Determination of a metric and a unitary connection from one boundary patch

Proof. Write g=ds^2+k(s,z), with k a tangential positive matrix, and put \rho=(k^{ab}\xi_a\xi_b)^{1/2}. Green’s formula identifies the measured operator with the density \nabla^A_\nu u\,dS_g. Relative to the coordinate density, its principal symbol is

Its square determines C=(\det k)k^{-1}. Since the tangential dimension is m=n-1,

Thus the boundary metrics and densities agree. Division by their common density gives equality of the ordinary covariant unit-normal DN operators locally on \Gamma. Localization on both sides by cutoffs supported in \Gamma gives equality of their full symbols.

We spell out the symbol calculation, extending the metric factorization of [16] to the present scalar-principal system. Recovery of a connection together with the metric by this even–odd symbol separation is also developed in [8, Theorem 1.1 and Section 3.3]; the calculation below uses complex Hermitian fibers. In normal gauge,

where P is the connection Laplacian on each tangential slice. With D_z=-i\partial_z, its degree-one symbol consists of the scalar metric term and -2ik^{ab}A_a\xi_b. Its degree-zero symbol uses only slice coefficients and their tangential derivatives. The local Dirichlet factorization has the form

where b\sim\rho I+b_0+b_{-1}+\cdots is a left symbol and

The positive root \rho I selects the decaying Dirichlet branch. The Poisson parametrix consequently identifies B(0), modulo a smoothing operator, with the outward normal derivative. This factorization is valid for smooth coefficients: its principal root is scalar and each subsequent matrix coefficient is solved by scalar division by 2\rho. The true inverse Dirichlet problem changes only the local smoothing remainder.

The part of b_0 involving the new jets is

This follows from \partial_s\rho=-(\partial_s k)(\xi^\sharp,\xi^\sharp)/(2\rho) and the degree-one factorization equation. At degree 1-j, j\ge2, the new normal derivatives arise only in \partial_s b_{2-j}/(2\rho). Inductively their contribution is

Every omitted term involves lower normal jets, with tangential derivatives. Differentiating \rho, a raised index, or a coefficient multiplying the preceding highest jet produces only such lower-jet expressions.

Subtract the two symbol recursions after their lower jets have been identified. The metric term in (2.3) is even in \xi, whereas the connection term is odd. The odd term determines all components of the new connection jet. If the even term vanishes for a symmetric tensor T, then T(v,v)=\operatorname{tr}_k(T) for every unit vector v. Polarization gives T=\operatorname{tr}_k(T)k, and taking trace gives (m-1)\operatorname{tr}_k(T)=0. As m\ge2, T=0. Beginning with the recovered boundary metric, this proves all normal jets by induction; their tangential derivatives recover all mixed jets. The remaining normal metric components and normal connection component are fixed by the coordinate and gauge choices.□

Proof. Choose a boundary chart compactly contained in \Gamma, and a nonnegative smooth bump b whose support is compact in that chart and whose positivity set P=\{b>0\} is nonempty and connected. Using the separate normal collars, move the boundary from s=0 to the graph s=-b(z), with b small enough to remain inside an extended collar. The common open cap is

Extend the first coefficient fields smoothly to negative s, retaining positive definiteness of the metric and skew-Hermitian values of the connection. Use the same exterior extension for the second pair. Lemma 2.1 makes the resulting pasting smooth to all orders on a neighborhood of \operatorname{supp}b. Restricting it to s\ge-b(z) gives smoothness also at the flat edges of the bump. The bundles remain trivial in the extended normal gauges.

For the energy comparison only, identify the extensions by the identity on the original M and by cap coordinates on E. In collar charts this map and its inverse are continuous and piecewise smooth with bounded first derivatives, and hence bi-Lipschitz. They identify the relevant H^1_0 spaces. Fix a vector source f compactly supported in E. The solution w_j=L_{j,D}^{-1}f is the unique minimizer of

Its trace on the original boundary is smooth and supported in \operatorname{supp}b\Subset\Gamma, because the original and extended boundaries coincide wherever b=0. Transport w_1 to N_2, and inside the original M replace it by the second harmonic extension of this trace. The replacing difference has zero trace on \partial M, so its zero extension is in H^1_0(N_2). The competitor is therefore admissible even at the narrowing edges of the cap.

On M, the source is zero and w_1 is the first harmonic extension of its trace. Equality of the measured quadratic forms makes its interior energy equal to that of the replacement. On E, both coefficients and the source pairing are unchanged. Thus \min\mathcal J_2\le\min\mathcal J_1. Reversing the argument gives equality, and uniqueness of the minimizer proves w_1=w_2 on E.

Normalize the Green matrix by

The source densities on E coincide. Varying the source gives equality of the distribution kernels on E\times E, hence pointwise equality off the diagonal. Self-adjointness of the Dirichlet inverse gives the adjoint symmetry in (2.4).□

Proof. The last assertion is smooth Dirichlet regularity. For the first two, a complex-linear functional on values or jets is represented by a dual-vector distribution T, supported at one or two points. Suppose it annihilates all source solutions, and define v=(L_D^{-1})^tT by distributional transposition. Then

Here the transpose and pairing are bilinear on the dual bundle; no complex conjugation convention is needed. Elliptic regularity makes v smooth off the support of T. The interior with one or two points removed is connected: a path can be diverted around each point in a punctured coordinate ball when n\ge2. Unique continuation from the nonempty set obtained by removing these points from U makes v supported at those points in the interior. If a point belongs to U, the asserted initial vanishing is distributional there as well.

At each support point a distribution is a finite sum of derivatives of the point mass. If its highest derivative order is k, its image under L^t has exact highest order k+2. Indeed, the highest homogeneous coefficient vector polynomial is multiplied by the scalar elliptic quadratic polynomial; this multiplication is injective. Consequently no nonzero order-zero distribution can be the image of a point-supported distribution. There are therefore no value annihilators, proving both surjectivity claims.

For a two-jet annihilator at p\notin\overline U, the image T has order at most two. Thus v=c\delta_p, with c a dual-fiber vector. Its image is precisely the functional u\mapsto c(Lu(p)). Conversely all such functionals annihilate the source solutions. Finite-dimensional duality gives the exact stated jet space. In particular, values and gradients may be prescribed freely, since the Hessian trace can be chosen to satisfy the equation.□

Proof. If two possible images of the same point were distinct, their evaluation identities would impose a fixed invertible linear relation between the evaluations at two distinct points of N_1^\circ. This contradicts joint surjectivity in Lemma 2.3. Once the image agrees, surjectivity at the source point determines J uniquely. The same argument using the evaluations in N_2 rules out two base points with the same image. Thus the maps glue on every overlap. Their union is connected because every domain contains E, and retains all the properties in (2.6).

For fixed p, the evaluation identity and the common source density give (2.7) when q\in U_0, away from the pole. As a function of q, the adjoint of each side solves the same pulled-back connection equation after the bundle identification. Injectivity of \Phi ensures that q=p is the only possible pole. Unique continuation on the connected set W\setminus\{p\} proves the identity everywhere claimed.□

Proof. Choose a small coordinate ball near an interior frontier and a point of W close enough to that frontier that its distance to the complement is attained inside the chart. The Euclidean ball with that center and radius lies in W and touches its complement at an interior point p. Its boundary gives the required smooth one-sided contact hypersurface.

If p_k\in W tends to p, compactness of N_1 and U(2), using the fixed global trivializations, gives a subsequence for which \Phi(p_k)\to q and J(p_k)\to J_0. Continuity of every source solution gives

The point q cannot lie on the boundary, where the right side would be zero, since I_2(p) is onto. Two distinct possible limits q contradict joint surjectivity on N_1; at a fixed q, surjectivity of I_2(p) determines J_0. The full limits therefore exist. Also q\notin E: otherwise its identity match on E and (2.9) would relate the evaluations at the distinct points q,p of N_2. Both p and q consequently avoid \overline{U_0}.

