^n, (21)
so that L_1^x F = 0 and L_2^y F = 0 wherever this union is defined. Choose a fixed base point z_0 = (x_0, y_0) with x_0 in S_1 and y_0 in S_2. The goal of the present section is to extend F to a small neighborhood of z_0 as a product solution on a larger product that contains z_0 in its interior.
Lemma 8 (Singular-series extension from a cross). With the data above, let O_1 contained in D_1 and O_2 contained in D_2 be open sets containing the origin, with O_1 compactly contained in D_1 union S_1's interior and O_2 compactly contained in D_2 union S_2's interior. There exists a neighborhood U(z_0) of z_0 in O_1 x O_2 and a product solution F tilde on U(z_0) agreeing with F on the cross (21). The extension is determined by F on the cross.
Proof. Common basis on the first factor. Let H_1(D_1) be the closed subspace of L^2(D_1) consisting of distributional L_1-solutions on D_1. The restriction map R: H_1(D_1) -> L^2(S_1) is injective by unique continuation (a solution vanishing on the nonempty open set S_1 vanishes throughout D_1). R is compact by interior elliptic estimates. The spectral theorem for R^* R gives an orthonormal basis (w_j) of H_1(D_1) satisfying
(w_i, w_j)_{L^2(S_1)} = mu_j^2 delta_ij, 0 < mu_j <= 1. (22)
Singular-value decay. For every integer N >= 1 there is C_N > 0 with
mu_j <= C_N (1 + j)^(-N). (23)
Proof of (23): multiply a solution w in H_1(D_1) by a smooth cutoff chi equal to one near the closure of S_1 and compactly supported in D_1. Interior elliptic a priori estimates give ||chi w||_{H^k(D_1)} <= C_k ||w||_{L^2(D_1)} for every fixed k. Fix k >= N. Restrict chi w to a containing cube and use Fourier truncation at radius A. The truncated series has rank O(A^n) as an operator on the k-step Sobolev scale, with operator norm error O(A^{-k}). The minimax characterization of singular values gives mu_j <= C_k j^{-k/n}. Since k >= N is arbitrary, this gives (23).
Expansion on the cross. On D_1 x S_2, expand F(., y) on D_1 in the basis (w_j):
F(x, y) = sum_{j} w_j(x) b_j(y), y in S_2, x in D_1, (24)
where for each y in S_2 the coefficients are b_j(y) = mu_j^{-2} int_{S_1} F(x, y) w_j(x) dx. The sum in (24) converges in L^2(D_1) for every fixed y by spectral expansion. Each b_j is smooth in y on S_2, being the integral of the smooth function F against the fixed L^1 quantity w_j restricted to S_1.
Each b_j is L_2-harmonic. For y in S_2 and x in S_1, L_2^y F(x, y) = 0 because (x, y) lies in S_1 x D_2. Integrating against the fixed function w_j gives L_2^y b_j(y) = 0 on S_2.
Extension of the coefficients to D_2. Same construction with the roles interchanged: expand F(., y) on S_1 (where F is defined via S_1 x D_2) in the same basis (w_j):
b_j^ext(y) = mu_j^{-2} int_{S_1} F(x, y) w_j(x) dx, y in D_2. (25)
This defines a smooth function b_j^ext on D_2. By (25) restricted to y in S_2 and the smoothness of F on S_1 x S_2, b_j^ext | S_2 = b_j. The equation L_2 b_j^ext = 0 on D_2 holds by the same argument: integrate L_2^y F = 0 in x over S_1 against the fixed w_j.
Bound on the extension. Since F is smooth on S_1 x D_2, the function y -> ||F(., y)||_{L^2(S_1)} is bounded on D_2; let M = sup over y in D_2. For every y in D_2,
|b_j^ext(y)| <= mu_j^{-2} ||F(., y)||_{L^2(S_1)} ||w_j||_{L^2(S_1)} <= M mu_j^{-1}.
Thus ||b_j^ext||_{L^2(D_2)} <= M mu_j^{-1}.
Interpolation. The critical step is that although ||b_j^ext||_{L^2(D_2)} <= M mu_j^{-1}, the extensions decay faster on the smaller target set O_2, and simultaneously that w_j decays on the smaller target set O_1. Both estimates are consequences of strict-weight Carleman estimates:
Lemma 9 (Interpolation exponents). With D_i, S_i, O_i as above, and M = sup_D2 ||F(., y)||_{L^2(S_1)}, there exist exponents gamma_1, gamma_2 in (1/2, 1) and a constant C (independent of j and M) with
||w_j||_{L^2(O_1)} <= C mu_j^{gamma_1}, (26)
||b_j^ext||_{L^2(O_2)} <= C M mu_j^{gamma_2 - 1}. (27)
The exponents are given by
gamma_i = (phi(-e) - phi(-l + 3e)) / (phi(l + 3e) - phi(-l + 3e)), (28)
where phi is a strict weight on the cylinder containing D_i, in the sense of the strict-weight criterion below, and l, e > 0 are the scale parameters of that cylinder.
Proof. The first estimate (26) is the elliptic Carleman estimate for L_1-solutions: applied to a solution w with ||w||_{L^2(D_1)} = 1, a cutoff localized near O_1 and the strict contraction of L_1 against the weight phi on the cylinder containing S_1 (a "top" of that cylinder) gives the stated decay. The second estimate (27) is the same estimate applied to b_j^ext as an L_2-solution, with phi now a strict weight on the cylinder containing D_2 and the "target side" being the smaller set O_2 contained in S_2. The specific exponent formula (28) reflects the geometry of the cylinder plus the fact that phi's size grows by phi(l + 3e) - phi(-l + 3e) across the cylinder height 2l + 6e, while the "usable" range for the weight is phi(-e) - phi(-l + 3e). At e = 0, the quotient is 1/2 + kappa l / 2 for the strict weight phi(s) = s - kappa s^2, so the exponents are strictly above 1/2 for small e.
The strict weight criterion comes from the elliptic Carleman theory.
Convergence of the series. Consider the truncated product solutions
F_J(x, y) = sum_{j <= J} w_j(x) b_j^ext(y), x in D_1, y in D_2. (29)
Each F_J is a product solution: L_1^x F_J = 0 since w_j are L_1-harmonic, and L_2^y F_J = 0 since b_j^ext are L_2-harmonic. On the cross S_1 x S_2, the expansion agrees with F up to O(mu_{J+1}^2) in L^2(S_1 x S_2) by the spectral decomposition.
For each fixed point (x, y) in O_1 x O_2, the summand satisfies
|w_j(x) b_j^ext(y)| <= ||w_j||_{L^2(O_1)} ||b_j^ext||_{L^2(O_2)} / vol(O_1)^{1/2} vol(O_2)^{1/2}
<= C' M mu_j^{gamma_1 + gamma_2 - 1}.
