Gluing and Localized Deformation

Gluing theorems exploit the underdetermined ellipticity of the constraints: one can join ends, insert a neck, or alter data inside a transition region while solving the constraints exactly. The operation is nevertheless conditional. Local Killing initial data create a finite-dimensional cokernel, and asymptotic charges must be balanced before a compactly supported correction can exist.

Required background. Analysis for Mathematical Relativity supplies elliptic estimates and implicit-function arguments, The Einstein Equation supplies the constraints, Hypersurface Normals and Tangential Projectors supplies $(h,K)$, and Differential Equations for Relativity supplies matching and boundary conditions.

Helpful background. Causality and Global Structure explains the developments being joined, while PDEs and Green Functions for Gravitation reviews elliptic parametrices.

Connected-sum gluing

Begin with vacuum $\Lambda=0$ constant-mean-curvature data $(\Sigma_a,h_a,K_a)$ satisfying the constraints, with the same constant mean curvature on both sides. Here $K_{ij}=-h_i{}^\mu h_j{}^\nu\nabla_\mu n_\nu$. Choose points away from boundaries and asymptotic singularities, remove small balls, and identify their spherical boundaries through a long neck. Conformal transplantation gives approximate data on the connected sum, but the neck introduces a small constraint error.

A representative connected-sum theorem assumes suitable regularity, CMC data, and a nondegeneracy condition excluding the relevant conformal or adjoint kernels. For sufficiently small neck parameter $\varepsilon$, one solves the coupled elliptic correction and obtains exact vacuum data on

$$\Sigma_1\#\Sigma_2.$$

Away from the neck the corrected data converge to the original data as $\varepsilon\to0$. They need not be literally equal there unless a stronger localized theorem is invoked. This distinction prevents “arbitrarily close” from being misreported as “unchanged.” The original conformal construction is Isenberg, Mazzeo, and Pollack 2002, Theorem 1 and §§7–8. In the compact boundaryless case its nondegeneracy conditions include nontrivial extrinsic curvature and restrictions on conformal Killing fields; the exact conditions depend on the version used.

The conformal splicing, long-neck correction, and convergence away from the neck are also explained in Bartnik and Isenberg 2004, §4.5. That discussion is useful for seeing where the elliptic solve enters: identifying two manifolds topologically does not identify their constraint solutions.

Compactly supported correction

Suppose trial data agree with an exact solution outside an annular transition region $\Omega$ but have a small residual $f=\Phi(h,K)$ inside it. Linearizing gives

$$D\Phi_{(h,K)}(\gamma,p)=-f+\text{nonlinear remainder}.$$

Because $D\Phi$ is underdetermined, one often seeks a correction through its adjoint, schematically

$$(\gamma,p)=\rho\,D\Phi^*u,$$

where the weight $\rho$ vanishes rapidly at $\partial\Omega$. The composed operator $D\Phi\,\rho D\Phi^*$ is elliptic because $D\Phi$ has surjective symbol and its adjoint has injective symbol, with mixed differential orders treated appropriately. Weighted estimates make $(\gamma,p)$ vanish at the boundary, so the corrected data can equal the prescribed interior and exterior data exactly.

Solvability requires the residual to be orthogonal to

$$\ker D\Phi^*|_\Omega,$$

the local KIDs. If this kernel is zero, a nonlinear inverse-function argument can close for sufficiently small error. If it is nonzero, the correction equation has finite-dimensional balancing conditions. One then varies available parameters—often mass, linear momentum, angular momentum, or center of mass—so that the residual’s KID projections vanish. Carlotto explains this local deformation architecture and its limitations in Carlotto 2021, §3.

A theorem that preserves the exterior exactly

Here is a precise localized result in three spatial dimensions. Let $(\widetilde\Sigma,\widetilde h,\widetilde K)$ be smooth vacuum data with $\Lambda=0$, and choose two disjoint open regions $\Omega_1,\Omega_2$ with smooth boundary and compact closure. Assume there are no nonzero local KIDs on either region. Choose $p_a\in\Omega_a$. For any prescribed finite differentiability order $k$ and sufficiently small $\varepsilon\gt0$, one can remove balls about the $p_a$, join them with a neck, and find smooth exact vacuum data on the resulting manifold such that:

• outside $\Omega_1\cup\Omega_2$, the data are exactly equal to the original data;
• away from the shrinking balls, they are $\varepsilon$-close in $C^k\times C^k$;
• the mean curvature need not have been constant in the original data.