Choose two source solutions whose columns give an invertible matrix F_1(q). Their corresponding second-side columns give F_2(p), invertible by (2.9). Shrink the charts so both matrices are invertible. In a harmonic frame, v=F_j^{-1}w satisfies an equation with scalar elliptic principal part and no zeroth-order term, since every column of F_j is harmonic. Lemma 2.3 is invariant under this smooth frame change. It therefore supplies matrices X_1^l with value zero and gradients equal to any chosen real coordinate covectors times I. Choose those covectors independently, and let X_2^l be the corresponding source matrices expressed in F_2.

Set x^l=\frac12\operatorname{Re}\operatorname{tr}X_1^l and y^l=\frac12\operatorname{Re}\operatorname{tr}X_2^l. The first functions form coordinates at q. On matched points, the source identity implies JF_2=F_1\circ\Phi and X_2^l=X_1^l\circ\Phi. Thus the scalar coordinates and the metric Gram matrices of their differentials agree under the isometry. Passing to p,q, their Gram matrices agree and are positive definite. Hence the y^l are coordinates at p. Fix a target chart neighborhood U_1 on which the x^l are coordinates. The full convergence of partner points permits a source chart neighborhood U_2 so small that \Phi(W\cap U_2)\subset U_1. Choose a small coordinate ball \Omega about zero contained in both x(U_1) and y(U_2), and restrict the two charts to their inverse images of \Omega. For a matched point in the restricted source chart, x(\Phi(p'))=y(p')\in\Omega; its partner therefore lies in the restricted target chart. Thus the old map is coordinate identity throughout its domain in these charts. The matrix gradient identity at q now reads \partial_iX_1^l(0)=\delta_i^lI; equality on the one-sided region and smoothness give the same identity for X_2^l.

Because the harmonic-frame equation has no zeroth-order term, a jet with zero value and gradient is constrained only by g_1^{ij}(0)H_{ij}=0. The jet lemma supplies a finite basis of these Hessians and corresponding source pairs. They agree on the known region, so their complete jets agree at the contact point. The same conclusion holds for every smooth coefficient or paired function already identified there.

Finally polar-orthonormalize the original harmonic frames: write F_j=H_jS_j, with H_j unitary and S_j positive Hermitian. On the matched set, JF_2=F_1 implies S_2=S_1 and JH_2=H_1. In the unitary frames H_j, the old bundle identification is therefore I; connection matrices and Green matrices agree there. We continue to write F_j for the harmonic-frame matrices in these unitary frames. A common real linear coordinate change makes the metric at zero Euclidean and the contact hyperplane horizontal. Apply the same linear change to the list of X_j^l; this preserves all identities in (2.8). Reverse the final normal coordinate if necessary to put the known side below its graph. In these coordinates the Hessian constraint is the one in the statement.□

Proof. Let \mathcal H_1(D_1) be the closed subspace of L^2(D_1;V_1) consisting of distributional \mathsf L_1-solutions. Restriction R:\mathcal H_1(D_1)\to L^2(S_1;V_1) is compact by interior elliptic estimates and is injective by unique continuation. The spectral theorem for R^*R gives an orthonormal basis (w_j) of \mathcal H_1(D_1) such that

For every N there is a uniform constant C_N with

Indeed, multiply a solution by a cutoff equal to one near \overline{S_1} and compactly supported in D_1. Interior estimates bound its H^k norm by its L^2(D_1) norm. Fourier truncation in a containing cube, component by component, gives an approximation of rank O(A^n) and error O(A^{-k}). The minimax characterization of singular values yields \mu_j\le C_kj^{-k/n}. The fixed fiber dimension changes only the constants.

For y\in S_2, expansion in the first variable gives

Contracting the first tensor index against \overline{w_j} extends the coefficients to D_2:

Here F_\alpha(x,y)\in V_2 are the first-index components in an orthonormal basis of V_1; the displayed contraction leaves a vector in V_2. Because the second system acts on that remaining index, \mathsf L_2b_j=0. Restriction orthogonality proves that this formula agrees with the original coefficient on S_2. The assumed interpolation gives

Thus the tensor series converges absolutely in L^2(O_1\times O_2): the norm of its jth term is at most CM\mu_j^{\gamma_1+\gamma_2-1}, and (3.3) makes these bounds summable. The limit solves both equations distributionally. Interior estimates for their elliptic sum give smoothness and the claimed bounds.

Agreement on the second arm requires a completeness check. Take z\in L^2(S_1;V_1) orthogonal to the closure of R\mathcal H_1(D_1). The V_2-valued function

solves the second system and vanishes on S_2, hence vanishes on connected D_2. Every second-arm slice therefore belongs to the closed restricted range. Its expansion in the orthonormal basis w_j/\mu_j is complete, with coefficients \mu_jb_j(y). Consequently the series equals F on S_1\times O_2; it converges there by the positive exponent \gamma_2. On O_1\times S_2 it is the original expansion, converging with exponent \gamma_1. These establish both required agreements. Only upper bounds for \mu_j have been used, so uniformity does not require continuous choices of singular bases or lower bounds on 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 3.1 to apply.

Splitting the weight between the factors. We seek symmetric forms P_i satisfying

for some K>0, with the vectors v_i as in (3.7). After division by |a_i|^2, the first inequality is the strict-weight condition for a function with gradient \nabla_i\rho and Hessian P_i. The second inequality will give the quadratic gap. To obtain these forms by convex separation, the dual tests are positive semidefinite forms D whose diagonal blocks are

This follows by separating the open convex cone consisting of positive definite forms plus the allowed P_1\oplus P_2; the individual dual cones are generated by a_i^{\otimes2}+v_i^{\otimes2}. Normalize \operatorname{tr} D=1. To obtain a suitable K, it suffices by compactness to prove H:D>0 on tests with D\,d\rho=0.

Represent such a D as a covariance matrix. 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

The case d=0 would force D=0. Among nonnegative tangential forms with these two fixed traces, every extreme point has rank one: on an image of rank r\ge2, a nonzero symmetric perturbation preserves the two traces because r(r+1)/2>2, and small perturbations of both signs preserve positivity. The rank-one tests are precisely those in (3.7), multiplied by d. The asserted positivity follows. Compactness of the normalized tests then supplies K, as required.

Prescribe partial covectors d_x\rho,d_y\rho and Hessians P_i for smooth functions \theta_i vanishing at the respective points. In small coordinates centered there, Taylor expansion gives

with c>0. Each \theta_i satisfies (3.5). The positive side of \widetilde\rho is the positive side of \rho in this neighborhood.

Placing the cross and obtaining interpolation. Use coordinates (s_i,z_i) with s_i=\theta_i. Fix a small \kappa>0 so that \phi(s_i)=s_i-\kappa s_i^2 remains a strict weight. Choose successively a small lateral radius R, a height l\ll cR^2, and e\ll\kappa l^2. Take

Use a cutoff equal to one on the inner lateral half and for -l+3e<s_i<l+2e, supported in the cylinder and in -l+2e<s_i<l+3e. Let S_i\Subset D_i include neighborhoods of its top and lateral derivative supports, where respectively

and an inner top slice near s_i=l+5e/4. The two closed product sets \overline{D_1\times S_2} and \overline{S_1\times D_2} lie strictly in the given side. For a top slice, the sum of the two s coordinates is positive. For a lateral slice, it is greater than -2l, which is dominated by the quadratic gap in (3.8). Making the sets slightly smaller if necessary gives a common \varepsilon>0 with \rho>\varepsilon on their closures.