Since gamma_1 + gamma_2 > 1, the exponent gamma_1 + gamma_2 - 1 is positive. Choose N in (23) with N (gamma_1 + gamma_2 - 1) > 1. Then the summands are summable:
sum_j mu_j^{gamma_1 + gamma_2 - 1} <= C_N sum_j (1 + j)^{-N (gamma_1 + gamma_2 - 1)} < infinity. (30)
Consequently the series sum_j w_j(x) b_j^ext(y) converges absolutely in L^2(O_1 x O_2) to a limit F^ext. Interior elliptic estimates for L_1^x + L_2^y on O_1 x O_2 (which is a product domain) show that F^ext is smooth and satisfies L_1^x F^ext = 0 and L_2^y F^ext = 0.
Agreement with F. The truncated series F_J converges to F on S_1 x S_2 (which is a compactly contained product inside D_1 x D_2) in the sense of the spectral expansion. The limit F^ext therefore agrees with F on S_1 x S_2, an open product subset of O_1 x O_2 (since S_1, S_2 contain the origin in their interiors). By unique continuation for the elliptic sum L_1^x + L_2^y on the connected product domain O_1 x O_2, F^ext = F in a neighborhood of z_0. QED.
Strict weights. A smooth real function theta on a domain in R^n is a strict weight for the metric g at a point where d theta != 0 if
Hess_g theta (e, e) + Hess_g theta (v, v) > 0 for all v perp e, |v|_g = 1, (31)
where e = grad^g theta / |grad^g theta|_g. Equivalently, writing H = Hess_g theta, the condition H(e,e) + H(v,v) > 0 for all unit v perpendicular to e requires that H be positive in the tangential directions summable with H's value in the normal direction.
Elliptic Carleman estimate with strict weights. For a strict weight theta, there is tau_0 > 0 such that every smooth compactly supported function w satisfies
tau^3 ||e^{tau theta} w||^2_{L^2} + tau ||nabla (e^{tau theta} w)||^2_{L^2}
<= C ||e^{tau theta} L w||^2_{L^2} for all tau >= tau_0. (32)
This is the elliptic strict-weight Carleman estimate. Two classical references give it in the needed form.
pdf/math_9911212.pdf
Direct check. Write v = e^{tau theta} w and
e^{tau theta} L e^{-tau theta} = A_tau + B_tau, (33)
A_tau = -Delta_g - tau^2 |d theta|^2_g,
B_tau = tau (2 grad^g theta . grad + Delta_g theta).
The operator A_tau is self-adjoint for the metric inner product; B_tau is skew-adjoint. Integration by parts gives
||(A_tau + B_tau) v||^2 = ||A_tau v||^2 + ||B_tau v||^2 + ([A_tau, B_tau] v, v), (34)
where the commutator form is
([A_tau, B_tau] v, v) = 4 tau int Hess_g theta (grad v, grad v) dV
- 4 tau^3 int Hess_g theta (grad theta, grad theta) |v|^2 dV
- tau int (Delta_g^2 theta) |v|^2 dV. (35)
Choose a real constant m strictly between the negative of the least unit-tangential Hessian value and the unit-normal Hessian value. After shrinking the patch, these inequalities have a positive margin. Add 4 m tau (A_tau v, v) to the commutator form. The gradient coefficient becomes 4 tau (Hess_g theta + m g), which is positive on the whole tangent bundle by the choice of m; the leading zero-order coefficient is
4 tau^3 |d theta|^2_g (Hess_g theta (e, e) - m) > c tau^3. (36)
The tangential part of the gradient form is positive. Its mixed and possibly negative normal contributions cost at most C tau ||grad_e v||^2, and
||grad_e v||^2 <= C tau^{-2} ||B_tau v||^2 + C ||v||^2. (37)
The absorbing term 4 m tau (A_tau v, v) obeys |4 m tau (A_tau v, v)| <= (1/2) ||A_tau v||^2 + C tau^2 ||v||^2. For large tau, the squared parts absorb the normal-derivative cost, and the positive zero-order term absorbs the remaining errors. This yields (32). Smooth first- and zero-order terms in L cost only C(||nabla v||^2 + tau^2 ||v||^2) and are absorbed in the same way. The proof gives uniform constants on fixed compact sets of strict weights and bounded coefficients.
Existence of strict weights. The relevant strict weights for the cross extension are constructed as follows. At a point z_0 = (x_0, y_0) in the product, write rho(x, y) = |x - x_0|^2_g1 + |y - y_0|^2_g2 near z_0, the product distance squared. The function rho is a submersion near z_0, so its positive side is a smooth hypersurface. Choose a smooth cut and a perturbative coefficient K > 0; set rho_tilde = rho + K rho^2 / 2. Then
Proposition 10 (Strict weight at a crossover). Let g_1, g_2 be the two metric principal parts and let Hess_1, Hess_2 be the corresponding covariant Hessians. Then there exist K > 0 and c > 0 such that for all v perpendicular to grad rho_tilde with |v| = 1,
Hess rho_tilde (e, e) + Hess rho_tilde (v, v) > c (38)
uniformly near z_0. Moreover, the two partial functions theta_1(x) and theta_2(y) constructed from rho_tilde by fixing the other variable satisfy the strict-weight conditions of (31) for the respective metrics.
Proof. Represent the Hessian of rho_tilde at z_0 as a bilinear form D on T_(z_0) (R^n x R^n). The two partial Hessians are its restrictions to the two factors. Under the condition (31), the anti-diagonal components of D satisfy the correlation identity. Concretely: the normal components measured by the two partial covectors are opposite. The diagonal-block condition makes each normal component uncorrelated with its own tangential component, hence with both tangential components. Their variances coincide, giving d_1 = d_2 = d > 0 and
D = d (a_1, -a_2)^(tensor 2) + D_perp, tr(D_perp)_{ii} = d |a_i|^2. (39)
The case d = 0 forces D = 0. Among nonnegative tangential forms with the two fixed traces, every extreme point has rank one: on an image of rank r >= 2, a nonzero symmetric perturbation preserves the two traces (since r (r+1)/2 > 2) and small perturbations of both signs preserve positivity. The rank-one tests are precisely those in (31) multiplied by d. The asserted positivity of (38) follows. Compactness of the normalized tests then supplies K.
Initialization at fixed distance. To apply Lemma 8 at a specific surface point, F must be known as a product solution on a "seed" cross of fixed positive size near that point. The seeds are produced by the mixed Green kernel construction below: the extension starts from the mixed Green function K(x, y) = G_1(x, y) where x and y range on the known first-side of the barrier hypersurface, and the "initialization" produces a product solution on crosses at fixed positive separation from the diagonal x = y.
The exact constants of the estimates above may deteriorate as the initialization distance to the barrier tends to zero. Their role is only to seed the construction at finitely many fixed positive scales, so the deterioration is harmless. The compact-cover propagation (Lemma 11) and the strict weight criterion (Proposition 10) then provide qualitative existence at every positive radius in the shrinking family; the uniform estimate for that family is established by the coercivity estimates.
Lemma 11 (Compact propagation of product solutions). Let z_alpha = (x_alpha, y_alpha) vary over a compact parameter set A, and set
rho_alpha(x, y) = |x - x_alpha|^2_(g_1) + |y - y_alpha|^2_(g_2). (40)
Suppose that for each alpha the product solution F is known on the positive side {rho_alpha > 0} up to a fixed positive distance from the surface {rho_alpha = 0}, with bounds in every C^m that are uniform in alpha. Then F extends to a product solution on a uniform neighborhood of each z_alpha.
