Proof. Fix \(p\in\partial M\) and apply Theorem 1 to \(\Gamma=\partial M\setminus\{p\}\). This is a nonempty proper open patch, dense in the entire boundary. Continuity makes the resulting \(\Phi\) the identity on \(\partial M\). Take \(F=\Phi^{-1}\). ◻

Proof. Fix a boundary chart compactly contained in \(\Gamma\), write \(m=n-1\ge2\), and let \(h\) denote the tangential metric matrix. Relative to the coordinate density, the energy operator has principal symbol \[e_1(z,\xi)=\sqrt{\det h(z)} \sqrt{h^{ab}(z)\xi_a\xi_b}.\] Thus its square determines the positive matrix \[C=(\det h)h^{-1},\qquad \det C=(\det h)^{m-1},\qquad h=(\det C)^{1/(m-1)}C^{-1}.\] The two induced metrics, and hence their boundary densities, are therefore equal. Dividing the common density-valued operator by this known density gives equality of the ordinary DN operators locally. Cutoffs supported in \(\Gamma\) and equal to one near a chosen point give equality of their complete symbols there.

We give the scalar symbol recursion to make its smooth and local content explicit. This boundary determination goes back to Lee and Uhlmann (1989); see also Joshi and Lionheart (2005, Theorem 1.1(ii) and Proposition 3.1). In inward normal coordinates \(s\ge0\), write \[g=ds^2+h(s,z),\qquad a=\frac12\mathop{\mathrm{tr}}(h^{-1}\partial_s h),\qquad L_g=-\partial_s^2-a\partial_s+P,\] where \(P=-\Delta_{h(s)}\) is tangential. Smooth boundary factorization gives a tangential operator \(B(s)\) of order one, with positive principal symbol, such that \[ L_g=(-\partial_s+B-a)(\partial_s+B) \pmod{\Psi^{-\infty}},\qquad B^2-aB-\partial_sB=P\pmod{\Psi^{-\infty}}. \tag{3}\] Here the remainders are tangentially smoothing and depend smoothly on \(s\). The local Poisson parametrix identifies \(B(0)\), modulo smoothing, with the outward DN operator: the decaying branch satisfies \((\partial_s+B)u=0\) and \(\partial_\nu=-\partial_s\). This is a local factorization for smooth coefficients; it makes no analyticity assumption.

For precision, use left symbols and \(D_z=-i\partial_z\), so \[b\mathbin{\#}c\sim \sum_\alpha\frac{(1/i)^{|\alpha|}}{\alpha!} (\partial_\xi^\alpha b)(\partial_z^\alpha c).\] Write \(b\sim b_1+b_0+b_{-1}+\cdots\) for the symbol of \(B\) and \(p=p_2+p_1\) for that of \(P\). Then \[b_1=\rho=(h^{ab}\xi_a\xi_b)^{1/2},\qquad b\#b-ab-\partial_sb=p.\] The equation of degree one is \[2\rho b_0+\frac1i\partial_\xi\rho\cdot\partial_z\rho -a\rho-\partial_s\rho=p_1.\] The terms \(p_1\) and the tangential composition term involve only \(h\) and its tangential derivatives. Since \[\partial_s\rho =-\frac{(\partial_s h)(\xi^\sharp,\xi^\sharp)}{2\rho},\] the part of \(b_0\) containing a normal derivative is \[\frac14\left(\mathop{\mathrm{tr}}_h\partial_s h -\frac{(\partial_s h)(\xi^\sharp,\xi^\sharp)}{\rho^2}\right).\] At each subsequent degree the new coefficient is obtained from \[2\rho b_{1-k}=\partial_s b_{2-k}+R_k,\qquad k\ge2,\] where \(R_k\) uses normal derivatives of \(h\) only through order \(k-1\) and finitely many tangential derivatives. Indeed all other terms in the symbol product use earlier coefficients, and tangential differentiation does not increase normal derivative order. Writing \(A_k=\partial_s^k h\), induction gives the highest-jet term \[ b_{1-k}= \frac{1}{4(2\rho)^{k-1}} \left(\mathop{\mathrm{tr}}_h A_k- \frac{A_k(\xi^\sharp,\xi^\sharp)}{\rho^2}\right) +\text{terms involving lower normal jets}. \tag{4}\] Differentiating \(\rho\), the raising of indices, or the coefficient in the preceding induction step contributes only lower-jet terms.

Suppose the two normal jets agree through order \(k-1\) as smooth functions on the boundary chart. Their tangential derivatives also agree. Equality of the symbols and Equation (4) give, for \(A=\partial_s^k(h_1-h_2)|_{s=0}\), \[\mathop{\mathrm{tr}}_h A-A(v,v)=0\qquad(|v|_h=1).\] Polarization yields \(A=(\mathop{\mathrm{tr}}_h A)h\). Taking the trace gives \((m-1)\mathop{\mathrm{tr}}_h A=0\), so \(A=0\). Starting from the already recovered boundary metric, induction proves all normal and mixed jet equalities. The normal metric components are \(g_{ss}=1\) and \(g_{sa}=0\), so these are the full metric jets.

Every step is local to a half-coordinate ball and uses densities. Although the differential-form formulation in (Joshi and Lionheart 2005) assumes orientability, this scalar coordinate proof requires none. Other boundary components affect only the local smoothing remainder. Finally, equality of smooth jets asserts no equality in an interior collar. ◻

Proof. Choose a boundary coordinate patch \(P\Subset\Gamma\) and a nonnegative smooth bump \(b\) with \(\mathop{\mathrm{supp}}b\Subset P\) and connected, nonempty positivity set \(P_0=\{b>0\}\). Use the separate inward normal collars of Lemma 4, and replace the boundary \(s=0\) by the graph \(s=-b(z)\). Take \(b\) sufficiently small that this graph lies in a collar. It gives smooth compact connected extended manifolds. Their common open added part is \[E=\{(z,s):-b(z)<s<0\}.\] Extend the first metric smoothly to negative \(s\) and use the same exterior metric for the second. A smooth extension of its coordinate coefficients exists in a collar and remains positive definite after shortening the collar. Their jets match on an open neighborhood of \(\mathop{\mathrm{supp}}b\). Pasting across the entire hypersurface \(s=0\) in that neighborhood is therefore smooth, to every order. Restricting this pasting to \(s\ge-b(z)\) proves smoothness also at the edges of the bump. We henceforth omit the superscript on the extended metrics.

For the following comparison, identify the two extensions by the identity on the original \(M\) and by the common collar coordinates on \(E\). This identification and its inverse are smooth up to each side of the seam, with bounded derivatives, and agree on the seam. In collar charts they are therefore bi-Lipschitz. In particular they identify the spaces \(H^1_0\) on the two extensions.