For an \mathsf L_i-solution normalized by \|w\|_{L^2(D_i)}=1, apply (3.6) to this cutoff times w. Interior estimates bound the bottom derivative terms using the large norm and the weight \phi(-l+3e). The remaining derivative terms use \|w\|_{L^2(S_i)} and a weight at most \phi(l+3e). On a smaller inner cylinder beginning at s_i=-e, the weight is at least \phi(-e). Balancing the two exponentials yields (3.1) with

Indeed at e=0 the quotient is 1/2+\kappa l/2. Values of the balancing parameter below \tau_0 are covered by increasing the constant. The output cylinder may be chosen to include a connected tube from the origin to the indicated top slice.

Constructing and identifying the extension. Lemma 3.1 now gives a product solution on a neighborhood of z_0. It agrees with the old solution there on the positive side: from a sufficiently near positive-side point, increase s_2 into the top slice. Since \partial_{s_2}\rho>0 locally, the segment stays on that side and in the larger constructed cylinder. Agreement on the slice and unique continuation along a neighborhood of the segment prove agreement at the original point. All choices have strict margins. Keeping their finitely many compact sets and shrinking their output neighborhoods once proves the stated perturbation uniformity by (3.6), Lemma 3.1 and interior elliptic estimates.□

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 3.2 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 \mathsf L_1^x+\mathsf 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. For the final comparison with the full original cross, fix at the outset

We construct the continuation for these enlarged location bounds and smaller separations, and then restrict to the stated target. We first cross the limiting plane, then deform variable-radius tubes to reach those separations. This follows the two-stage construction of [18, Section 3, “Fixed-separation initialization”]. We first choose the Euclidean geometric ratios, including the opening of the region reached in the first stage. We next choose the entire tube cylinder and a bounded positive-separation range containing it. The first-stage constructions are then made on that range. Only after all these fixed sets have been chosen do we decrease \lambda.

Crossing the plane and obtaining a common seed. We first extend slightly across the plane in one variable while keeping the other at a comparable positive separation. The first interpolation exponent can be made close to one; this compensates for the fixed positive exponent obtained by propagation in the separated second variable.

For the Euclidean equations, the following choices are invariant under translations parallel to the plane and under a common change of spatial scale. We make the geometric choices on a unit-scale template. They therefore give an opening constant c_0>0 independent of the bounded range on which the construction is subsequently used. On any fixed compact range of centers and scales 0<s_{\min}\le s\le s_{\max}, the same strict weights and geometric fractions work for the dilated systems once \lambda is sufficiently small. The lower-order coefficients are bounded uniformly there (and converge to zero under the connection dilation).

Fix a point x_0 of height zero and a separation scale s>0. In the second factor take

and an observed ball S_2 about x_0-se_n. In the first take a ball of radius 3\vartheta_0s centered at x_0-\vartheta_0se_n, with \vartheta_0>0 fixed small enough that the two large domains are separated. Its observed ball has radius (1-\alpha)\vartheta_0s, with \alpha>0 to be chosen. These observed balls lie strictly in the lower half-space. On a fixed compact scale range their heights are bounded above by a common negative number, since s\ge s_{\min}>0. Thus, after a finite covering, one choice of small \lambda places every observed ball in W_\lambda with room. No neighborhood of the whole limiting plane is assumed to belong to W_\lambda.

There is a two-constants estimate from S_2 to a slightly enlarged closed annulus 4s/5\le|y-x_0|\le6s/5, with an exponent \gamma_2>0 independent of \alpha. Here is a direct way to obtain it with room for perturbations. On a Euclidean annulus the weight |z-z_*|^{-1} is strict: its radial Hessian is 2|z-z_*|^{-3} and each tangential Hessian value is -|z-z_*|^{-3}. Radial cutoffs in (3.6), followed by exponential balancing, propagate smallness from an inner ball to an intermediate ball using the norm on an outer ball. A finite chain of overlapping balls with enlarged balls inside D_2 connects S_2 to the target annulus. The annulus is connected since n\ge3. Multiplying the finitely many positive exponents gives the asserted \gamma_2. The same weights and margins work for sufficiently small perturbations of the Euclidean metric.

In the first factor use this radial argument centered at x_0-\vartheta_0se_n. Keep the outer cutoff a fixed distance beyond radius \vartheta_0s, and put the inner cutoff just inside radius (1-\alpha)\vartheta_0s. Take a concentric target ball of radius (\vartheta_0+c)s, with c>0 small; it contains the observed ball and a ball of radius cs about x_0. To see the resulting exponent, let R_{\rm in} be an inner cutoff radius, chosen just inside the observed radius, let R_{\rm tar}=(\vartheta_0+c)s, and let R_{\rm out}>R_{\rm tar} be a fixed starting radius for the outer derivative support. With \theta(R)=R^{-1}, exponential balancing gives

The inner core is bounded directly by the observed norm. As \alpha and c decrease, choose the inner cutoff so that both R_{\rm in} and R_{\rm tar} approach \vartheta_0s, while R_{\rm out} stays a fixed distance beyond that radius. Thus \gamma_1\to1. Choose these constants so that \gamma_1+\gamma_2>1. Lemma 3.1 extends the kernel to a small ball about x_0 times the target annulus. On these fixed separated sets the rescaled Green kernels and their derivatives are uniformly bounded: the same bounds hold entrywise for the smooth systems used here. To see this, an interior inverse parametrix has order -2 and frequency pieces bounded by C_N2^{j(n-2)}(1+2^j|x-y|)^{-N}, up to a smooth kernel. Summing gives the local bound C|x-y|^{2-n} for n\ge3. The normalization in (3.10) preserves this bound; interior estimates for the uniformly elliptic dilated equations give all fixed derivative bounds away from the pole.

The extensions agree with both lower pieces on their actual comparison domains. For y\in S_2, the series equals G_{1,\lambda} for all x in its first output ball. Fix any such x in a truncated lower half-space contained in W_\lambda. On S_2 both variables then lie in W_\lambda, so this value also equals G_{2,\lambda}. Unique continuation for the second system on the connected output annulus proves agreement with G_{2,\lambda} throughout that annulus. In the other direction, fix y in the output annulus and in W_\lambda. On the first observed ball S_1, agreement with the second arm gives G_{2,\lambda}=G_{1,\lambda}. Unique continuation for the first system on its connected output ball gives agreement with G_{1,\lambda} there. Each comparison stays a positive distance from the diagonal.

We make the overlap comparison precise because the two variables may play different roles in two local constructions. Inside the output just obtained choose a larger comparison product of the form

where \kappa,\zeta>0 are fixed geometric fractions. The first ball is contained in the actual concentric first output ball, and the annulus is contained in the actual second output. Keep the full original outputs for the preceding lower-piece comparisons. For coverage retain only

The transposed products are used when y is the near-plane variable. If a point belongs to two retained products, their scales are comparable to each other and to |x-y|, with fixed constants. Choose \epsilon_1 small enough, relative to \kappa and \zeta-\zeta', that a downward translation of size at most a fixed multiple of \epsilon_1 times either scale stays inside each larger comparison product whenever it acts on a retained near-plane variable. This follows directly from the triangle inequality for the ball and annular distances. On the chosen compact scale range take \eta_0>0 much smaller than the remaining margins times s_{\min}, and with \eta_0\le\eta.

For products of the same orientation, lower only their common near-plane variable by

(or the analogous expression in y). The other variable remains fixed. For products of opposite orientations, both x and y lie in retained near-plane balls; lower both by

The scale comparability and the preceding choices keep the entire path in the intersection of the two larger comparison products. Its endpoint has the near-plane variable below -\eta_0 in the first case, and both variables below -\eta_0 in the second. For small enough \lambda these endpoint neighborhoods lie in the original lower cross, where the preceding comparisons identify both germs. Unique continuation for the elliptic sum on the component of the larger intersection containing the path identifies the germs at the starting point. The path need not remain in the retained products. This proves agreement on every retained overlap without a connectedness assumption on that overlap.