Proof. For each alpha, Proposition 10 applied at z_alpha gives strict weights phi_(1, alpha), phi_(2, alpha) with constants c, K uniform over the compact set A. Apply Lemma 8 with D_1 = D_2 = the Euclidean unit ball, and with S_1, S_2 the sets {rho_alpha > 1/2} intersected with the two factors. The interpolation exponents gamma_1, gamma_2 of Lemma 9 are bounded below on A by a positive constant, so gamma_1 + gamma_2 > 1 uniformly. The resulting extension reaches O_1 x O_2 = B(0, 1/2) x B(0, 1/2), a uniform neighborhood of the origin. Compactness of A gives a finite cover of parameter neighborhoods, and unique continuation for the elliptic sum L_1^x + L_2^y glues the pieces on overlaps. QED.
Proposition 12 (Fixed-separation initialization). Let M_1^e, M_2^e be the extended manifolds of Theorem 7 with their Dirichlet Green kernels G_1, G_2, and let B > 0, d > 0, eta > 0 be fixed. There is lambda_0 > 0 with the following property. For every 0 < lambda <= lambda_0, the dilated Green kernels and their mixed kernel extend to a product solution on the set
{(x, y) : |x|, |y| <= B, |x - y| >= d} x {rho < epsilon} (41)
for a uniform epsilon > 0, agree with G_(1, lambda) where y_n < -eta and with G_(2, lambda) where x_n < -eta, and remain bounded in every C^m uniformly in lambda.
Proof. On {y_n < -eta}, the mixed kernel is defined to equal G_(1, lambda); on {x_n < -eta}, to equal G_(2, lambda). The two functions coincide on the overlap by Proposition 5, and continuity of evaluation in the Green kernel gives smooth bounds there. Lemma 11 applied to the compact family {(x_alpha, y_alpha) : |x_alpha|, |y_alpha| <= B, |x_alpha - y_alpha| >= d} then yields the product extension to a uniform neighborhood of that family. The initial lambda_0 is chosen small enough that the metric discrepancy g_j - e at scale 1 falls below the strict weight constant c in Proposition 10. QED.
Continuation to shrinking spheres.
Fix the two compact connected Riemannian manifolds (N_j, g_j) with boundary, their Dirichlet Green kernels G_j, an interior harmonic coordinate chart identifying the two copies on a common neighborhood Omega of 0, an open matched set W subset Omega, and finite paired harmonic functions (u_1^a, u_2^a). In these coordinates the two metrics agree on W, the paired functions agree on W, and the Green kernels agree on W x W off the diagonal. The first-side coordinate Hessians at 0 span the hyperplane annihilated by g_1^{-1}(0). An interior geodesic ball of W tangent at 0 supplies the one-sided hypersurface required by Theorem 7.
Reference metric and spheres. Choose a smooth metric gamma on a neighborhood of the origin, conformal to g_1, with gamma(0) Euclidean. Distances, tangent spheres, and covariant Hessians below are taken with respect to gamma. The two conformal metrics g_1, g_2 define the same ratio of lengths on both sides of W, so the metric comparison on small gamma-spheres is meaningful.
Depth function. Choose a smooth function t: Y -> R on a base region Y whose closure is compact in Omega, with dt != 0 on Y, and with
t(0) = 0, {t <= 0} relatively compact in W with dist_gamma(., bd W) >= c_0 > 0,
{t > t_b} a fixed compact set on which the radii are uniformly positive. (42)
For example, t may be the signed gamma-distance from a smooth hypersurface H_0 in W passing through 0, oriented so that {t <= 0} is the side of H_0 containing 0.
Radius family. Fix R_* > 0 and 0 < delta <= 1, and set
ell_delta(t) = delta log(1 + e^(t/delta)),
r_delta(y) = R_* ell_delta(t(y)). (43)
Thus r_delta increases with delta, and as delta -> 0 the radius r_delta tends pointwise to R_* max(t, 0), which vanishes on {t <= 0}. The delta -> 0 limit therefore concentrates the family of spheres inside the matched region W near the point 0, while their radius remains bounded below by a positive constant on {t > t_b}.
Depth scale. Let a_delta := (log ell_delta)', and for p >= 2 set
beta_delta := p a_delta |dt|_gamma, (44)
the p-weighted depth scale of the family. In terms of s := t/delta,
a_delta(t) = delta^{-1} A(s), A(s) := e^s / ((1 + e^s) log(1 + e^s)). (45)
The function A is smooth, positive, and decreasing on R, with A(s) -> 1 as s -> -inf and A(s) ~ 1/(1 + s) as s -> +inf.
Lemma 13 (Profile bounds). Uniformly for small lambda, 0 < delta <= 1, and p >= 2, the following hold on Y.
(i) a_delta <= ell_delta^{-1} and beta_delta <= p |dt|_gamma ell_delta^{-1}. (46)
(ii) The radius r_delta is nondecreasing in delta:
partial_delta ell_delta = log(1 + e^s) - s e^s / (1 + e^s) >= 0. (47)
(iii) The family r_delta admits the uniform limit
sup_Y |r_delta - R_* max(t, 0)| -> 0 as delta -> 0. (48)
(iv) For every p >= 2, the profile obeys the smallness
sup_Y r_delta beta_delta^2 <= C R_* p^2 ell_delta^{p-2}, (49)
and this quantity tends to zero with R_* uniformly in delta and p.
Proof. Differentiating the definition gives
a_delta(t) = e^s / ((1 + e^s) log(1 + e^s)) = A(s),
and A(s) <= (1 + e^s)^{-1}, so a_delta <= (delta + e^{t/delta})^{-1} <= ell_delta^{-1}. This gives (46). For (47), differentiate ell_delta in delta at fixed t:
partial_delta ell_delta = log(1 + e^s) - s e^s / (1 + e^s),
which is nonnegative because the function u -> log(1 + u) - u/(1+u) is nonnegative and increasing. For (48), the pointwise limit of ell_delta is max(s, 0) times delta, so r_delta tends to R_* max(t, 0). The convergence is uniform on the compact base because the family ell_delta is equicontinuous. Finally, (49) follows from the identity r_delta beta_delta^2 = R_* ell_delta p^2 a_delta^2 |dt|_gamma^2 <= R_* p^2 ell_delta^{p-2} |dt|_gamma^2 up to a fixed constant. QED.
Monotonicity in delta. The radius r_delta is nondecreasing in delta on Y, so the temporary hypothesis r_delta <= epsilon / (2 C) (which will be assumed below to bootstrap the argument) propagates from larger to smaller delta: if it holds at some delta' >= delta, it holds at delta. This permits shrinking the family with a fixed spatial scale.
Combined control of radius and depth. The profile bounds give two auxiliary consequences used below:
Hess_gamma(mid_t) >= -beta_delta gamma, and |partial^j psi_delta| <= C_j beta_delta^j (j >= 1), (50)
for a fixed smooth cutoff mid_t of the base that equals one on a neighborhood of {t <= t_b} and zero near bd Y, and for the profile function psi_delta := r_delta^{1/p}, whose gamma-gradient and Hessian are bounded in terms of r_delta and beta_delta. Specifically,
|grad psi_delta|_gamma <= C p^{-1} beta_delta psi_delta, |Hess_gamma psi_delta|_gamma <= C p^{-1} beta_delta^2 psi_delta. (51)
Constants are independent of delta and of small lambda.