Fix a real \(F\in C_c^\infty(E)\) and let \(w_j\) minimize \[\mathcal J_j(w)=\frac12\int_{M_j^e}|dw|_{g_j}^2\,dV_{g_j} -\int_E Fw\,dV_{g_j} \quad\text{on }H^1_0(M_j^e).\] The minimizers are smooth up to the extended boundary and solve \(-\Delta_{g_j}w_j=F\). Their traces \(h_j\) on the original boundary are smooth and supported in \(\mathop{\mathrm{supp}}b\Subset\Gamma\): wherever \(b=0\), that boundary is also the new zero-data boundary. Transport \(w_1\) by this identification and replace it inside the original manifold by the \(g_2\)-harmonic extension of \(h_1\), leaving it unchanged on \(E\). The difference between the replacement and the transported \(w_1\) is the zero extension of a function in \(H^1_0(M)\); thus the replacement is an admissible competitor in \(H^1_0(M_2^e)\). This argument does not require the narrowing cap to have a regular boundary at its edges.

The interior energies before and after replacement are the equal quadratic energies of \(h_1\) for the two measured maps. The cap metric and source pairing are common. Hence \(\min\mathcal J_2\le\min\mathcal J_1\). Reversing the roles gives equality. Strict convexity of the Dirichlet form makes the replacement equal to \(w_2\), proving \(w_1=w_2\) on \(E\).

Normalize the symmetric Dirichlet Green kernels by \[ -\Delta_{g_j,x}G_j(x,y)=\delta_y^{g_j}(x),\qquad G_j(\,\cdot\,,y)|_{\partial M_j^e}=0, \qquad w_j(x)=\int_{M_j^e}G_j(x,y)F(y)\,dV_{g_j}(y), \tag{5}\] where the point mass has unit mass relative to metric volume. The source densities are identical on \(E\). Varying \(F\) gives equality of the distribution kernels on \(E\times E\), and therefore pointwise equality off the diagonal. ◻

Proof. Continuity and the zero boundary values follow from smooth Dirichlet regularity. For a nonnegative nonzero \(f\in C_c^\infty(U)\), the maximum principle and its strong form give \(Rf>0\) throughout \(N^\circ\). Thus the interior functionals are nonzero.

We prove the jet assertion by finite-dimensional duality. Let \(T\) be a linear functional on second jets at \(p\) annihilating all \(Rf\), represented as a distribution of order at most two supported at \(p\). Define \(w=RT\) by transposition with the metric density. The Dirichlet inverse is self-adjoint, so \[\langle w,f\rangle=\langle T,Rf\rangle=0, \qquad f\in C_c^\infty(U).\] Thus \(w=0\) on \(U\) and \(L_gw=T\). Away from \(p\) it is smooth and harmonic by elliptic regularity. The punctured interior is connected: a path meeting \(p\) can be diverted in a small punctured coordinate ball, whose spheres are connected in dimension at least two. Unique continuation consequently gives \(\mathop{\mathrm{supp}}w\subset\{p\}\).

A distribution supported at one point is a finite sum \[w=\sum_{|\alpha|\le k}c_\alpha\partial^\alpha\delta_p.\] If \(w\ne0\) and \(k\) is its highest derivative order, then \(L_gw\) has exact order \(k+2\). Indeed its highest coefficient polynomial is the product of the nonzero degree-\(k\) coefficient polynomial and the nonzero elliptic quadratic principal symbol. Such a product cannot vanish. Since \(T\) has order at most two, \(w=c\delta_p\). Hence the annihilator consists precisely of multiples of \(u\mapsto L_gu(p)\). That functional does annihilate the source solutions because \(p\notin\overline U\). Finite-dimensional duality proves that their jet range is the whole stated hyperplane.

To prove separation, suppose \(I(p)=I(q)\) for distinct interior points. Apply the same transposition argument to \(T=\delta_p-\delta_q\). The solution \(RT\) vanishes on \(U\), and UCP on the connected interior with these two points removed shows that it is supported at those points. The argument still applies if a point lies in \(U\), since the initial vanishing on \(U\) is distributional. At each support point a nonzero point-supported distribution has an elliptic image of order at least two. It cannot have the order-zero image \(\delta_p-\delta_q\). This contradiction proves separation.

Finally the equation constraint allows arbitrary first derivatives, by adjusting the Hessian. Choose \(n\) source solutions with independent differentials; their tuple is a harmonic coordinate chart. In this chart the first-order coefficients of \(\Delta_g\) vanish, so the ordinary coordinate Hessians have precisely the constraint \(g^{ab}\partial_{ab}u=0\). The jet conclusion supplies a basis for that hyperplane, with zero first derivatives if desired. A chosen finite basis remains spanning, with a positive Gram bound, after shrinking the chart. ◻

Proof. Let \(\mathcal H_1(D_1)\) be the closed subspace of \(L^2(D_1)\) consisting of distributional \(L_1\)-solutions. Restriction \(R:\mathcal H_1(D_1)\to L^2(S_1)\) is compact by interior elliptic estimates and injective by unique continuation. The spectral theorem for \(R^*R\) gives an orthonormal basis \((w_j)\) of \(\mathcal H_1(D_1)\) such that \[ (w_i,w_j)_{L^2(S_1)}=\mu_j^2\delta_{ij}, \qquad 0<\mu_j\le1. \tag{8}\] For every \(N\), \[ \mu_j\le C_N(1+j)^{-N}. \tag{9}\] Indeed, multiply a solution by a cutoff equal to one near \(\overline{S_1}\) and compactly supported in \(D_1\). Its \(H^k\) norm is bounded by its \(L^2(D_1)\) norm. Fourier truncation in a containing cube gives a rank \(O(A^n)\) approximation with operator error \(O(A^{-k})\). The minimax characterization of singular values yields \(\mu_j\le C_kj^{-k/n}\). All these constants have the stated uniformity.

For \(y\in S_2\) expand \(F(\cdot,y)\) on \(D_1\): \[F(x,y)=\sum_j w_j(x)b_j(y),\qquad \|b_j\|_{L^2(S_2)}\le M.\] Equation (8) continues these coefficients to \(D_2\) by the formula \[ b_j(y)=\mu_j^{-2}\int_{S_1}F(x,y)\overline{w_j(x)}\,\mathrm dx, \qquad \|b_j\|_{L^2(D_2)}\le M\mu_j^{-1}. \tag{10}\] The integral solves \(L_2b_j=0\) and agrees with the original coefficient on \(S_2\). Applying (7) gives \[\|w_j\|_{L^2(O_1)}\le C\mu_j^{\gamma_1},\qquad \|b_j\|_{L^2(O_2)}\le CM\mu_j^{\gamma_2-1}.\] Consequently the series converges absolutely in \(L^2(O_1\times O_2)\): its summands are bounded by \(CM\mu_j^{\gamma_1+\gamma_2-1}\). Choose \(N\) in (9) so that \(N(\gamma_1+\gamma_2-1)>1\). The limit solves both equations in distributions. Interior estimates for their elliptic sum give the asserted smoothness and bounds on smaller sets.