Apply the same construction with the variable roles and their tensor factors interchanged, and then return the values to the original tensor order. The resulting germs solve the same two equations as the first orientation. The overlap comparison therefore applies to them and produces, for some fixed c_0>0, a single-valued continuation in

on every fixed compact range of bounded locations and positive separations. The estimates are uniform in \lambda on that range. For coverage near the plane, take x_0=(x',0) and s=|y-x_0| when the first variable has the smaller nonnegative height; then x is in the retained first ball and y in the retained annulus if c_0 is small; for instance c_0/(1-c_0)<\epsilon_1 guarantees the required ball inclusion. Points already below the plane are covered by the same construction near it or by the given cross farther below it. The preceding overlap argument makes these choices compatible.

Sweeping tubes inward from the common seed. The region (3.12) is now the seed for the second stage. At fixed center y and direction \omega, we move x along the ray from y: large radii lie in the seed, and decreasing the radius will reach the target pairs. Choose M>\widehat B+3 and introduce

where \sigma,e_0>0 will be small. The base is

and a decreases through a fixed interval [a_{\min},a_{\max}]. The term e_0|y'|^2 makes this base compact. The parameter a decreases while (y,\omega) remains fixed, as required by the compact-cylinder propagation lemma. Put m=M-\widehat B-1>2. Reserve the bounds 0<\sigma\le1, e_0\le(1+\widehat B^2)^{-1}, and e_0T_0\le1/4. Once \sigma is chosen below, take

and then choose

For every target center, t\le M+\widehat B+1; hence its target separations exceed r_{a_{\min}}(y). On the initial level, c_0r_{a_{\max}}>2t>y_n, so this level lies in (3.12). The bottom base boundary has y_n<0, and on the face t=T_0 we have c_0r_a>2t>y_n for all allowed a. Thus the initial level and both base boundary faces are supplied by the seed. The power 1+\sigma>1 is what permits this last inequality at the top face. This known region is upward closed in a: since \min(x_n,y_n)=y_n+r\min(\omega_n,0), its defining inequality is y_n<(c_0-\min(\omega_n,0))r, whose radius coefficient is positive.

It remains to check the strict extension criterion outside that region. At the Euclidean metrics use

Away from \omega+b=0, multiply the vectors in (3.7) by the common positive factor r_a and write their real and imaginary parts as

The identity \omega\cdot(U_1-U_2)=b\cdot U_2 cancels the normal part of the distance Hessian. Consequently the test is

For bounded b, outside fixed small neighborhoods of b=-\omega and b=-2\omega, the first term is at least c|b|^2. To see this, a zero forces the real part of U_2 to be parallel to \omega, and equality of the projected imaginary lengths gives |\omega+b|=1. Thus b=0 or b=-2\omega. Near b=0, the real projected difference controls |\pi_{\omega^\perp}b| to first order; also

so the imaginary projected difference controls |\omega\cdot b| to first order. This proves the quadratic lower bound near zero; compactness proves it on the remaining range.

Outside (3.12), y_n\ge c_0r_a, so r_a/t\le1/c_0 and

has |b|\le4/c_0 under the reserved bounds. This fixes a compact range for b before \sigma, the a-interval, T_0, and e_0 receive their final values. When \nabla t is close to e_n, the two excluded neighborhoods already lie in (3.12). Indeed,

which is arbitrarily small near b=-\omega and negative near b=-2\omega as \sigma and the gradient error tend to zero. On the complement |U_2| is bounded and differentiation of (3.13) gives

Here the choices can be made in a definite order. First choose a neighborhood radius \delta_0>0 for b=-\omega and b=-2\omega, in terms of c_0, and upper bounds on \sigma and |\nabla t-e_n| so that their closures satisfy x_n/r_a<c_0/2. On the complementary bounded b-range let c>0 be the constant in the quadratic lower bound. There |U_2|^2\le2/\delta_0^2. Direct differentiation sharpens (3.16) to

Choose \sigma>0 small enough that its contribution to this bound is less than c\delta_0^2/8. Choose the a-interval and T_0 as above, and finally take e_0>0 sufficiently small that all reserved bounds hold, |\nabla t-e_n|\le2\sqrt{e_0T_0} has the required smallness, and \sigma+2e_0T_0<c\delta_0^2/4. The second term of (3.15) is then at most (c/2)|b|^2 in absolute value. The whole test is bounded below by

where continuation is needed. Also the two partial gradients of \rho have lengths at least r_a and \delta_0r_a, respectively. This supplies fixed strict margins before any metric perturbation.

All these sets are bounded and stay at separation at least a_{\min}(M-\widehat B-1)^{1+\sigma}>0. Smooth convergence of the metrics therefore preserves the strict tests for sufficiently small \lambda. At this point the tube parameters and its compact range have all been fixed. Apply the first-stage construction on a slightly larger bounded range of locations and separations containing this cylinder, and then choose \lambda small enough for both its data and the strict tube tests. Lemma 3.3 continues the kernel through the cylinder, with uniform estimates and agreement on the seed. A target pair has a=|x-y|/t^{1+\sigma}\ge a_{\min}; if a>a_{\max} it is already in the coarse region. Hence every target pair is covered.

It remains to identify this extension with both pieces of the full cross (3.11), including parts of W_\lambda not in a truncated half-space. For |y|\le B put

The excluded closed ball lies strictly inside B(0,\widehat B), so D_y is connected; it contains the fixed lower ball A, which is disjoint from every excluded ball and lies below x_n=-\eta. For fixed y\in W_\lambda in the target range, the constructed kernel and G_{1,\lambda} solve the same first-variable system on D_y. On A the constructed kernel equals G_{2,\lambda}, which equals G_{1,\lambda} because both variables are in W_\lambda. Unique continuation on D_y proves agreement on the target overlap. Interchanging the variables proves agreement with the other piece. The inequalities defining \widehat B and \widehat d leave open room around the original target, so this also gives the claimed neighborhood and its compact-subset estimates.□

Proof. Write a_\delta=(\log\ell_\delta)'. With s=t/\delta,

The function A is positive and decreasing: setting u=e^s gives

At the negative end A tends to one, and at the positive end it is comparable to (1+s)^{-1}. Differentiating its convergent expressions at the two ends gives |A^{(j)}|\le C_jA^{j+1}; positivity gives the same bounds on the remaining compact interval. Consequently, on the fixed range of t,

Since |dt|_\gamma is bounded above and below, \beta=p_*a_\delta|dt|_\gamma. The identity

and (4.5) prove the first two bounds in (4.7). The chain rule gives the derivative bounds; their constants can be independent of p_* because p_*a_\delta^j\le p_*^j a_\delta^j for j\ge1.

The estimate 0<\ell_\delta'<1 proves the Lipschitz assertion. Moreover,

The displayed expression tends to zero at both ends and its derivative in s is -s e^s/(1+e^s)^2, so it is strictly positive at finite s. The limit of \ell_\delta is \max(t,0). Finally,

and \ell_\delta is uniformly bounded on the fixed depth interval. This proves (4.8).□

Proof. Put b=\nabla^\gamma r, so that |b|=r\beta. Parallel transport along the radial \gamma-geodesic identifies the endpoint tangent spaces; let \omega be its unit direction from y to x. Multiply the normalized vectors in Proposition 3.2 by the common factor r. Their real and imaginary parts can be written as

All lengths in the following calculation use \gamma. Set s=\omega+b. In a \gamma-orthonormal frame at y, the exact conditions on the second vector are

Here the matrix of a is the identity. Therefore

The assumptions make b and h small. Both partial gradients of \rho are consequently nonzero, and |U_2| is bounded above and below.