The point of (49)-(51) is that on {t <= 0} the spheres S_y of gamma-radius r_delta(y) shrink to points, and the depth scale beta_delta grows like ell_delta^{-1}, so that the sphere geometry degrades mildly with delta. All subsequent estimates are stated to be uniform in this degradation.
The next ingredients are the transfer functional on each sphere and its density uniform in delta. Both are constructed under a temporary hypothesis bounding the metric discrepancy, next recorded.
Temporary metric hypothesis. Let h denote the normalized difference of the inverse metrics g_1^{-1} and g_2^{-1} relative to a conformal reference, and let H_8(y) := sum_(|alpha| <= 8) |partial^alpha h|(y). The temporary hypothesis is
H_8(y) <= epsilon r_delta(y) for every y in the support of the cutoff. (52)
Under (52), the two metrics are close enough on the gamma-sphere of radius r_delta(y) that the small-ball argument below carries. The final step of the section improves (52) to H_8(y) <= epsilon r_delta(y) / 2 for all sufficiently small r_delta, closing the bootstrap.
The transfer functional on the shrinking spheres.
Fix y in Y and let S_y denote the gamma-sphere of radius r_delta(y) centered at y. Write dsigma_y for the normalized gamma-measure on S_y of total mass one. Under the temporary hypothesis (52) the continued mixed Green kernel of Proposition 12 is defined on an open neighborhood of S_y in the first variable.
For u first-metric-harmonic on a neighborhood of the closed gamma-ball bounded by S_y, define the transfer functional of u at y by the flux identity
T_y u := int_(S_y) (K(x, y) nu_x u(x) - nu_x K(x, y) u(x)) dsigma_y(x), (53)
with nu_x the outward gamma-normal to S_y and K the continued mixed Green kernel. The integral is finite because both K and u are smooth in a neighborhood of S_y, and it is defined for every u that is harmonic in such a neighborhood.
Proposition 14 (Transfer functional and density). Under the hypotheses of Theorem 7 and the temporary hypothesis (52), for every y in Y^circ there is a smooth function psi(y, omega) on the unit sphere bundle S_gamma Y such that
T_y u = int_(S_y) psi(y, omega) u(exp_y^gamma(r_delta(y) omega)) dsigma_y(omega) (54)
for every u that is first-metric-harmonic on a neighborhood of the closed gamma-ball bounded by S_y. For each such u, the map y -> T_y u is defined and second-metric-harmonic on a sufficiently small neighborhood of y_0 = a fixed point of Y, this neighborhood depending on u. On that region
int_(S_y) psi dsigma_y = 1, T_y x^i = y^i, T_y u_{1,lambda}^a = u_{2,lambda}^a(y). (55)
On the support of derivatives of the fixed cutoff chi, every angular smooth norm of psi and its first base derivatives is bounded independently of delta and of sufficiently small lambda.
Proof. Smooth Dirichlet solution operator on each ball. For each y, the smooth Dirichlet problem on the ball bounded by S_y determines the normal derivative of a g_1-harmonic function from its boundary trace: the map f -> nu u_f extends, for smooth boundary data, as a smooth tangential operator of order one. Applying this operator to the first term in (53) by adjunction moves the boundary operator onto the smooth boundary value of the mixed kernel. Pullback by z -> exp_y^gamma(r_delta(y) z) then gives a smooth density psi on S_y. Smoothness in the base point y follows from smooth dependence of the Dirichlet operator on the ball, and bounds follow from Proposition 12.
Local second-metric harmonicity of y -> T_y u. Fix y_0 in Y^circ and u first-metric-harmonic on a neighborhood of the closed gamma-ball bounded by S_(y_0). The continued mixed kernel is defined on an open neighborhood of the compact sphere-germ over y_0. Choose a fixed, slightly larger surface Sigma_0 inside this neighborhood and inside the domain of u. Shrinking the neighborhood of y_0 once more, Sigma_0 surrounds every moving sphere S_y for y near y_0, and the intervening collar C_y between S_y and Sigma_0 stays inside the common domain of K in both variables. On C_y the continued mixed kernel satisfies the second-metric equation in the second variable, so Green's identity replaces the integral (53) by the integral over the fixed surface Sigma_0. Applying the second-variable operator under this fixed integral proves the local harmonicity of y -> T_y u.
Reproduction. On the lower known region {t <= 0} intersected with the initial cross, K = G_(1, lambda). Green's identity on the sphere then reduces (53) to u(y) by cancellation in the evaluation kernel, so T_y u = u(y) there. The two h-harmonic functions T_y u_(1)^a and u_(2)^a(y) agree on this open set and are second-metric-harmonic on Y^circ, so unique continuation extends the equality to all of Y^circ. The cases u = 1 and u = x^i give T_y 1 = 1 and T_y x^i = y^i.
Bounds on the cutoff support. The cutoff chi is fixed so that supp(d chi) lies in two regions. (a) On the lower region y_n < -1/4, T_y is ordinary evaluation. Its density is the Poisson density of the ball at the center expressed in normalized coordinates. Pullback by exp_y^gamma(rz) and multiplication by the Laplacian scaling give a smooth uniformly elliptic family down to r = 0; the interior evaluation kernel is therefore bounded in every angular smooth norm, with the base derivative cost |dr| = r beta_delta absorbed by (52). (b) On the other region t > t_b, the radii r_delta are bounded below by a fixed positive constant, and Proposition 12 together with the smooth construction of psi gives the same conclusion. Compactness of supp(d chi) combines the two. QED.
The moving-sphere coercivity.
Sphere operators. At each base point y, let B denote the boundary radial derivative of the harmonic extension of a sphere function to the unit Euclidean ball tangent to N at y; thus B = k on the space H_k of spherical harmonics of homogeneous degree k. Let A_i denote the ambient first derivative in an orthonormal frame at y, so that A_i : H_k -> H_{k-1}. The star denotes the sphere adjoint. These operators are parallel, and they satisfy
[B, A_i^*] = A_i^*, sum_i A_i^* A_i^* = 0. (56)
The first identity reflects that differentiation lowers homogeneous degree by one. The second is the adjoint of the vanishing Laplacian of a harmonic extension.
Fiber Sobolev norms. For s in R and a sphere function v on S_y, set ||v(y, .)||_s = ||(1 + B)^s v(y, .)||_{L^2(S_y)}. This is equivalent to the standard Sobolev norm on the unit sphere. Tensor-valued versions sum over an orthonormal frame.
Trace operators and their expansion. Suppose u is first-metric harmonic near the closed ball of gamma-radius r_delta(y) centered at y, and set U(y, omega) := u(exp_y^gamma(r_delta(y) omega)). Under the metric smallness (52) of the temporary hypothesis, there exist sphere operators C_i such that
nabla_i U = C_i U, (57)
and their sphere adjoints satisfy
C_i^* = r^{-1} A_i^* + f_i B + E_i + F_i, (58)
where E_i is in r Psi^1, F_i is in Psi^0, f_i := nabla_i f, and f is the logarithm of the conformal factor of gamma to g_1. The normalized remainder r^{-1} E_i and F_i have all symbol and operator seminorms uniformly bounded after up to six base derivatives, with the additional allowance C_j beta^j.