We verify agreement with the second given piece; no assertion that an arbitrary local solution extends to \(D_1\) is needed. Let \(z\) be orthogonal to the closure of \(R\mathcal H_1(D_1)\) in \(L^2(S_1)\). The function \[y\longmapsto\int_{S_1}F(x,y)\overline{z(x)}\,\mathrm dx\] solves the second equation and vanishes on \(S_2\), so it vanishes on connected \(D_2\). Thus each second-piece slice belongs to that closed range. Its expansion in the orthonormal basis \(w_j/\mu_j\) is complete, and its coefficients are \(\mu_jb_j(y)\) by (10). It follows that the series is \(F\) on \(S_1\times O_2\); the series converges there also by the positive exponent \(\gamma_2\). On \(O_1\times S_2\) it is the original expansion, with positive exponent \(\gamma_1\). This proves both agreements. Only upper bounds for the singular values were used, so uniformity does not require continuous choices of singular bases or positive lower bounds for their singular values. ◻

Proof. We first split the product Hessian condition into a strict weight in each factor. Their sum will lie below a defining function for the given side, with a quadratic gap. That gap places two intersecting product sets in the given side; the weights then yield interpolation exponents greater than one half, allowing Lemma 9 to apply.

Proof. Starting from the top, let \(a_*\) be the infimum of levels down to which an extension agreeing with the seed has been obtained. In the uniform version include uniform bounds in this property. At a non-seed point of \(B\times\{a_*\}\), Proposition 10 uses compact data strictly above that level. Take a finite cover of the level by such output neighborhoods and seed neighborhoods. Their data are contained above \(a_*+\eta\) for one \(\eta>0\), so an already reached level supplies them with the needed bounds.

Shrink the outputs in \((z,a)\) coordinates to boxes \(B_i\times(a_*-d_i,a_*+d_i)\). Every connected component of an overlap contains vertical segments to a common height greater than \(a_*\). Both germs equal the preceding extension there. Unique continuation for \(L_1^x+L_2^y\) makes them identical on that overlap component. The same argument gives agreement with \(\mathcal S\), because a vertical segment moving upward from a seed point stays in the seed. Boundary neighborhoods use the given solution. A finite subcover has a positive minimum output width and therefore passes the putative last level, or includes the endpoint \(a_-\). This proves continuation on the whole cylinder. The same finite constructions retain all strict margins under small perturbations. Their Carleman, series and interior estimates prove the uniform assertion. Data bounded up to a limiting surface are never required: each local construction uses compact sets strictly above it. ◻

Proof. We first cross the limiting plane, then deform variable-radius tubes to reach the prescribed separations. All the sets and positive separation margins used in the two steps are fixed before choosing \(\lambda\).

Proof. Write \(a_\delta=(\log\ell_\delta)'\). In terms of \(s=t/\delta\), \[a_\delta(t)=\delta^{-1}A(s),\qquad A(s)=\frac{e^s}{(1+e^s)\log(1+e^s)}.\] The function \(A\) is positive and decreasing. At the negative end it tends to one, and at the positive end it is comparable to \((1+s)^{-1}\). Differentiating the formula, or its convergent expressions at the two ends, gives \(|A^{(j)}(s)|\leq C_jA(s)^{j+1}\). On the remaining compact interval the same inequalities follow by positivity. On the fixed \(t\) interval this gives a positive lower bound for \(a_\delta\) and \[a_\delta'\leq0,\qquad |\partial_t^j\log\ell_\delta|\leq C_j a_\delta^j, \qquad a_\delta\leq\ell_\delta^{-1}.\] Since \(|dt|_\gamma\) is bounded above and below, \(\beta_\delta=p a_\delta|dt|_\gamma\). Equation (26) and \[\operatorname{Hess}_\gamma f_\delta =p a_\delta\operatorname{Hess}_\gamma t +p a_\delta'\,dt\otimes dt\] prove the Hessian bound. The chain rule proves the other derivative bounds, with constants independent of \(p\geq2\) because \(p a_\delta^j\leq p^j a_\delta^j\).

The bound \(0<\ell_\delta'<1\) gives the asserted Lipschitz estimate for \(r_\delta^{1/p}=R_*^{1/p}\ell_\delta\circ t\). Moreover, \[\partial_\delta\ell_\delta =\log(1+e^s)-\frac{s e^s}{1+e^s}>0.\] The function on the right increases up to \(s=0\) and decreases afterward, with limit zero at both ends. The pointwise limit of \(\ell_\delta\) is \(\max(t,0)\). Finally, \[r_\delta\beta_\delta^2 \leq C R_*p^2\ell_\delta^{p-2},\] and \(\ell_\delta\) is uniformly bounded on the fixed interval. This proves (29). ◻

Proof. We check Proposition 10 at each intermediate radius \(r=r_{\delta'}\). Put \(f=\log r\), \(\beta=|df|_\gamma\), and \(b=\nabla^\gamma r\), so \(|b|=r\beta\). Use the defining function \[\rho(x,y)=\tfrac12\bigl(d_\gamma(x,y)^2-r(y)^2\bigr).\] The positive side is the outside of the sphere. Parallel translation along its radial geodesic identifies the two tangent spaces for \(\gamma\); write \(\omega\) for the unit vector from \(y\) to \(x\).

In the normalization of the strict test, the normal and tangential vectors can be combined into complex vectors \[U_1=\omega+i v,\quad v\perp\omega,\quad |v|=1, \qquad U_2=q+i w,\] where the conditions for the second principal metric \(Q^{-1}\) give \[ (\omega+b)\cdot U_2=1,\qquad q=\frac{\omega+b}{|\omega+b|^2}+O(|h|),\qquad |w|=|\omega+b|^{-1}+O(|h|). \tag{33}\] Here and below lengths and dot products in these formulas use \(\gamma\) and its parallel identification. The first equation includes \((\omega+b)\cdot w=0\). Since \(|b|\) and \(|h|\) are small, both partial gradients of \(\rho\) are nonzero and \(|U_2|\) is bounded above and below.

The Hessian of half the squared distance, with the product \(\gamma\)-connection, evaluated on these endpoint vectors is \[|U_1-U_2|^2+O(r^2)(|U_1|^2+|U_2|^2).\] To see the error size, parametrize the radial geodesic on \([0,1]\). Its Jacobi field with prescribed endpoint values differs, in a parallel frame, from the affine interpolant by \(O(r^2)\) in \(C^1\), by the Jacobi equation and bounded curvature. The second variation formula then gives the displayed bound. Replacing the second connection by that of \(Q^{-1}\) costs \(O(r^2)\) as well: the connection difference is \(O(|h|+|\partial h|)=O(\varepsilon r)\) and \(|d\rho|=O(r)\).