With the product \gamma-connection, the endpoint Hessian of half the squared distance has value

To obtain the error, parametrize the radial geodesic on [0,1]. In a parallel frame its Jacobi field with prescribed endpoint values differs from the affine interpolant by O(r^2) in C^1, because the curvature is bounded and the velocity has size r. The second variation formula gives (4.15). Replacing the second connection by that of Q^{-1} contributes only O(r^2): its difference from the \gamma-connection is O(|h|+|\partial h|)=O(\varepsilon r), while |d\rho|=O(r).

The relation s\cdot U_2=1 implies \omega\cdot(U_1-U_2)=b\cdot U_2. Also

Thus the Hessian sum in the criterion, multiplied by r^2, equals

For a real bilinear form, evaluation on a complex vector and its conjugate means the sum of the evaluations on its real and imaginary parts, as in the criterion.

We record the cancellation quantitatively. The real projected difference is -\pi_{\omega^\perp}b+O(|b|^2+|h|). The imaginary part obeys

Comparing this length with |v|=1 and squaring gives

The real and imaginary directions of U_2 are almost orthonormal; hence

Using |b|=r\beta, |h|\le\varepsilon r, and (4.7), expression (4.16) is bounded below by

Divide the error terms by r^2\beta. Their ratios are bounded by C/\beta, Cr\beta^2, C\varepsilon, and C\varepsilon^2/\beta, respectively. Increasing p_* controls the first, and then decreasing \varepsilon controls the others by (4.12). This proves strict positivity.□

Proof. Deform \eta from 1 down to \delta, using (y,\omega) in the unit sphere bundle of \gamma as base coordinates and

The base over the closed cylinder Y is compact. The radius has strictly positive derivative in \eta, so these variables give cylinder coordinates on every fixed interval \delta\le\eta\le1. The positive side of a level is the direction of increasing \eta. Since r_\eta\ge r_\delta, condition (4.11) holds at every intermediate radius. Lemma 4.3 applies there.

At \eta=1 the separations have a positive minimum on Y. The same is true throughout t>t_b by (4.10). Proposition 3.4 therefore supplies the top and positive-depth data, with open room and uniform bounds at any fixed order. In the lower strip y_n<-1/4, use the original kernel G_1; this strip lies in the dilated known side for small \lambda. The initialization agrees with this kernel on their overlap.

These two base regions cover every lateral face of Y: its bottom face is in the lower strip, and on a remaining lateral or upper face outside that strip one has t_0\ge3/4. Both regions are independent of \eta and hence upward closed. Together with a neighborhood of the top level they supply exactly the seed required by Lemma 3.3. That lemma and the strict sphere test extend the kernel through the whole cylinder, preserving agreement on the seed. Normal-neighborhood room and positive radii on each fixed interval ensure that all surfaces remain off the diagonal.□

Proof. The Dirichlet problem on each ball is invertible. In the unitary frame this follows from the positive connection energy: a zero-boundary null solution is parallel and hence zero. Invertibility is preserved by the harmonic-frame change and by positive scalar multiplication of the equation. Its Dirichlet solution operator determines the boundary covariant normal derivative from the boundary value. Transpose this boundary operator onto the smooth matrix coefficient in the first term of (4.18), then multiply by the frame factors and pull the surface density back under \omega\mapsto\exp_y^\gamma(r(y)\omega). This gives the smooth matrix density in (4.20). Smooth dependence of the ball Dirichlet problem on its center and positive radius gives smooth dependence of \psi.

To prove the local harmonicity assertion, fix y_0 and u=F_1v. The kernel is defined on an open neighborhood of the compact sphere over y_0. Choose a fixed slightly larger surrounding surface inside that neighborhood and inside the domain of u. After restricting the center neighborhood, this surface surrounds all the moving spheres and the intervening collars remain in the common domain. Green’s identity on each collar replaces (4.18) by its integral over the fixed surface. Both first-variable equations are homogeneous in the collar. Applying the second-variable operator under the fixed integral now proves that S(y;u) is second-system harmonic: the columns of \mathcal G(x,y)^* solve that system. Equation (4.19) proves the assertion in harmonic frames. This argument applies to arbitrary local harmonic inputs; they need not extend throughout the coordinate region.

If y_n<-1/4, the kernel is G_1 and the Green representation formula gives S(y;u)=u(y). On that region F_1=F_2, so T_yv=v(y). For a fixed pair (v_1,v_2), both T_yv_1 and v_2(y) solve the second harmonic-frame equation on connected Y^\circ and agree in the lower strip. Unique continuation proves their equality. Finally constant columns are harmonic in both harmonic frames and agree there, so the same argument yields the first identity in (4.21).□

Proof. On the lower cutoff region, T_y is evaluation at the center in the harmonic frame. Normalize its equation to principal matrix a=\gamma^{-1}, use a smooth \gamma-orthonormal frame to pull it back under z\mapsto\exp_y^\gamma(rz), and multiply by r^2. With center and radius regarded as independent parameters, the resulting Dirichlet systems extend smoothly to r=0, are uniformly elliptic, and have the Euclidean Laplacian as limit; their lower-order coefficients vanish at r=0. The Dirichlet inverses are uniformly bounded, either by their limiting inverse and perturbation or by the positive energy before the change of frame. Elliptic boundary regularity, applied after differentiation in the parameters, shows that the evaluation density at the center has uniform angular smooth bounds and smooth center–radius dependence. The center remains a fixed positive distance from the unit sphere in these coordinates. Substituting the varying radius costs only |dr|_\gamma=r\beta, which is uniformly bounded by (4.12) and the lower bound for \beta.

This evaluation density is the density constructed in Proposition 4.5. Indeed both represent the same functional on traces of functions harmonic near the closed ball. These traces are dense in smooth boundary data. For a prescribed smooth trace \varphi(\omega), solve on the ball of radius r(1+\eta) with boundary value \varphi transported along the radial rays. On pulling this problem back to the closed unit ball, its solution z_\eta converges in every C^k norm to the solution z_0 at radius r, by smooth dependence of the Dirichlet inverse and boundary regularity. Its trace on the original sphere is z_\eta(\omega/(1+\eta)), which converges in C^k(S_y) to z_0(\omega)=\varphi(\omega). The corresponding physical solutions are harmonic near the original closed ball. Thus the two smooth density matrices agree.

On the other cutoff region t>t_b, the radii have the positive lower bound (4.10); all their required derivatives are uniformly bounded in \delta. Proposition 3.4 gives uniform kernel bounds there. The smooth Dirichlet construction of the density therefore gives the same bounds for \psi_\delta and its first base derivatives. Compactness of the cutoff derivative support completes the proof.□

Proof. First regard the radius s as an independent parameter. Pull the first scalar equation back by z\mapsto\exp_y^\gamma(sz) and multiply by s^2. Its principal coefficients are \delta^{ij}+O(s^2) and its first-order coefficients are O(s). The same statement holds for the system, with scalar principal coefficients times I. Both Dirichlet operators extend smoothly to s=0, where they equal the Euclidean Laplacian. On each fixed Sobolev scale their Dirichlet inverses are uniformly bounded and smooth for sufficiently small s, by elliptic Dirichlet estimates and invertibility at zero.

For these smooth strongly elliptic Dirichlet families on the unit ball, the boundary radial derivative operator is classical of order one. Its principal symbol depends only on the principal coefficients and the differentiating vector. The boundary symbol construction, obtained by choosing the decaying normal root and solving the successive transport equations, is smooth in (y,s); see, for example, Joshi and Lionheart [12, Section 3, Proposition 3.1]. Their factorization is for the Hodge Laplacian; its construction applies to the present Dirichlet systems because their principal symbol is scalar times I, while the transport equations allow matrix coefficients. The true Dirichlet inverse corrects the parametrix by a smoothing operator. Differentiating this inverse on arbitrarily high fixed Sobolev scales bounds every required smooth kernel seminorm of the correction. Thus these assertions hold in the full symbol topology, including smoothing remainders and parameter derivatives.