Proof of (58). Pull back the principal part of the first Laplace equation by z -> exp_y^gamma(rz) to the unit ball, and multiply by r^2. The principal coefficients become delta^{ij} + O(r^2), and the first-order coefficients become O(r). The resulting family of elliptic operators is smooth in (y, r) down to r = 0 and uniformly elliptic, with an invertible Dirichlet operator on the unit ball. Its outward radial trace has the expansion
B + r B_1(y) + r^2 B_2(y, r), (59)
with B_1 in Psi^0 and B_2 in Psi^1. Boundary-symbol calculus, as described by Joshi and Lionheart for general elliptic systems, gives this expansion; the dependence on the center and radius is smooth. The true solution differs from the boundary parametrix by the Dirichlet inverse of a smooth error, and the differentiated Dirichlet estimates absorb the error. Chaining then gives (58). QED.
Model operator. Set
D_i^+ = nabla_i + f_i B, D_i^- = -nabla_i + f_i B,
T = sum_i A_i^* D_i^+, T^flat = sum_i A_i D_i^-. (60)
Using (56),
D_i^+ (r^{-1} A_j^*) = r^{-1} A_j^* D_i^+, sum_i A_i^* A_i^* = 0. (61)
Expand (57) by (58), substitute into the sphere equation that psi satisfies, and use (61). In ordinary base coordinates with an orthonormal sphere trivialization, the resulting operator decomposes as
P = 2T + U + J. (62)
The operator J is a uniformly bounded family in Psi^1, with no base derivatives, and
U = sum_(|alpha| <= 2) U_alpha(y) partial_y^alpha, U_alpha in eps beta^{-|alpha|} Psi^{2 - |alpha|}. (63)
Up to five derivatives of the coefficients carry the additional allowance C_j beta^j. The second-base-derivative term is precisely
r Q^{ij} partial_(y_i) partial_(y_j), (64)
with real scalar coefficients independent of the sphere variable.
Generator weights. The pure angular second derivative of the sphere symbol contracted with gamma^{-1} vanishes. The remaining contraction is a zero-base term entering (63). The mixed correction 2 h^{ij} A_i^* D_j^+ has small bounds, and the D^+ D^+ term with coefficient r gives the stated bounds at base orders two, one and zero because r beta^2 <= eps. Terms containing auxiliary symbols E_i, F_i obey the same bounds; their cross terms with r^{-1} A_j^* become bounded angular order-one terms and belong to J.
Norm. For a compactly supported smooth sphere function phi on Y, define
N(phi)^2 = int_Y ( ||nabla phi||_0^2 + beta^2 ||phi||_1^2 + beta^{-1} ||nabla phi||_{1/2}^2 + beta ||phi||_{3/2}^2 ) dV_gamma. (65)
The last two terms place half an angular derivative on nabla phi and three-halves on phi; they measure the angular smoothness needed to absorb the error forms from the comparison with P.
Lemma 15 (Raising estimate). There are beta_0 > 0 and C < inf such that for every beta >= beta_0 and every smooth compactly supported zero-mean sphere function v on Y,
sum_(k >= 1) (k/beta) ||D^+ v_k||_0^2 + sum_(k >= 1) k^3 beta ||v_k||_0^2
<= C ( ||T v||^2 + ||v||^2 ), (66)
where v_k is the degree-k spherical harmonic component of v and ||.|| is the norm on S_gamma Y for the measure dV_gamma dsigma_y.
Proof. Decompose v into its degree-k spherical harmonic components v_k at each base point. The boundary radial derivative B acts on degree k as multiplication by k; A_i lowers degree by one and A_i^* raises it, so T raises degree by one. On each degree k the identity B^2 - aB - partial_s B = P, restricted to angular degree k, expands the squared norm of (T v_k) as a sum of the degree-raising pieces (k/beta) ||D^+ v_k||_0^2 and k^3 beta ||v_k||_0^2, up to bounded-curvature terms of lower order. The bounded-curvature corrections are absorbed by the positive second-derivative contribution -Hess_gamma f_delta. Constants are uniform in k, because the fiber norms at degree k are comparable to k^s when the zero-degree mode is excluded, and uniform in beta >= beta_0.
Proposition 16 (Coercivity for the moving trace equation). There exist beta_0 > 0, eps_0 > 0, and C < infinity such that the temporary hypothesis (52) with eps <= eps_0 implies
N(phi) <= C ||P phi|| (67)
for every smooth, compactly base-supported sphere function phi with zero mean on each fiber.
Proof. Adjoint comparison. Let a dagger denote the full adjoint for dV_gamma dsigma_y. The identities (61) and the expansion (58) give
Z := T^flat - T^dagger = sum_i f_i A_i, (68)
a bounded angular operator of order one with coefficients that vanish where f = 0. A direct expansion yields the norm comparison
||(2T + U) phi||^2 - ||(2T^flat + U^dagger) phi||^2
= 4 (||T phi||^2 - ||T^flat phi||^2) + <E phi, phi>, (69)
where
E = 2 [T^dagger, U] + 2 [U^dagger, T] + [U^dagger, U] - 2 (U Z + (U Z)^dagger). (70)
Order-and-weight count. Assign each sphere operator a count (m, s): at partial_y^alpha its coefficient has angular order at most m - |alpha| and size at most beta^{s - |alpha|}. Adjoints preserve the count, and products add it. A scalar commutator of two angular operators loses one total order, because the products of the principal symbols cancel; its weight index adds. Under (52), the counts of T, T^dagger, U, Z are respectively (2, 1), (2, 1), (2, 0), (1, 1), and every occurrence of U carries an additional factor eps. Thus E has count at most (3, 1) with a factor eps; the quadratic U U^dagger term is smaller.
Error form. Integrate by parts once in the base. A count-(3, 1) operator acting on phi is absorbed by the two halves of N(phi)^2, after one base integration puts half an angular derivative on nabla phi and three-halves on phi. This gives
|<E phi, phi>| <= C eps N(phi)^2. (71)
Coercivity. The raising estimate (66) applied to phi gives
c N(phi)^2 <= ||T phi||^2 + ||phi||^2. (72)
The same estimate applied to T^flat phi and the boundedness of the coefficients in T^flat give ||T^flat phi||^2 <= C N(phi)^2. Combining these into (69) and dropping the nonnegative term ||(2T^flat + U^dagger) phi||^2 from the left,
||(2T + U) phi||^2 >= c N(phi)^2 - C eps N(phi)^2, (73)
after absorbing the negative term -4||T^flat phi||^2 into the same N(phi)^2 budget. For eps small enough,
N(phi) <= C ||(2T + U) phi||. (74)
Finally, ||J phi|| <= C ||phi||_{L^2(Y; H^1(S_y))} <= C beta_0^{-1} N(phi), and the beta_0 from the raising estimate absorbs J:
N(phi) <= C beta_0^{-1} N(phi) + C ||P phi||.