Now \(\omega\cdot(U_1-U_2)=b\cdot U_2\) and \(\operatorname{Hess}(r^2/2)=2\,dr\otimes dr+r^2\operatorname{Hess}f\). Consequently the strict Hessian expression is \[ |\pi_{\omega^\perp}(U_1-U_2)|^2-|b\cdot U_2|^2 -r^2\operatorname{Hess}_\gamma f(U_2,\overline U_2)+O(r^2). \tag{34}\] For a real Hessian, its value on a complex vector and its conjugate is the sum of its values on the real and imaginary parts, as required by the criterion.

The real part of the projected difference is \(-\pi_{\omega^\perp}b+O(|b|^2+|h|)\). For its imaginary part, (33) gives \(|\pi_{\omega^\perp}w|=1-\omega\cdot b+O(|b|^2+|h|)\); compare this with \(|v|=1\). Squaring these two bounds yields \[|\pi_{\omega^\perp}(U_1-U_2)|^2 \geq |b|^2-C\bigl(|b|^3+|b||h|+|h|^2\bigr).\] The real and imaginary directions of \(U_2\) are almost orthonormal, so \[|b\cdot U_2|^2\leq \bigl(1+C(|b|+|h|)\bigr)|b|^2.\] By Lemma 13, the expression (34) is therefore bounded below by \[c r^2\beta -C\bigl(r^2+r^3\beta^3+\varepsilon r^2\beta +\varepsilon^2r^2\bigr).\] Choose \(p\) so that \(\inf\beta\) absorbs the \(Cr^2\) term. Then choose \(\varepsilon\) small. The inequality \(r\beta^2\leq\varepsilon\) absorbs \(r^3\beta^3\), and the expression is strictly positive.

It remains to specify the starting and boundary data for this local test. Deform \(\delta'\) from \(1\) to \(\delta\), using \((y,\omega)\) as base variables. The inequality \(H_8\leq\varepsilon r_\delta\) implies its counterpart for every \(\delta'\geq\delta\), because the radii increase with \(\delta'\). At \(\delta'=1\) the required separations have a fixed positive minimum. The same is true wherever \(t>t_b\), since \(r_{\delta'}\geq R_*t_b^p\). Proposition 12 supplies uniformly bounded data on these fixed separation ranges, with open room. Where \(y_n<-1/4\), use the original \(G_{1,\lambda}\); for small \(\lambda\) this is part of the known lower cross. These regions cover the lateral and end boundaries of the cylinder that are not supplied by the initial parameter, and their prescriptions agree on overlaps.

The base sphere bundle over the closed cylinder is compact. On each fixed interval \(\delta\leq\delta'\leq1\), the radii are positive, all surfaces lie in normal neighborhoods, and the strict tests just proved apply away from the prescribed regions. Lemma 11 therefore supplies compatible extension germs on the whole family. Its finite-cover gluing also retains agreement with the prescribed regions. This is an existence argument for each finite interval; it does not assert uniform continuation bounds when the lower endpoint tends to zero. ◻

Proof. For each ball, its smooth Dirichlet solution operator determines the normal derivative of a harmonic function from its boundary trace. Move this boundary operator in the first term of (35) onto the smooth boundary value of \(\mathcal G\) by adjunction, and then pull back to \(S_y\). The resulting smooth density is \(\psi\). Smooth dependence of the Dirichlet problem and of the sphere parametrization gives its smooth dependence on \(y\) for each positive radius. This construction initially applies to smooth boundary values and hence to every harmonic function used in (36).

Fix a center \(y_0\) and a function \(u\) harmonic on a neighborhood of its closed ball. The continued kernel is defined on an open neighborhood of the compact sphere over \(y_0\). Choose a fixed slightly larger surrounding surface inside this kernel neighborhood and the domain of \(u\). After restricting the center neighborhood, it surrounds every moving sphere and the whole intervening collar remains in that common domain. Green’s identity on the collar replaces the moving integral in (35) by the integral over this fixed surface, since both first-variable equations are homogeneous there. Applying the second-variable operator under the fixed integral proves the stated local harmonicity. This argument uses arbitrary local harmonic functions, not only the source-generated pairs.

On the lower known region \(\mathcal G=G_{1,\lambda}\), so Green’s formula gives \(T_yu=u(y)\). Unique continuation on the connected set \(Y^\circ\) now proves all identities in (37), first for the constant and coordinate pairs and then for each given harmonic pair.

For the uniform bounds on the cutoff region, distinguish the two sets in (31). On the known lower set, \(T_y\) is ordinary evaluation. Its density is the Poisson density at the center of the ball, expressed in normalized coordinates. After pulling the equation back by \(z\mapsto\exp_y^\gamma(rz)\) and multiplying by the Laplacian scaling, the coefficients form a smooth uniformly elliptic family down to \(r=0\), whose limiting equation is the Euclidean Laplace equation. The interior evaluation kernel for the Dirichlet problem consequently has uniform angular smooth bounds and smooth center–radius dependence. This follows directly by differentiating the Dirichlet equation and using its uniform elliptic estimates; the center stays a fixed positive distance from the unit sphere. The chain rule for a base derivative costs \(|dr|=r\beta\), which is uniformly bounded by (32).

On the other cutoff set, \(t>t_b\). There the radii have a fixed positive lower bound and all their required derivatives are bounded uniformly in \(\delta\). The fixed-separation bounds of Proposition 12, followed by the smooth Dirichlet construction of \(\psi\), give the same conclusion. Compactness of the cutoff derivative supports completes the proof. ◻

Proof. Normalize the principal part of the first equation to \(\gamma^{-1}\) and pull it back by \(z\mapsto\exp_y^\gamma(rz)\) to the unit ball. After multiplication by \(r^2\), its principal coefficients are \(\delta^{ij}+O(r^2)\) and its first-order coefficients are \(O(r)\). The resulting smooth family remains uniformly elliptic and has an invertible Dirichlet problem, including at \(r=0\). Its radial derivative operator therefore has the expansion \[ B+r B_1(y)+r^2 B_2(y,r),\qquad B_1\in\Psi^0,\quad B_2\in\Psi^1. \tag{43}\] Here is the parameter justification. The classical boundary-symbol construction is described in (Joshi and Lionheart 2005, Theorem 1.1(i) and Proposition 3.1); we keep the center and radius dependence explicitly. In boundary coordinates the decaying normal root of the principal symbol constructs the Dirichlet parametrix. Successive lower-order transport equations construct its symbols smoothly in \((y,r)\). The first normal derivative trace is an operator of order one, whose principal symbol depends only on the principal coefficients and the differentiating vector. Those coefficients have zero first \(r\)-derivative at \(r=0\), so that derivative of the trace operator has order zero. Taylor’s formula in this symbol class gives (43) to smoothing error. The true solution differs from the parametrix by the Dirichlet inverse of a smooth error; differentiated Dirichlet estimates control this remainder in every fixed seminorm. More explicitly, on a compact center set the rescaled Dirichlet operator is smoothly invertible between each fixed pair of Sobolev spaces, uniformly for \(0\le r\le r_0\). Differentiating that inverse controls every fixed number of parameter derivatives of the smoothing correction; increasing the Sobolev orders controls its smooth kernel seminorms. Thus the Taylor expansion holds in the complete classical symbol topology, including the smoothing remainder, and not merely in one Sobolev operator norm. This proves the stated expansion and its parameter bounds.