At s=0 the radial operator is \mathsf B, and the first s-derivative of its principal symbol vanishes because the principal coefficients have no linear term in s. Taylor expansion in the symbol topology therefore gives, for the scalar equation,

The system and scalar radial operators have identical principal symbols for every s and coincide at zero. Their difference is consequently s times a smooth family in \Psi^0.

Differentiating the center of the exponential map while transporting \omega parallel gives a Jacobi field with initial derivative zero. In the unit-ball coordinates its velocity is

putting s=r(y) adds f_i\omega. Tangential derivatives act directly on boundary data and are identical for the scalar and matrix equations. For the radial component use (5.8). The leading terms are r^{-1}\mathsf A_i+f_i\mathsf B; the order-one errors have size O(r) and the order-zero errors are bounded, since r\beta\le\varepsilon/\beta. This proves (5.6) after sphere adjunction. The radial difference r\Psi^0, multiplied by the center velocity O(r^{-1}+\beta), proves (5.7). Base derivatives cost at most one factor \beta each by the profile bounds. Sphere transposition and commutation with the smooth angular connection fields preserve these orders and bounds.□

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

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 [20, 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

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

Changing from Fischer adjoints to sphere adjoints gives

where \Pi_{k+1} is harmonic projection in homogeneous degree k+1. Consequently N_k times the left side of (5.16) is

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

Covariant integrations by parts give

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 (5.19) 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 (5.20). For complex functions all the displayed forms are understood through their real parts.

Substituting (5.18) into (5.17) rewrites that expression as

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 (5.19)–(5.20). On the remaining fraction use the pointwise convex combination

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 (5.21), the coefficient retained for |D^+v|^2 is

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

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 (5.22) by choosing c_1 sufficiently small. We have therefore retained positive multiples of

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

No differentiation of \beta^{-1} is required. Dividing by N_k, and then summing the orthogonal degree components, proves (5.16); 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

A direct expansion yields

where

We will prove

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,\mathcal U,Z are, respectively,

and every occurrence of \mathcal U carries the factor \varepsilon. Thus (5.28) has count at most (3,1) and a factor O(\varepsilon); the quadratic \mathcal 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

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

In particular it has base order at most one. In [T^\dagger,\mathcal U] and its partner, the possible third-base-derivative coefficient is a commutator of an angular coefficient with rQ^{ij}, which is zero by (5.14). For [\mathcal U^\dagger,\mathcal U], first write it as [\mathcal U^\dagger-\mathcal U,\mathcal U]. The operator \mathcal U^\dagger-\mathcal U has no second base derivative, since the coefficient in (5.14) 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, \mathcal U Z already has at most two base derivatives. It follows that \mathcal E is a sum of terms of precisely the following three permitted types:

This argument also covers lower orders, since \beta\ge1.

For the first type in (5.30), 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

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 (5.29).

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 5.2. Dropping the nonnegative second squared norm in (5.27) gives \mathcal N(\varphi)\le C\|(2T+\mathcal U)\varphi\|. Finally,

so a sufficiently large \beta_0 absorbs \mathcal J and proves (5.25).

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 (5.13). Forming these coefficients uses at most one base derivative of a trace operator; six trace derivatives therefore suffice. Since its weight dependence is through r and f_i, profile derivatives through order seven suffice, within the eight allowed in (5.2). Differentiating a factor such as h/r five times uses only five derivatives of h, with the remaining factors bounded by the permitted powers of \beta. Higher angular symbol seminorms depend on the fixed smooth first-metric family: the tensor h is evaluated at the center, so angular differentiation introduces no further center derivatives of it. The scalar drifts have fixed smooth background bounds. Thus the eight smallness derivatives in (5.3) suffice in every fixed dimension; no smallness of all derivatives of h is used.□

Proof. Put q^i=b_2^i-b_{2,0}^iI and use D_i=D_{i,0}+K_i from (5.7). Let L_M denote left multiplication by a matrix M. Expansion, without interchanging any factors, gives the identity

Thus D_{i,0}K_j denotes composition and includes differentiation of K_j. There is no derivative of q^i or b_2^i in this formula. Left multiplication commutes with the row action of a sphere operator when its matrix depends only on the center, but no such commutation is needed in (5.31).

The trace expansion gives \mathcal C_{i,0}^*\in(r^{-1}+\beta+1)\Psi^1. Together with K_i\in\Psi^0, \nabla K_i\in\beta\Psi^0, and bounded drifts, (5.31) consequently yields

To see the weights explicitly, every term with a derivative of \varphi in the base is r\Psi^0\nabla\varphi. Products of K_i with the leading r^{-1}\mathsf A_j^* leave a bounded \Psi^1 term after multiplication by r; products with f_j\mathsf B have coefficient O(r\beta). Differentiating K_j gives r\beta\Psi^0. The remaining products have smaller orders and bounded coefficients. Since r\beta\le \varepsilon/\beta, all constants in (5.32) are uniform as the lower threshold \beta_0 increases.

Apply Proposition 5.3 to the four entries of \varphi. By the definition of \mathcal N,

It follows that

Increase the fixed lower threshold so that the last coefficient is at most 1/2. This proves the assertion. In the parameter choices for the construction, this increase is made when p_* is chosen, before \varepsilon and the radius amplitude are fixed.□

Proof. The reproduction identity \int\psi_\delta\,d\sigma_y=I makes \varphi=\chi(\psi_\delta-I) mean zero on each sphere. It is smooth and compactly supported in the base. Equation (5.9) gives

The first term has the uniform bound just established. For the second, (5.33) requires only one angular derivative and r\nabla\psi_\delta on the cutoff derivative region. The uniform bounds of Lemma 4.6 supply these: in the lower region the density is the normalized ball evaluation kernel, and in the other region the radii have a fixed positive lower bound. System coercivity bounds \mathcal N(\varphi), hence \|\varphi\|_{L^2}, uniformly. Adding the fixed matrix function \chi I proves (5.34).□

Proof. As \lambda\to0, the coordinate columns converge smoothly to y^l e_\alpha, 1\le l\le n, 1\le\alpha\le2, where e_\alpha is the standard basis of \mathbb C^2. The additional columns converge to the trace-free quadratic polynomials specified by their Hessians at the origin. At any point y, the latter give arbitrary trace-free Hessians, and the affine columns correct their gradients to any prescribed values. Thus the limiting jets span all (p,H) with \sum_iH_{ii}=0, uniformly for y\in Y.

The spaces in (6.3) form a smooth vector bundle of constant rank: contraction with the positive definite tensor a is onto \mathbb C^2. They can be trivialized by the gradient and the trace-free part of the Hessian, the remaining trace being determined by (6.3). In this trivialization the selected jets give a smooth matrix of constant full row rank at \lambda=0. Compactness of Y preserves this rank, with a positive lower bound on its nonzero singular values, for small \lambda. If this matrix is T, the right inverse T^*(TT^*)^{-1} has the asserted smoothness and uniform bounds. Each selected column satisfies (6.3), so its range is precisely the stated space.□

Proof. Writing \rho(y)=\|\psi(y,\cdot)\|_{L^2(S_y)}, uniform normal coordinates and Cauchy–Schwarz give |M(y)|\le Cr(y)\rho(y). Taylor expansion in the fixed chart, followed by (6.5), gives

This calculation uses only the uniform second derivatives of the selected columns and |x-y|\le Cr; the coordinate displacement is a scalar, so no matrix factors have been interchanged.