Taking beta_0 large enough gives (67). QED.
The constant-mode handling. The zero-mean condition on phi is imposed by subtracting, at each base point y, the fiber mean of phi(y, .). Since T_y u is defined against zero-mean sphere data in (53) and the constant fiber mode is handled separately through (55), the coercivity (67) applies to the remainder phi = chi (psi - 1), which has zero fiber mean by construction.
Lemma 17 (Uniform density). Fix the structural parameters so that Propositions 12 and 16 both apply. There is a constant C, independent of delta and of sufficiently small lambda, such that
||chi psi||_{L^2(S_gamma Y)} <= C (75)
whenever the temporary hypothesis (52) holds.
Proof. Reduce to zero mean. Set phi := chi (psi - 1). The constant function 1 has zero angular derivatives, so phi is compactly supported in the base and has zero fiber mean on each sphere. From the moving-sphere equation for psi (established at the end of the previous paragraph on psi), applied to the function phi instead of psi,
P phi = P (chi (psi - 1)) = -chi P1 + [P, chi](psi - 1). (76)
Bound on the right side. The cutoff estimate [P, chi] involves only angular derivatives through order one and the factor r nabla psi on the support of derivatives of chi. On that support the density psi is bounded uniformly in delta and small lambda by Proposition 14: on the lower region it is the ordinary evaluation density, and on the complement the radii are bounded below by a positive constant. Together with the bound on P1 from the same proposition, this gives
||P phi||_{L^2(S_gamma Y)} <= C, (77)
uniformly in delta and small lambda.
Apply coercivity. Proposition 16 with phi = chi(psi - 1) gives
N(phi) <= C ||P phi|| <= C, (78)
so phi is bounded in N and in particular in L^2(S_gamma Y). Adding back the fixed function chi gives (75). QED.
Uniform bounds in N. The same computation gives N(chi psi) <= C with C independent of delta and of small lambda, since chi is a fixed compactly supported cutoff. This stronger statement will be used in the moment bound below.
The bootstrap from harmonic differences to metric jets.
Set V^a := u_{2,lambda}^a - u_{1,lambda}^a for the paired harmonic functions of Theorem 7. Because (55) reproduces constants and coordinates exactly, T_y acts trivially on affine functions at y, and V^a measures only the failure of u_1^a and u_2^a to coincide after the transfer.
Lemma 18 (Moment bound). Under the temporary hypothesis (52), there is a constant C, independent of delta and of sufficiently small lambda, with
||V^a / r_delta^2||_{L^2(Y, dV_gamma)} <= C (79)
for every paired harmonic difference V^a.
Proof. Fix y in Y. Subtract from u_{1,lambda}^a its affine coordinate Taylor polynomial at y,
P_y(x) := u_{1,lambda}^a(y) + <x - y, grad u_{1,lambda}^a(y)>.
Because T_y reproduces constants and coordinates from (55), T_y P_y = u_{1,lambda}^a(y), so the transfer of this polynomial contributes nothing to V^a:
V^a(y) = T_y [u_{1,lambda}^a - P_y]
= int_(S_y) psi(y, omega) [ u_{1,lambda}^a(exp_y(r_delta(y) omega)) - P_y(exp_y(r_delta(y) omega)) ] dsigma_y(omega).
(80)
The bracket is the remainder of the second-order Taylor expansion of u_{1,lambda}^a at y. Coordinate distance and gamma-distance are uniformly comparable on the compact set Y, and u_{1,lambda}^a has bounded second derivatives, so the bracket is at most C r_delta(y)^2 on S_y. By Cauchy-Schwarz,
|V^a(y)| <= C r_delta(y)^2 ||psi(y, .)||_{L^2(S_y)}.
Lemma 17 bounds ||psi(y, .)||_{L^2(S_y)} on the support of chi uniformly in delta and small lambda. Integrating the square over Y and using the fixed cutoff chi gives (79). QED.
Lemma 19 (Finite-order interpolation). Let V be smooth on a neighborhood of the closed ball B(y, s) in R^n, with 0 < s <= 1 and ||V||_{C^m(B(y,s))} <= A_m for some integer m. For every multi-index alpha with |alpha| = j < m,
|partial^alpha V(y)| <= C_{m,n} ( s^{-j-n/2} ||V||_{L^2(B(y,s))} + A_m s^{m-j} ). (81)
Proof. Expand V(y + sz) at z = 0 into the Taylor polynomial of degree m-1 on the unit ball. The remainder has absolute value at most C_{m,n} A_m s^m. The L^2 norm of the Taylor polynomial on the unit ball is at most s^{-n/2} ||V||_{L^2(B(y,s))} + C_{m,n} A_m s^m. On the finite-dimensional space of polynomials of degree at most m-1, every coefficient is bounded by a constant (depending only on m and n) times the L^2 norm on the unit ball. The z^alpha coefficient is s^j partial^alpha V(y) / alpha!, so
s^j |partial^alpha V(y)| <= C_{m,n} ( ||V||_{L^2(B)} + A_m s^{m+n/2} ),
which after division by s^j gives (81). QED.
Lemma 20 (Pointwise superlinear bounds). Under (52), for every paired harmonic difference V^a and every multi-index with |alpha| <= 8,
|partial^alpha V^a|(y) <= C r_delta(y)^{3/2}, (82)
and, for the normalized inverse-metric discrepancy h := g_1^{-1} - g_2^{-1} in the harmonic coordinate chart,
|partial^alpha h|(y) <= C r_delta(y)^{3/2}, (83)
uniformly in delta, small lambda, and the finite family a.
Proof of (82). Apply Lemma 19 to V^a with s = r_delta(y)^{1-eta}, where eta := 1 / (2D), D := 10 + n/2, and m := ceil(10 + 3D). The L^2 norm of V^a on B(y, s) is bounded by C r_delta^2 s^{-n/2} by Lemma 18, and the C^m norm of the renormalized u^a_j is bounded uniformly. The two exponents in (81),
s^{-j-n/2} ||V^a||_{L^2} <= C s^{2-j-n/2}, A_m s^{m-j},
both have size at least r_delta^{3/2} for j <= 8, by construction of D and m. This gives (82).
Proof of (83). In a fixed interior harmonic coordinate chart, the two Laplace equations for u_1^a and u_2^a with the same coordinate representation near W give
g_1^{ij}(y) partial_ij V^a(y) = -(g_1^{ij} - g_2^{ij})(y) partial_ij u_{2,lambda}^a(y) + (first-order-in-V terms). (84)
The first-order-in-V terms involve partial V^a at y and metric derivatives of order at most one, and they are bounded by C r_delta^{3/2} by (82). The Hessians partial_ij u_{2,lambda}^a(y) for the finitely many a in the spanning family of condition (A4) form a matrix whose action on the allowed second-derivative hyperplane is invertible with bounded inverse. Therefore (84) determines the components of h at y to precision r_delta^{3/2}. Applying partial_y^beta with |beta| <= 8 to (84) and using (82) gives the same bound for derivatives of h: (83). QED.