Differentiate the exponential map in its center, transporting \(\omega\) parallel to the center velocity. The Jacobi field has initial derivative zero. After inverting the differential in the \(z\) variable, the velocity acting on the unit-ball solution is \[r^{-1}\bigl(e_i+O(r^2)\bigr)+f_i\omega.\] The coefficient error is smooth in \((y,r,\omega)\). Tangential derivatives act directly on the boundary value; the radial derivative is (43). At \(r=0\) the first term is \(r^{-1}A_i\). The order-one errors are \(O(r)\), and the order-zero errors are bounded: in particular \(r f_i B_1\) is bounded because \(r\beta\le\varepsilon/\beta\). Taking sphere adjoints gives (41). Differentiating the radius costs at most one factor \(\beta\) per derivative, by (38); the asserted parameter estimates follow.

For fixed \(u\), differentiate \(T_yu=\langle\psi(y,\cdot),U(y,\cdot)\rangle_{S_y}\). Metric compatibility and \(\nabla_iU=\mathcal C_iU\) put the differentiated operator on \(\psi\) as \(\nabla_i+\mathcal C_i^*\). The local second-metric harmonicity in Proposition 15 proves that (42) pairs to zero with the trace of every function harmonic near the fixed closed ball. No global extension or source-family approximation is used in this step. At a fixed sphere these traces are dense among smooth boundary data: solve the smooth Dirichlet problem on slightly enlarged balls, using boundary values transported from the limiting sphere, and use smooth dependence of the solution on the radius. Thus the paired equation is the stated equation for \(\psi\). ◻

Proof. Both operators change angular degree by exactly one, so it suffices to prove the estimate at one input degree \(k\ge1\). Identify that component with a harmonic homogeneous polynomial \(v(z)\) of degree \(k\). Use the Fischer inner product, in which the monomials \(z^\alpha\) are orthogonal with squared norm \(\alpha!\). Polynomial differentiation \(a_i=\partial_{z_i}\) is adjoint to multiplication \(M_i=z_i\). On harmonic polynomials of degree \(k\) the Fischer squared norm is the normalized squared sphere norm multiplied by \[N_k=n(n+2)\cdots(n+2k-2),\qquad N_0=1.\] For completeness, integration against a standard real Gaussian identifies the Fischer norm on trace-free homogeneous polynomials with their Gaussian \(L^2\) norm; integrating radially then gives the factor \(N_k\). These classical identities are also recorded in (Render 2008, Equation (2.1) and Theorem 2.1).

Set \(d=k+n/2-1\) and use \(D_i^\pm=\pm\nabla_i+k f_i\) at this fixed degree. For harmonic polynomials \(q_i\) of degree \(k\), the Laplacian of \(\sum_iM_iq_i\) is \(2\sum_i a_iq_i\). Since \[\Delta_z(|z|^2s)=4d\,s\qquad(s\in\mathcal H_{k-1}),\] its harmonic projection is \(\sum_iM_iq_i-|z|^2\sum_i a_iq_i/(2d)\). Orthogonality of this projection and its complement shows that its squared norm is \[\sum_i\|q_i\|^2+\sum_{i,j}(a_jq_i,a_iq_j) -d^{-1}\Bigl\|\sum_i a_iq_i\Bigr\|^2.\] Changing from Fischer adjoints to sphere adjoints gives \[A_i^*|_{\mathcal H_k} =\frac{N_{k+1}}{N_k}\Pi_{k+1}M_i,\] where \(\Pi_{k+1}\) is harmonic projection in homogeneous degree \(k+1\). Consequently \(N_k\) times the left side of (49) is \[ \begin{split} (2d+2)\left[\|D^+v\|^2 +\sum_{i,j}(a_jD_i^+v,a_iD_j^+v) -\frac1d\Bigl\|\sum_i a_iD_i^+v\Bigr\|^2\right] -2d\Bigl\|\sum_i a_iD_i^-v\Bigr\|^2. \end{split} \tag{50}\] All norms in this calculation include integration over the base and use the polynomial inner product in the fibers. The factors are \(N_{k+1}/N_k=2d+2\) and \(N_k/N_{k-1}=2d\).

Write \(C_{ij}=2k(\operatorname{Hess}_\gamma f)_{ij}\) and \[C[av]=\int_Y\sum_{i,j}C_{ij}\langle a_iv,a_jv\rangle\,dV_\gamma.\] Covariant integrations by parts give \[\begin{align*} \sum_{i,j}(a_jD_i^+v,a_iD_j^+v) &=\Bigl\|\sum_i a_iD_i^-v\Bigr\|^2-C[av] +O(k^2)\|v\|^2,\tag{51}\\ \Bigl\|\sum_i a_iD_i^+v\Bigr\|^2 &\le k\|D^-v\|^2-C[av]+O(k^2)\|v\|^2,\tag{52}\\ \|D^-v\|^2 &=\|D^+v\|^2+\int_Y\mathop{\mathrm{tr}}C\,|v|^2\,dV_\gamma. \tag{53}\end{align*}\] Here and below a form denoted \(O(k^2)\|v\|^2\) has absolute value at most a fixed constant times that quantity. To verify the first two formulas, move a \(D_j^+\) across the pairing and commute \(D_j^-D_i^+\) to \(D_i^+D_j^--C_{ij}\). The reordered term in (52) is at most \(k\|D^-v\|^2\), because \(\sum_i|a_iw|^2=k|w|^2\) for \(w\in\mathcal H_k\). The extra commutator is curvature: it acts as a sum of rotations of norm \(O(k)\) at degree \(k\), and the two contractions supply one more factor \(k\). This proves the claimed error bounds. Expanding the two squares proves (53). For complex functions all the displayed forms are understood through their real parts.