Apply the same calculation to the coordinate matrices. Subtracting (6.6) shows that the uniformly invertible map defining D sends D-M to an error bounded by Cr^2\rho. Therefore

Multiplication by \chi and integration prove (6.8).□

Proof. By Lemma 4.1, r^{1/p_*} has a uniform Lipschitz constant. Set s=r(y)^\eta. Since \eta>1/p_*, for r(y) sufficiently small every z\in B(y,s) satisfies

Consequently c r(y)\le r(z)\le C r(y) on this ball, with constants depending only on the fixed p_*. Because \chi=1 on a neighborhood of the compact set K, the ball lies in that neighborhood after a further fixed reduction of \rho_0. The weighted bounds yield

For any smooth vector- or matrix-valued function U with a bounded C^m norm, Taylor expansion to degree m-1 and equivalence of norms on the finite-dimensional polynomial space give

Indeed the Taylor remainder has L^2 norm at most C_m s^{m+n/2}, and, after rescaling the ball to unit radius, each coefficient of its Taylor polynomial is bounded by that polynomial’s L^2 norm. This proves the displayed inequality componentwise.

For |\alpha|\le10, the first term of (6.11) applied to D is at most Cr(y)^{1-\eta d_*}=Cr(y)^{1/2}; for R_\nu it is at most Cr(y)^{2-\eta d_*}=Cr(y)^{3/2}. The second term is at most C_m r(y)^{\eta(m-10)}\le C_m r(y)^{3/2}, taking \rho_0\le1. This proves (6.10).□

Proof. Denote the right-hand side of (6.12) for the column v_{1,\nu} by Z_\nu. Its derivatives through order eight are bounded by Cr^{3/2} on the stated set. For \mathcal L_2R_\nu this uses derivatives of R_\nu through order ten in (6.10). Every derivative of either remaining product contains a derivative of h of order at most eight and a derivative of D of order at most nine or eight, respectively. Thus (6.4) and (6.10) give the factor r\,r^{1/2}. The other factors are fixed smooth derivatives of the selected columns.

At each point, Lemma 6.1 permits linear combinations of the identities (6.12) for which the combined gradient is zero and the combined Hessian is any prescribed a-trace-free symmetric covariant tensor with values in \mathbb C^2. Such Hessians satisfy (6.3) with zero gradient. They annihilate the \mathcal Y term and test precisely the \gamma-trace-free part of the coefficient of the Hessian.

More explicitly, choose a smooth uniformly bounded basis H_s of scalar symmetric Hessians with a^{ij}(H_s)_{ij}=0, and use the right inverse in Lemma 6.1 to realize H_se_\alpha, for \alpha=1,2, with zero gradient. The resulting combinations of Z_\nu give the two columns of the contraction of the projected coefficient with H_s. Inverting this spatial duality defines \mathcal E and proves (6.15). The coefficients realizing these prescribed jets are smooth functions of y and give zero combined gradient identically. We form these combinations before differentiating, so the \mathcal Y term is identically zero. The resulting formula expresses \mathcal E using only the Z_\nu and coefficients with uniformly bounded derivatives. Differentiating that formula through order eight proves (6.16), without introducing derivatives of \mathcal Y.□

Proof. Differentiate the equation twice. At a point, its highest-order part is a linear map from the tensor of third derivatives of W to the twice-differentiated trace-free symmetric gradient. We first show that this map is injective. A constant linear change of coordinates sets a^{ij}=\delta^{ij} at the point, and each real component of V can be considered separately.

A tensor in the kernel defines a homogeneous cubic vector field Z whose trace-free symmetric gradient vanishes: that gradient is homogeneous quadratic and all of its second derivatives are zero. Hence

Differentiating and combining the three permutations of the indices gives

Taking divergence in k yields

Its trace gives (2n-2)\Delta\sigma=0; since n\ge3, the Hessian of \sigma is zero. The preceding formula therefore gives \partial_i\partial_j\partial_lZ_k=0, so the homogeneous cubic Z vanishes. This proves injectivity.

Let T_a denote this injective map on third-derivative tensors, using fixed Euclidean inner products on the finite-dimensional tensor spaces. It has the smooth left inverse (T_a^*T_a)^{-1}T_a^*. Uniform ellipticity and compactness of bounded coefficient ranges give uniform bounds for this inverse. All other terms in the twice-differentiated equation contain at most two derivatives of W or of E. Applying the left inverse proves the claim, including the smooth coefficient bounds after further fixed numbers of differentiations.□

Proof. Apply Lemma 6.5 to (6.18) with V=\mathcal V_0. For each multi-index \alpha of order three, it gives an identity of the form

with uniformly smooth coefficient maps.

Fix y\in K with r(y)<\rho_0. By the core geometry of Section 4.2, the vertical segment

stays in K, and r(z(s))\le r(y). At its initial point, D and all its derivatives vanish: the paired coordinate matrices agree on an open neighborhood in the known lower region. The same holds for W.

Let \mathcal W(s) consist of all coordinate derivatives of W through order two at z(s). Differentiating this vector with respect to s either gives another one of its components or a third derivative, to which (6.21) applies. Thus

The bound for F follows from (6.16), since the entire segment has radius below \rho_0. Its length is uniformly bounded, so the elementary integral form of Gronwall’s inequality gives

This argument holds at every point of the stated set. Differentiating (6.21) k-3 times, for 3\le k\le9, expresses the derivatives of W of order k in terms of its derivatives through order k-1 and derivatives of \mathcal E_0 through order k-1. Induction and (6.16) therefore give the same bound through order nine. Notice that the last step uses precisely eight derivatives of \mathcal E_0.

Equation (6.17) now gives the same estimates for d and hence for D. Finally, differentiate (6.15) through order eight. Its right-hand side uses derivatives of D through order nine and of \mathcal E through order eight, all bounded by Cr^{3/2}. This proves (6.20).□

Proof. First fix the reference geometry and uniform smooth bounds for all sufficiently small dilations. Choose p_* large enough for the sphere geometry, the angular estimate, and Lemma 6.3. Next choose \varepsilon sufficiently small and R_* sufficiently small to satisfy those results, including r\beta^2\le\varepsilon. As explained in Remark 4.2, every fixed-separation range needed for the continuation and cutoff bounds is then fixed before \lambda. Proposition 5.5 and all estimates of this section have constants uniform for sufficiently small \lambda.

Choose 0<\rho_*<\rho_0, with \rho_*\le1, so small that C\rho_*^{1/2}\le\varepsilon/2 in (6.20). Thus, wherever that estimate applies and r_\delta<\rho_*, it improves (6.4) to H_8\le\varepsilon r_\delta/2.

As \lambda\to0, h\to0 in C^8(Y). The numbers

are positive and fixed independently of \lambda. We can therefore choose one sufficiently small \lambda such that

We also make the known lower region contain \{y_n<-1/4\} on the fixed ranges, as in the sphere construction. Fix this \lambda for the rest of the proof.

Let

It contains 1 by (6.22). For any \delta\in\mathcal I, the preceding estimates improve the inequality to H_8\le\varepsilon r_\delta/2 throughout Y. Indeed, Proposition 6.6 handles the points in K with r_\delta<\rho_*. Equation (6.22) handles all points with r_\delta\ge\rho_*. At a point outside K, either y_n<-1/2, where equality of the coefficients is already known, or t>2t_b. In the latter case \ell_\delta(t)\ge t, so r_\delta\ge R_*(2t_b)^{p_*} and (6.22) again applies.

The set \mathcal I is relatively closed in (0,1] by continuity and compactness of Y. It is also relatively open: at any positive \delta, the radius has a positive minimum on Y, so the strict factor 1/2 just proved persists for nearby parameter values in the weaker defining inequality. Since (0,1] is connected, \mathcal I=(0,1].