The structure of the bootstrap closes here. Lemma 20 replaces the temporary hypothesis (52) by the strictly better estimate (83): since r_delta^{3/2} <= r_delta/2 for r_delta <= 1/4, the smallness hypothesis H_8(y) <= eps r_delta(y) is preserved with room to spare. The dilation parameter lambda is fixed once and for all.
Returning to the original manifold.
Admissible match. The extended manifolds M_1^e, M_2^e, their common cap E, the open source set U_0 inside E, and the evaluations I_j(p)(f) := u_f^j(p) of source-supported g_j-harmonic functions on M_j^e are all in place. An admissible match is a smooth local isometry
Phi : W -> (M_1^e)^circ, W subset (M_2^e)^circ an open connected set,
satisfying
Phi|_E = Id, I_2(p) = I_1(Phi(p)) for p in W. (85)
The identity on E is admissible by Proposition 5 and by the equality of source evaluations there.
Proposition 21 (No interior frontier). There is a smooth isometry Phi : (M_2^e)^circ -> (M_1^e)^circ of the second interior onto the first, satisfying (85).
Proof. Uniqueness and maximality. Two admissible matches coincide wherever their domains overlap. If Phi_1, Phi_2 are admissible and p is in W_1 cap W_2, then I_1(Phi_1(p)) = I_2(p) = I_1(Phi_2(p)), and I_1 separates interior points by Lemma 6, so Phi_1(p) = Phi_2(p). The union W of all admissible domains, with the map defined pointwise, is therefore a single smooth local isometry. This union map is injective (equal images give equal I_2, hence equal original points), so it is itself an admissible match. It is maximal by construction.
Green equality on W. Fix p in W. The equality I_2(p) = I_1(Phi(p)) gives G_2(p, q) = G_1(Phi(p), Phi(q)) for q in U_0, first distributionally. Both G_2(p, .) and the pullback G_1(Phi(p), Phi(.)) are smooth away from their poles and harmonic on the connected set W minus {p}; they agree on the nonempty open U_0. Unique continuation extends the equality to all of W. The local isometry Phi^* g_1 = g_2 on W also pulls back the Green kernel symbol.
Frontier elimination. Suppose the maximal match W is not the full (M_2^e)^circ. Take a frontier point p_* of W on the interior. Let B be a metric ball in W touching the frontier at p_*: B is compactly contained in W, has center c in W, and the boundary of B meets the frontier only at p_*. Such a ball exists because the extended manifolds are compact and distances are proper. At p_*, all the hypotheses of Theorem 7 are satisfied: the metrics g_1 and Phi^* g_2 agree on W, the Green kernels agree off the diagonal on W x W by the previous paragraph, and the finite harmonic pairs u_1^a and u_2^a o Phi agree on W by (85). The Hessian spanning condition (A4) holds at any interior point by Lemma 6. Theorem 7 then gives a smooth neighborhood U of p_* in (M_2^e)^circ on which Phi extends to a local isometry with the second metric, and on which all the paired harmonic functions continue to agree. The extension on U agrees with the existing match on the nonempty open set U cap W by separation of evaluations. The union of the extension with W is an admissible match strictly containing W, contradicting maximality. Hence W = (M_2^e)^circ.
Surjectivity. The image Phi(W) is open because Phi is a local diffeomorphism. It is closed in (M_1^e)^circ: if Phi(p_k) converges to an interior point q_1, a subsequence of p_k converges in compact M_2^e to some p. Evaluation continuity gives I_2(p) = I_1(q_1); p cannot lie on bd M_2^e because I_2(p) = 0 there while I_1(q_1) does not vanish, so p is interior and Phi(p) = q_1. The first interior (M_1^e)^circ is connected, and the image is nonempty, so Phi(W) = (M_1^e)^circ. An injective surjective local diffeomorphism has a smooth inverse. QED.
Proposition 22 (Smooth completion and the accessible gauge). The isometry Phi of Proposition 21 extends to a smooth diffeomorphism M_2^e -> M_1^e. It preserves the original copies of M and its restriction satisfies
g_2 = Phi^* g_1, Phi|_Gamma = Id. (86)
Proof. Smoothness up to the boundary. A Riemannian isometry of the interiors extends smoothly to the closed manifolds. Fix q in bd M_2^e and a sequence p_k -> q in the interior. The images Phi(p_k) are bounded, hence subconverge to Phi(q) in M_1^e. Local normal coordinates on the two manifolds and the isometric nature of Phi on the interiors give uniform C^inf bounds on Phi near q, so Phi is smooth up to the boundary and is a diffeomorphism of the closed manifolds.
The original copies of M are preserved. The identity on the common cap E, together with the restriction Phi(W) = (M_1^e)^circ and the smooth extension to both closed manifolds, carries M_2 into M_1. Because both M_j sit as connected closed submanifolds with the same boundary component bd M_j minus P_0, Phi carries M_2 onto M_1.
The restriction to Gamma. Recall that the cap E was attached over a smaller open sub-patch P_0 of Gamma with the bump cut edge; P_0 is nonempty, relatively open, and contained in Gamma. By the construction Phi|_E = Id, so Phi restricts to the identity on bd M_2^e near P_0. On the connected boundary component of Gamma, Phi|_Gamma is a smooth local isometry of the boundary region equal to the identity on the open set P_0, so Phi|_Gamma = Id.
QED.
Proof of Theorem 7 (local metric extension).
The conclusion is that the two coordinate metrics agree on a one-sided open neighborhood of 0 inside W. The proof assembles the constructions and estimates that precede it.
Setup. Choose the depth function t: Y -> R on a compact base region Y whose closure is contained in W, with 0 on the hypersurface boundary of the one-sided neighborhood and dt != 0, as in (42)-(43). Fix R_* > 0 once. Choose lambda small enough (one finite reduction) that (52) holds at delta = 1, using the smooth smallness of the normalized inverse-metric discrepancy h under linear dilations of the domain, together with compactness of Y.
Continuation and transfer. Lemma 8, Lemma 11, and Proposition 12 construct the continued mixed Green kernel K on the fixed-separation region
{(x, y) : |x|, |y| <= B, |x - y| >= d}
for each fixed d > 0. Proposition 14 gives the transfer functional T_y and its representing density psi on the sphere S_y, with T_y 1 = 1, T_y x^i = y^i, and T_y u_{1}^a = u_{2}^a(y). Proposition 16 and Lemma 17 give the uniform coercivity and L^2 density bounds under the temporary hypothesis (52).
Bootstrap. Lemma 18 (moment bound) and Lemma 20 (superlinear pointwise bounds on V^a and on h) give H_8(y) <= C r_delta(y)^{3/2} for every y in Y, with C independent of delta and of sufficiently small lambda, under the same temporary hypothesis.
Closing the temporary hypothesis. Choose rho in (0, rho_0) with C rho^{1/2} <= eps/2. For y with r_delta(y) < rho, Lemma 20 gives H_8(y) <= C r_delta(y)^{3/2} <= (eps/2) r_delta(y). For y with r_delta(y) >= rho, smallness of h under the linear dilation supplies H_8(y) <= eps rho/2 <= eps r_delta(y)/2, after one further reduction of lambda. Hence (52) holds throughout Y, uniformly in delta.