Substituting (51) into (50) rewrites that expression as \[ \begin{split} &(2+2/d)\left[d\|D^+v\|^2 -\Bigl\|\sum_i a_iD_i^+v\Bigr\|^2-dC[av]\right]\\ &\qquad+2\Bigl\|\sum_i a_iD_i^-v\Bigr\|^2 +O(k^3)\|v\|^2. \end{split} \tag{54}\] We must control the remaining negative divergence square while retaining both an ordinary derivative estimate and an additional half-order angular derivative estimate. The latter requires a coefficient of size \(k/\beta\) in front of \(|D^+v|^2\), including when \(k\) is much larger than \(\beta\). Choose a small constant \(\vartheta>0\). On the fraction \(1-\vartheta\) of its negative divergence square use (52)–(53). On the remaining fraction use the pointwise convex combination \[ \begin{split} \Bigl|\sum_i a_iD_i^+v\Bigr|^2 &\le(1-\zeta)(k+n-1)|D^+v|^2\\ &\quad+\zeta\left(2\Bigl|\sum_i a_iD_i^-v\Bigr|^2 +8k^3\beta^2|v|^2\right), \qquad \zeta=c_1/\beta. \end{split} \tag{55}\] The fraction \(1-\vartheta\) in the integrated estimates is constant; the \(y\)-dependent \(\zeta\) is used only in this pointwise inequality. The first bound follows from the adjoint multiplication operators, since the contraction \((q_i)\mapsto\sum_i a_iq_i\) has squared norm at most \(k+n-1\) on polynomials of degree \(k\). The second follows from \(D_i^++D_i^-=2k f_i\) and \(|\sum_i f_i a_i v|^2\le k\beta^2|v|^2\).

Combining these bounds in (54), the coefficient retained for \(|D^+v|^2\) is \[ (2+2/d)\bigl[d-k-\vartheta(n-1) +\vartheta\zeta(k+n-1)\bigr]. \tag{56}\] Since \(d-k=(n-2)/2\), a sufficiently small fixed \(\vartheta\) makes this at least a fixed positive constant plus a positive multiple of \(c_1k/\beta\). The coefficient of \(|\sum_i a_iD_i^-v|^2\) is nonnegative when \(\beta_0\) is sufficiently large. The Hessian contribution is exactly \[ -(2+2/d)\left[(d-1+\vartheta)C[av] +(1-\vartheta)k\int_Y\mathop{\mathrm{tr}}C\,|v|^2\,dV_\gamma\right]. \tag{57}\] For \(k\ge1\) and \(n\ge3\), both coefficients inside this expression are positive. The Hessian hypothesis bounds it below by \(c k^3\int\beta|v|^2\). This absorbs the curvature error \(O(k^3)\|v\|^2\) by increasing \(\beta_0\), and absorbs the last term of (55) by choosing \(c_1\) sufficiently small. We have therefore retained positive multiples of \[\|D^+v\|^2,\qquad \int_Y\frac{k}{\beta}|D^+v|^2\,dV_\gamma, \qquad k^3\int_Y\beta|v|^2\,dV_\gamma.\] The first controls both \(\|\nabla v\|^2\) and \(k^2\int\beta^2|v|^2\), since its cross term is \(-k\int\Delta_\gamma f\,|v|^2\ge0\). For the weighted derivative use the pointwise inequality \[\frac{k}{\beta}|\nabla v|^2 \le2\frac{k}{\beta}|D^+v|^2+2k^3\beta|v|^2.\] No differentiation of \(\beta^{-1}\) is required. Dividing by \(N_k\), and then summing the orthogonal degree components, proves (49); the weights \((1+k)^s\) are comparable to \(k^s\) because the zero degree is excluded. ◻

Proof. Let a dagger denote the full adjoint for \(dV_\gamma d\sigma\). Covariant integration and the degree shift give \[ Z:=T^\flat-T^\dagger=\sum_i f_iA_i. \tag{59}\] A direct expansion yields \[ \begin{split} &\|(2T+U)\varphi\|^2- \|(2T^\flat+U^\dagger)\varphi\|^2\\ &\qquad=4\bigl(\|T\varphi\|^2-\|T^\flat\varphi\|^2\bigr) +\langle\mathcal E\varphi,\varphi\rangle, \end{split} \tag{60}\] where \[ \mathcal E=2[T^\dagger,U]+2[U^\dagger,T]+[U^\dagger,U] -2\bigl(UZ+(UZ)^\dagger\bigr). \tag{61}\] We will prove \[ |\langle\mathcal E\varphi,\varphi\rangle| \le C\varepsilon\mathcal N(\varphi)^2. \tag{62}\]

Here are the complete order and weight counts needed for this claim. An operator has count \((m,s)\) if its coefficient at \(\partial_y^\alpha\) is angular order \(m-|\alpha|\), of size \(O(\beta^{s-|\alpha|})\), with the derivative allowances already specified. Adjoints preserve the count, and products add the counts. When a base derivative hits a coefficient it consumes one differential order and adds a factor \(\beta\), so it preserves the weight index. A commutator of scalar angular operators loses one angular order: the products of their principal symbols cancel. Hence a scalar commutator loses one in total order while preserving the sum of the weight indices. The counts of \(T,T^\dagger,U,Z\) are, respectively, \[(2,1),\quad(2,1),\quad(2,0),\quad(1,1),\] and every occurrence of \(U\) carries the factor \(\varepsilon\). Thus (61) has count at most \((3,1)\) and a factor \(O(\varepsilon)\); the quadratic \(U\) term is smaller.

There is a further cancellation that the total count alone does not express: no third base derivative survives. To check it explicitly, use an orthonormal sphere trivialization and write the base measure as \(\mu(y)\,dy\). The sphere probability measure is then fixed. Write \[U=a^{ij}\partial_i\partial_j+\mathcal B^i\partial_i+\mathcal C, \qquad a^{ij}=rQ^{ij}=a^{ji}\in\mathbb R.\] The coefficients \(\mathcal B^i\) and \(\mathcal C\) are angular operators. Replacing a horizontal derivative by a coordinate derivative plus its angular connection term changes only base orders at most one, so these are all the second-base coefficients. If a star denotes the sphere adjoint, direct integration by parts gives \[\begin{align*} U^\dagger-U ={}&\left(2\mu^{-1}\partial_j(\mu a^{ij}) -\mathcal B^i-(\mathcal B^i)^*\right)\partial_i\\ &+\mu^{-1}\partial_i\partial_j(\mu a^{ij}) -\mu^{-1}\partial_i\bigl(\mu(\mathcal B^i)^*\bigr) +\mathcal C^*-\mathcal C. \end{align*}\] In particular it has base order at most one. In \([T^\dagger,U]\) and its partner, the possible third-base-derivative coefficient is a commutator of an angular coefficient with \(rQ^{ij}\), which is zero by (47). For \([U^\dagger,U]\), first write it as \([U^\dagger-U,U]\). The operator \(U^\dagger-U\) has no second base derivative, since the coefficient in (47) is real. Taking adjoints against the smooth base density only produces lower base orders. Its remaining possible third-base-derivative coefficients commute for the same reason. Finally, \(UZ\) already has at most two base derivatives. It follows that \(\mathcal E\) is a sum of terms of precisely the following three permitted types: \[ \varepsilon\beta^{-1}\Psi^1\partial_y^2, \qquad \varepsilon\Psi^2\partial_y, \qquad \varepsilon\beta\Psi^3. \tag{63}\] This argument also covers lower orders, since \(\beta\ge1\).