On the fixed open set K^\circ\cap\{t<0\}, which contains the origin, r_\delta\to0 as \delta\to0. The coefficients h,D,R_\nu are independent of \delta. The provisional bound and (6.10) therefore imply

there. In particular all selected pairs agree. Subtracting their equations now gives

The gradient projection of the space (6.3) is all of (\mathbb C^2)^n: for any gradient one can choose the trace of the Hessian to satisfy that constraint. Lemma 6.1 therefore implies B_1=B_2. Together with Q=a, this proves equality of the normalized harmonic-frame operators. The definition of Q shows that the two inverse metrics are positive scalar multiples, so the metrics are smoothly conformal. Undoing the fixed dilation proves the stated conclusion in the original contact coordinates.□

Proof. Let

These are the connection matrices in the harmonic frames; they need not be skew-Hermitian in the standard matrix inner product. The negative of the actual connection Laplacian, expressed in such a frame, is

Its zeroth-order term is zero, because the columns of F_j are harmonic. The common-principal-part hypothesis therefore says

Indeed their second-order coefficients agree after the indicated conformal normalization, their first-order coefficients agree by assumption, and both zeroth-order coefficients vanish.

Put \phi=(n-2)\omega/2. The conformal change of the Christoffel symbols gives

Comparison of first-order terms with (7.1) yields C_2=C_1-d\phi\,I. Substitution, with matrix multiplication in its displayed order, gives

Here \Delta_{g_1}=\operatorname{div}_{g_1}\nabla_{g_1}. The zeroth-order terms in (7.1) consequently imply

On the known open side the metrics already agree, so e^\phi=1. Unique continuation on \Omega gives e^\phi=1 everywhere. Since n\ge3 and \omega is real, \omega=0, and hence C_1=C_2.

The last equality is equivalent to d_{A_1}(F_1F_2^{-1}u)=F_1F_2^{-1}d_{A_2}u. Although the harmonic frames are not unitary, J_{\mathrm{loc}}=F_1F_2^{-1} is. To see this, parallel transport the intertwining identity along any path starting in the known side. Both connection transports are unitary, and J_{\mathrm{loc}} is the old unitary identification at the initial point. It remains unitary along the path; connectedness of \Omega proves the assertion everywhere.

For an arbitrary source f\in C_c^\infty(U_0;\mathbb C^2), both w_{1,f} and J_{\mathrm{loc}}w_{2,f}, in the common coordinate domain, are solutions of the first homogeneous connection equation. The charts avoid \overline{U_0}. Their difference vanishes on the known side, so unique continuation makes it zero on \Omega. Thus all evaluation identities hold, not only those for the finite family used in the local argument. On an overlap with the old match, the two proposed first-side images of a point, together with their invertible fiber maps, would impose an invertible linear relation between the two first-side evaluation maps. Joint surjectivity in Lemma 2.3 forces the image points to coincide. Surjectivity at the second-side point then forces the fiber maps to coincide as well. Thus the local identification agrees with the old match on every overlap.□

Proof. If its domain had an interior frontier, Proposition 2.5 and the dilation in Section 2.5 would supply the local data used in Sections 4–6. Proposition 6.7 gives conformally equal metrics and equal normalized harmonic-frame equations on a neighborhood of the contact point in the original charts. Apply Lemma 7.1. Coordinate identity and J_{\mathrm{loc}} extend the old match to a neighborhood containing the frontier point. The concluding assertion of Lemma 7.1 ensures agreement on every overlap. This contradicts maximality. The domain, already nonempty and open, has no relative frontier in the connected interior, and is therefore all of N_2^\circ.

The image is open because \Phi is a local isometry. To see that it is relatively closed, suppose \Phi(p_k)\to q\in N_1^\circ. Compactness gives a subsequence with p_k\to p\in N_2 and J(p_k)\to J_0\in U(2). If p\in\partial N_2, continuity of the evaluations and (2.6) would give I_1(q)=J_0I_2(p)=0, contrary to Lemma 2.3. Thus p is interior, and continuity gives \Phi(p)=q. Connectedness of N_1^\circ makes the image all of that interior. Injectivity was proved in Lemma 2.4; the inverse is smooth because the map is a bijective local diffeomorphism.□

Proof. An onto interior isometry preserves lengths of curves in both directions and therefore preserves intrinsic distance. The metric completion of the interior of a compact smooth Riemannian manifold with boundary is the manifold with its length metric. In particular, a curve touching the boundary can be pushed slightly into an inward collar with arbitrarily small change of length; uniform local metric comparisons identify the resulting completion with the original compact topology. Thus \Phi and its inverse extend to mutually inverse homeomorphisms of N_2,N_1, mapping boundary to boundary.

The extensions are smooth. In a sufficiently thin uniform collar, the distance s_j to \partial N_j is smooth and its gradient generates inward unit normal flow. The isometry preserves distance to boundary, so s_1\circ\Phi=s_2, and in the interior it intertwines these gradient fields. On a fixed small level s_2=s_0>0, the map is already smooth. For 0\le s\le s_0, express it using that smooth level map and the smooth normal flows on the two sides. This gives a smooth extension to s=0; the inverse has the same construction. Parallel transport for A_1,A_2 along these normal curves similarly extends J smoothly, retaining unitarity and the connection intertwining identity.

The maps are identity on the open cap, hence on its closure by continuity. With closure taken in N_j, the original interiors satisfy

Bijectivity and identity on \overline E preserve these complements, and hence their closures, the original copies of M. The restricted diffeomorphism fixes the attachment patch P=\{b>0\}, and J=I there.

We must recover the boundary identity on every part of \Gamma, which may have more than one component. For f\in C_c^\infty(\Gamma;\mathbb C^2), let u_j now denote the harmonic solutions on the original manifolds with boundary value f. On the second manifold define \widetilde u_1=J^{-1}(u_1\circ\Phi). The metric and connection identities make \widetilde u_1 second-harmonic. On P, it has the same boundary value as u_2. It also has the same covariant normal derivative there. Indeed equality of the measured forms, tested against arbitrary functions supported in P, gives the equality of smooth densities

The densities agree by Lemma 2.1, the isometry transports the outward normal, and the bundle map equals I on P. Thus \widetilde u_1-u_2 has zero Cauchy data there. Boundary and interior unique continuation give \widetilde u_1=u_2 throughout M.

Taking boundary traces yields

If p\in\Gamma and \Phi(p)\ne p, choose f nonzero at p, supported in a small neighborhood of p avoiding \Phi(p). Equation (7.2) contradicts invertibility of J(p). Hence \Phi(p)=p on \Gamma. Arbitrary vector values of f(p) then give J(p)=I there.□

Proof of Theorem 1.1. The boundary-identity gauges V_j of Section 2 put the connections in normal gauge. Lemma 2.1 and Proposition 2.2 produce common exterior Green data. Lemmas 2.3 and 2.4 define the maximal match, and Proposition 2.5 supplies the local contact data wherever an interior frontier is present. Proposition 6.7, followed by Lemma 7.1, removes that frontier. Propositions 7.2 and 7.3 give a smooth global metric and connection identification fixing the entire measured patch.

To distinguish the original connections in this final step, write A_j^{\mathrm{orig}} for them, so the connections used in the proof were

In the original trivializations the bundle map from second to first fiber is

It is smooth and unitary, and equals I on \Gamma, since both original gauges equal I on the entire boundary. The intertwining identity is d_{\Phi^*A_1^{\mathrm{orig}}}(Uu)=U d_{A_2^{\mathrm{orig}}}u. Expanding it gives

These are the asserted identities. The argument uses densities and a single nonempty cap; it requires neither orientability nor connectedness of the boundary or of \Gamma.□