This is Chruściel, Isenberg, and Pollack 2005, Theorem 1.1; proof in §4, specialized to $\Lambda=0$ and dimension three. It imposes no global compactness, completeness, or asymptotic-flatness condition. If the points are on different components it makes a connected sum; on one component it adds a handle. The no-KID hypothesis is local to the chosen regions, not a demand that the entire development have no global symmetries.

The proof first makes a small local deformation that removes KIDs on the needed collars, then chooses CMC hypersurface pieces inside local spacetime developments. It performs the conformal neck construction on those pieces and uses localized deformation to restore agreement on their outer collars. Finally it chooses the desired spacelike slice in the resulting local spacetime. These steps, detailed in §4 of the paper, obtain exact agreement with the original data outside the chosen regions. That conclusion cannot be inferred merely from convergence of the conformal factor to one.

Proof architecture and scale separation

The nonlinear proof separates three estimates. First, a parametrix or coercive estimate controls the adjoint-composed operator on the complement of the KID space. Second, the finite-dimensional projection is solved by adjusting balancing parameters. Third, the quadratic remainder

$$\Phi(h+\gamma,K+p)-\Phi(h,K)-D\Phi(\gamma,p)$$

is bounded in a weaker weighted norm and absorbed by contraction. The transition region must be large or the neck sufficiently long so that the initial residual is smaller than the inverse estimate permits.

This scale dependence is part of the hypotheses. Shrinking a transition annulus can enlarge derivatives of cutoff tensors even when the two data sets are pointwise close. Conversely, moving an asymptotic annulus outward makes the data closer to their model end and supplies a small parameter. A valid gluing theorem specifies which geometric scale produces smallness and whether its constants remain uniform as that scale changes.

The correction also has to preserve positive definiteness of $h$, the declared asymptotics, and any boundary condition. Solving $\Phi=0$ algebraically in a norm is not enough if the corrected tensor leaves the manifold of Riemannian metrics or changes the retained end.

Exact model exteriors

An important asymptotically flat application starts with vacuum data and modifies them in a distant annulus so that the exterior is exactly a member of a stationary model family, such as Schwarzschild in a time-symmetric setting or Kerr-type data in a general setting. The model parameters are not decorative: they span the obstruction directions needed to match the conserved charges.

The resulting statement is not that an arbitrary gravitational field can be hidden with no cost. The transition carries a genuine deformation, the hypotheses include decay and smallness in the far region, and the chosen exterior parameters must be close to the original asymptotic charges. Positive mass and other global restrictions remain in force.

There are therefore three levels of claim:

1. approximate splicing: interpolate tensors but leave a small constraint error;
2. gluing with small global change: solve the constraints and converge to the originals away from a neck;
3. localized gluing: solve with a correction supported in a controlled region, subject to KID balancing.

Only the third justifies “exactly unchanged outside.” The local theorem above allows non-CMC data under its no-KID hypothesis, but it does not prescribe an arbitrary final mean-curvature profile. Matter fields, apparent-horizon boundary conditions, low regularity, or additional control of the final geometry require separate theorems. Ellipticity makes gluing powerful, not assumption-free.

Check your understanding

Why does a flat annulus obstruct an arbitrary compactly supported constraint correction?

Solution

Flat time-symmetric data admit nontrivial KIDs inherited from Minkowski translations, rotations, and boosts. These lie in $\ker D\Phi^*$ on the annulus. Integration by parts shows that the desired source must be orthogonal to every such KID. If it is not, no compactly supported linear correction exists; model parameters must be varied to satisfy the finite-dimensional balance equations.

References

• Bartnik, Robert, and Jim Isenberg. “The Constraint Equations.” In The Einstein Equations and the Large Scale Behavior of Gravitational Fields, 1–38. Birkhäuser, 2004. arXiv:gr-qc/0405092.
• Carlotto, Alessandro. “The General Relativistic Constraint Equations.” Living Reviews in Relativity 24 (2021): 2. DOI.
• Chruściel, Piotr T., James Isenberg, and Daniel Pollack. “Initial Data Engineering.” Communications in Mathematical Physics 257 (2005): 29–42. arXiv:gr-qc/0403066.
• Isenberg, James, Rafe Mazzeo, and Daniel Pollack. “Gluing and Wormholes for the Einstein Constraint Equations.” Communications in Mathematical Physics 231 (2002): 529–568. DOI. Open PDF.