For the first type in (63), integrate once in the base. The principal form is bounded by \(C\varepsilon\int\beta^{-1}\|\partial_y\varphi\|_{1/2}^2\). A derivative of its coefficient costs one factor \(\beta\) and leaves an \(\varepsilon\Psi^1\partial_y\) term. This includes differentiation of the weight itself: \(|d\beta|\le C\beta^2\) implies \(|d(\beta^{-1})|\le C\). The second type has the form bound \[C\varepsilon\int_Y \|\partial_y\varphi\|_{1/2}\|\varphi\|_{3/2}\,dV_\gamma \le C\varepsilon\int_Y\left( \beta^{-1}\|\partial_y\varphi\|_{1/2}^2 +\beta\|\varphi\|_{3/2}^2\right)dV_\gamma.\] The third type is bounded by \(C\varepsilon\int\beta\|\varphi\|_{3/2}^2\). The lower-order term from differentiating a coefficient satisfies the same bound. Replacing coordinate derivatives by horizontal derivatives adds a bounded angular first derivative, again controlled by these norms. A fixed finite partition of the base region has the same harmless lower-order errors. This proves (62).

The constants in the preceding estimates are uniform for all \(\beta\ge\beta_0\) once a fixed lower threshold is met. Choose \(\varepsilon_0\) small relative to the constant in Lemma 17. Dropping the nonnegative second squared norm in (60) gives \(\mathcal N(\varphi)\le C\|(2T+U)\varphi\|\). Finally, \[\|J\varphi\|\le C\|\varphi\|_{L^2(Y;H^1(S_y))} \le C\beta_0^{-1}\mathcal N(\varphi),\] so a sufficiently large \(\beta_0\) absorbs \(J\) and proves (58).

All these operations use angular symbol estimates pointwise in the base, and ordinary base differential operators of degree at most two. Formal adjoints, the commutator expansion and the one integration by parts use at most the five base coefficient derivatives specified in (46). Construction of these coefficients differentiates the moving trace operators once and the second metric to only finitely many orders below eight. Higher angular symbol seminorms depend on the fixed smooth first-metric family; differentiating in the angular variable does not differentiate the center-dependent tensor \(h\). Thus the eight smallness derivatives in (40) suffice in every fixed dimension. No smallness assumption on all derivatives of \(h\) is used. ◻

Proof. Set \(\varphi=\chi(\psi-1)\), which is compactly supported in the base and has zero fiber mean. For the actual moving-sphere operator \(P\), Equation (42) gives \[ P\varphi=P\bigl(\chi(\psi-1)\bigr) =-\chi P1+[P,\chi](\psi-1). \tag{66}\] The cutoff estimate (64) involves only angular derivatives through order one and \(r\nabla\psi\) on the supports of derivatives of \(\chi\). Proposition 15 bounds these there uniformly: on the lower region the density is that of ordinary evaluation, and on the other region the radii stay at a fixed positive separation. Together with the bound for \(P1\), this gives a uniformly bounded \(L^2\) norm for the right side of (66). Proposition 18 bounds \(\varphi\); adding the fixed function \(\chi\) proves (65). ◻

Proof. Fix a center \(y\) and subtract from \(v_{1,\lambda}^a\) its affine coordinate Taylor polynomial at \(y\). Exact constant and coordinate reproduction in (37) cancels this polynomial in \(T_yv_{1,\lambda}^a-v_{1,\lambda}^a(y)=V^a(y)\). The remainder on \(\partial B_\gamma(y,r)\) is bounded by \(Cr^2\), since coordinate distance and \(\gamma\)-distance are uniformly comparable and the functions have uniform second derivatives. Hence \[|V^a(y)|\leq C r_\delta(y)^2 \|\psi(y,\cdot)\|_{L^2(S_y)}.\] Lemma 19 proves (68). The coordinate volume measure and \(dV_\gamma\) are uniformly comparable. No derivative in \(y\) of the continued density is used here. ◻

Proof. Expand \(V(y+sz)\) at \(z=0\) through 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 is therefore 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 times the \(L^2\) norm on the unit ball. Its \(z^\alpha\) coefficient is \(s^j\partial^\alpha V(y)/\alpha!\), giving the claim. ◻

Proof. Put \(r=r_\delta(y)\) and \(s=r^\eta\). By Lemma 13, \(r_\delta^{1/p}\) has a fixed Lipschitz constant. Since \[\frac{s}{r^{1/p}}=r^{\eta-1/p}\longrightarrow0,\] a sufficiently small threshold \(\rho_0\) ensures \[\tfrac12 r^{1/p}\leq r_\delta(z)^{1/p} \leq\tfrac32 r^{1/p} \quad(z\in B(y,s)).\] The same fixed threshold ensures that this ball lies where \(\chi=1\), because \(K\) has a fixed buffer inside that set. The upper comparison and Lemma 20 imply \[\|V^a\|_{L^2(B(y,s))}\leq C r^2.\] Apply Lemma 21 and the fixed uniform \(C^m\) bounds of the renormalized functions. For \(j\leq10\) it gives \[|\partial^\alpha V^a(y)| \leq C_m\bigl(r^{2-\eta(j+n/2)}+r^{\eta(m-j)}\bigr) \leq C_m r^{3/2}.\] Both exponents are at least \(3/2\) by (72). Equation (70) completes the proof. ◻

Proof of Theorem 8. Make the linear normalization and spatial dilation of Section 4. All harmonic functions and their finite spanning family are chosen before the dilation. We now specify the remaining choices and hold each one fixed before making the next.

First choose the fixed conformal geometry, cylinder, depth function, and cutoff, keeping uniform background smooth bounds for \(0<\lambda\leq\lambda_0\). Choose \(p,\eta,m\) as in (72), with \(p\) large enough for the geometric and angular estimates. Choose \(\varepsilon\) for those estimates and then \(R_*\) so that the balls lie in normal neighborhoods and \(r_\delta\beta_\delta^2\leq\varepsilon\) for every \(\delta\). These choices fix every separation range needed to start the continuation and to bound the cutoff terms. Restrict \(\lambda_0\) further so that Proposition 12 applies to these fixed ranges, with bounds uniform for \(0<\lambda\leq\lambda_0\).