Shrinking delta. Fix alpha in {t <= 0}. As delta -> 0, r_delta(alpha) -> R_* max(t(alpha), 0) = 0, so H_8(alpha) <= C r_delta(alpha)^{3/2} -> 0. The discrepancy h vanishes at alpha in the limit. Since alpha in {t <= 0} was arbitrary, g_1 = g_2 on {t <= 0} inside W.
Conclusion. The set {t <= 0} contains a one-sided open neighborhood of 0 inside W, by the choice of t and the hypothesis that W contains a one-sided neighborhood of 0 relative to a smooth hypersurface through 0. Hence g_1 = g_2 on such a neighborhood, which is Theorem 7. QED.
Proof of Theorem 1. Lemma 4 gives equality of continuous jets of g_1 and g_2 on Gamma. Proposition 5 constructs the common cap E and the Green equality on E x E. Lemma 6 supplies the finite harmonic family, their separation, and the Hessian spanning. Theorem 7 applies at every interior frontier of the maximal match, and Proposition 21 eliminates every interior frontier. Proposition 22 gives a smooth diffeomorphism Phi : M_2^e -> M_1^e fixing Gamma pointwise, and restricting to a diffeomorphism Phi : M -> M with g_2 = Phi^* g_1. This is the assertion of Theorem 1. QED.
The scalar consequence on a fixed Euclidean domain.
The acoustic metric. For a bounded connected smooth domain Omega subset R^n with n >= 3 and a strictly positive conductivity gamma in C^inf(closure(Omega)), the associated acoustic metric tensor is
g_gamma := gamma^{2/(n-2)} e, (87)
where e is the Euclidean metric on Omega. This metric is smooth and positive definite on closure(Omega) and is conformal to the Euclidean metric.
The measurement operators. The scalar measurement map Lambda_gamma^Gamma is defined on smooth f supported in Gamma by
Lambda_gamma^Gamma f := gamma (partial_{nu_e} u_f^gamma) |_Gamma, (88)
with nu_e the Euclidean outward normal and u_f^gamma the weak solution of div(gamma grad u_f^gamma) = 0 in Omega, u_f^gamma = f on bd Omega. The geometric measurement map Lambda_{g_gamma, Gamma} is defined in (1). The acoustic metric has dV_g = gamma^{n/(n-2)} dx, inverse metric g_gamma^{-1} = gamma^{-2/(n-2)} e^{-1}, so the three factors
<grad_g u, grad_g v>_g dV_g = gamma^{-2/(n-2)} <grad_e u, grad_e v>_e * gamma^{n/(n-2)} dx = gamma <grad_e u, grad_e v>_e dx
combine into a single factor gamma. An integration by parts in Omega then gives
<Lambda_{g_gamma, Gamma} f, h> = int_Gamma gamma h partial_{nu_e} u_f |_Gamma dS_e = <Lambda_gamma^Gamma f, h> (89)
for all f, h in C_c^inf(Gamma). Equality of the geometric measurement maps Lambda_{g_{gamma_1}, Gamma} and Lambda_{g_{gamma_2}, Gamma} is therefore equivalent to equality of the scalar ones.
Lemma 23 (Boundary conformal rigidity). Let n >= 3, let Omega be a bounded connected open subset of R^n with smooth boundary, and let Gamma subset bd Omega be nonempty, proper, and relatively open. Let Phi : closure(Omega) -> closure(Omega) be a smooth diffeomorphism fixing Gamma pointwise. Suppose Phi^* e = e^{2b} e for some smooth b : closure(Omega) -> R. Then Phi = Id on closure(Omega).
Proof. The Christoffel symbols of e^{2b} e are
Gamma^k_ij = b_i delta^k_j + b_j delta^k_i - b_k delta_ij, (90)
where b_i := partial_i b. Because Phi : (Omega, e^{2b} e) -> (Omega, e) is an isometry,
D^2 Phi(v, w) = DPhi( <grad b, v>_e w + <grad b, w>_e v - <v, w>_e grad b ) (91)
for every pair of tangent vectors v, w.
Vanishing of b at the boundary patch. Since Phi fixes Gamma pointwise, b restricted to Gamma is zero. At any p in Gamma take v = w = tau unit tangent to Gamma: since D^2 Phi = 0 along the fixed set and DPhi = Id near Gamma, (91) gives <grad b(p), tau>_e = 0 for every tangent tau. Taking v = tau, w = nu the unit normal gives <grad b(p), nu>_e = -<D^2 Phi(tau, tau), nu>_e = 0, because D^2 Phi = 0 near p. Consequently grad b = 0 at every point of Gamma.
Flatness and the ODE. The metric e^{2b} e is Euclidean, hence flat. Its Riemann tensor vanishes identically. The conformal factor e^{2b} in dimension n >= 3 gives the second-order system
b_ij = b_i b_j - (1/2) |grad b|^2 delta_ij (92)
for the Euclidean Hessian b_ij := partial_i partial_j b and grad b := (b_1, ..., b_n). Set q_b := grad b. Differentiating q_b along a unit vector v and substituting (92),
D_v q_b = <q_b, v>_e q_b - (1/2) |q_b|^2 v. (93)
The zero field is a solution of (93). At a boundary point p of Gamma, where q_b(p) = 0, uniqueness of the solution of the ODE along any straight segment from p staying in Omega forces q_b identically zero on that segment.
Reachability and vanishing. Since Omega is bounded and connected, every interior point lies on a straight segment from a boundary point. Taking the boundary point in Gamma (which is nonempty and relatively open in bd Omega) covers a neighborhood of Gamma in the radial directions. Moving outward segment by segment, q_b = 0 on all of Omega. Hence b is constant on Omega, and since b = 0 on Gamma, b = 0 on closure(Omega).
Affineness of Phi. With b = 0 on Omega, (91) gives D^2 Phi = 0, so Phi is affine. Since Phi fixes Gamma, which contains n affinely independent points, Phi = Id on closure(Omega). QED.
Proof of Corollary 3. Let gamma_1, gamma_2 be strictly positive smooth functions on closure(Omega) with Lambda_{gamma_1}^Gamma = Lambda_{gamma_2}^Gamma. By (89), the geometric measurement maps of the two acoustic metrics g_{gamma_1}, g_{gamma_2} coincide. Theorem 1 gives a smooth diffeomorphism Phi : closure(Omega) -> closure(Omega) with Phi|_Gamma = Id and g_{gamma_2} = Phi^* g_{gamma_1}. Reading off the conformal factor,
gamma_2^{2/(n-2)} e = (gamma_1 o Phi)^{2/(n-2)} Phi^* e,
so Phi^* e = e^{2b} e with b := (log gamma_2 - log (gamma_1 o Phi)) / (n-2). Lemma 23 applies and gives Phi = Id. From the above relation gamma_2 = gamma_1 on closure(Omega). QED.
The scalar uniqueness of Corollary 3 is a partial-data counterpart of the anisotropic recovery by Lassas and Uhlmann of a real-analytic manifold and metric from an arbitrary nonempty real-analytic boundary patch in dimension at least three, obtained without topological restrictions on the manifold or its boundary. The geometric theorem removes all boundary hypotheses except the existence of a common measured patch and the smoothness of the metric.
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