The constants \(C,\rho_0\) of Lemma 22 are now fixed. Choose \[ 0<\rho<\rho_0,\qquad C\rho^{1/2}\leq\varepsilon/2. \tag{74}\] At points with \(r_\delta<\rho\), that lemma improves the coefficient bound to \(H_8\leq\varepsilon r_\delta/2\) whenever the temporary bound holds. At points with \(r_\delta\geq\rho\), the same improvement will hold if \[ \sup_Y H_8\leq\varepsilon\rho/2. \tag{75}\] Since \(h\to0\) smoothly under dilation, choose one \(\lambda\leq\lambda_0\) satisfying (75) and the additional initial bounds described next. This \(\lambda\) is kept fixed for the rest of the proof.

There is a uniform reason the coefficient bound holds off \(K\) for every \(\delta\). Outside \(K\), either \(y_n<-1/2\), where \(h=0\) after the fixed dilation is sufficiently small, or \(t>2t_b\), where \[r_\delta\geq R_*(2t_b)^p>0.\] Smooth smallness of \(h\) therefore gives \(H_8\leq\varepsilon r_\delta/2\) off \(K\) for every \(\delta\) by reducing the same dilation once. At \(\delta=1\), the radius has a positive minimum on all of \(Y\), so we can also ensure the initial coefficient bound there. These conditions involve only fixed positive constants, and are compatible with (75).

For this fixed metric pair on the dilated neighborhood, set \[\mathcal A=\{\delta\in(0,1]: H_8(y)\leq\varepsilon r_\delta(y)\text{ for every }y\in K\}.\] It contains \(1\). At any \(\delta\in\mathcal A\), the bound already established off \(K\) gives the full hypothesis (32). Proposition 14 and the uniform density estimate of Lemma 19 then apply. The moment bound, metric system, and interpolation argument give Lemma 22. The small-radius improvement (74) and the large-radius bound (75) give \[H_8\leq\tfrac12\varepsilon r_\delta\quad\text{on }K.\] For every fixed positive \(\delta\), \(r_\delta\) has a positive minimum on the compact set \(K\). Thus \(\delta\mapsto\max_K(H_8/r_\delta)\) is continuous on \((0,1]\). The set \(\mathcal A\) is relatively closed, and the strict improvement makes it relatively open. Connectedness implies \(\mathcal A=(0,1]\).

Finally let \(\delta\downarrow0\), with the spatial dilation, harmonic family, cutoff and all structural parameters fixed. On the fixed open neighborhood of \(0\) where \(t<0\), the radii tend to zero. Hence \(h=0\) there. By (24), \(g_{2,\lambda}\) is conformal to \(g_{1,\lambda}\) on that neighborhood. Write \(g_{2,\lambda}=e^{2\omega}g_{1,\lambda}\). The conformal change formula for the Laplace–Beltrami operator is \[\Delta_{e^{2\omega}g}u =e^{-2\omega}\bigl(\Delta_g u +(n-2)\langle d\omega,du\rangle_g\bigr).\] Apply it to every common harmonic coordinate. Since their differentials form a basis and \(n-2\ne0\), it forces \(d\omega=0\). Since \(g_{1,\lambda}(0)=g_{2,\lambda}(0)=I\), this constant is zero. This is the local harmonic-morphism rigidity mechanism of (Fuglede 1978, sec. 8), in the restricted-family form used in (Lassas et al. 2020, Proposition 4.1). The metrics therefore agree on a connected neighborhood of \(0\); undoing the dilation proves the theorem. ◻

Proof. An interior isometry preserves lengths of curves and hence the intrinsic distances. The metric completion of the interior of a compact smooth Riemannian manifold with boundary is the manifold itself with its length metric. To see the only boundary issue, use a collar to push a boundary-touching curve an arbitrarily small distance into the interior; its length changes arbitrarily little. Compactness and local metric comparison then identify the completion with the original compact manifold. Thus \(\Phi\) and its inverse extend to mutually inverse homeomorphisms of the two completions. Boundary is mapped to boundary, since the interiors already correspond bijectively.

These extensions are smooth. In a sufficiently small uniform boundary collar, the distance \(s\) to the boundary is a smooth function, and its unit gradient is the inward normal flow. The isometry preserves \(s\) and, in the interior, intertwines its gradients. Fix a small positive level \(s=s_0\). The map is already smooth on that interior hypersurface. Expressing points with \(0\le s\le s_0\) by normal flow from this level expresses \(\Phi\) as the composition of this smooth level map with smooth normal flows on both sides. This extends smoothly to \(s=0\). The same argument applies to the inverse.

Since \(\Phi\) is the identity on \(E\), continuity makes it the identity on \(\overline E\). The original interiors satisfy \[M_j^\circ=(M_j^e)^\circ\setminus\overline E.\] The extended map preserves these complements, and hence their closures, the original manifolds. Its restriction is therefore a smooth diffeomorphism \(\Phi:M\to M\) with \(g_2=\Phi^*g_1\). It fixes the attached original patch \(P_0=\{b>0\}\) pointwise.

To prove the full boundary assertion, take any \(f\in C_c^\infty(\Gamma)\) and let \(v_f^j\) now denote the harmonic functions on the original manifolds with boundary value \(f\). On the second manifold both \(v_f^2\) and \(v_f^1\circ\Phi\) are \(g_2\)-harmonic. Their difference has zero boundary value on \(P_0\), where \(\Phi\) is the identity. It also has zero normal derivative there. In fact the equal energy maps give \[(\partial_{\nu_1}v_f^1)\,dS_{g_1} =(\partial_{\nu_2}v_f^2)\,dS_{g_2} \quad\text{on }P_0.\] The boundary densities are the same by Lemma 4, and the smooth isometry transports the outward unit normal. Boundary unique continuation, followed by interior continuation, therefore gives \[v_f^2=v_f^1\circ\Phi\quad\text{throughout }M.\] Taking boundary traces yields \[f(q)=f(\Phi(q))\qquad (q\in\partial M,\ f\in C_c^\infty(\Gamma)).\] For \(q\in\Gamma\), these test functions distinguish \(q\) from every other boundary point: choose a function equal to one at \(q\) with support in a small neighborhood avoiding the other point. Thus \(\Phi(q)=q\) for all \(q\in\Gamma\). ◻

Proof of Theorem 1. Lemma 4, Proposition 5, and Lemma 6 provide the common cap, Green kernels, and finite harmonic tests. Theorem 8 applies at every interior contact in Proposition 23. Proposition 24 gives the required diffeomorphism. All integrations use densities; the argument required neither an orientation nor connectedness of the boundary or of \(\Gamma\). ◻