Characteristic Gluing with $Λ$: III. High-differentiability nonlinear gluing

Technical Summary: Characteristic Gluing with $\Lambda$: III. High-differentiability nonlinear gluing

Problem Statement
This paper addresses the construction of vacuum gravitational fields by "gluing" two distinct characteristic initial data sets along a common null hypersurface. Specifically, the authors investigate the gluing of data sets defined on overlapping subsets $N_1$ and $N_2$ of a smooth hypersurface $N$ (typically $u=0$) within $(n+1)$-dimensional spacetimes ($n \ge 3$). The background spacetimes are assumed to be near Birmingham-Kottler metrics (which include Minkowski, de Sitter, anti-de Sitter, and Myers-Perry backgrounds) with an arbitrary cosmological constant $\Lambda \in \mathbb{R}$.

The primary motivation stems from the limitations of previous work by Aretakis, Czimek, and Rodnianski [1, 2], who established a $C^2$-gluing construction near light cones in four-dimensional Minkowski spacetime. While that construction ensured continuity of initial data and the first two transverse derivatives, the resulting spacetimes suffered from poor differentiability properties when evolved, limiting their utility for further analysis. This paper seeks to generalize the gluing procedure to allow for an arbitrary finite number of transverse derivatives ($C^k$-gluing) and to extend the result to arbitrary spacetime dimensions and cosmological constants.

Methodology
The authors employ a nonlinear analysis framework based on the Bondi gauge formalism. The methodology proceeds through several key stages:

1. Function Spaces and Regularity: The analysis is conducted in carefully tailored function spaces, specifically Hölder spaces $C^{k_\gamma, \lambda}(S)$ and Sobolev spaces $W^{k_\gamma, p}(S)$ on the cross-sections $S$ of the null hypersurface. The regularity index $k_\gamma$ is chosen to ensure sufficient differentiability to solve the elliptic equations arising from the constraint system while accommodating the loss of derivatives inherent in the characteristic Cauchy problem.

2. Interpolating Fields: The core of the construction involves defining an interpolating metric field $g_{AB}$ on the gluing region $N[r_1, \hat{r}]$. This field is constructed as a weighted sum of the background metric, the first data set ($g_1$), a deformed extension of the second data set ($E(\Psi^* g_2)$), and a set of "free fields" $\phi_{AB}$ multiplied by radial cutoff functions $\kappa_i$. The free fields are used to compensate for obstructions to gluing.

3. Coordinate Transformations and Deformations: To handle the gauge freedom and the nonlinearity of the Einstein equations, the authors introduce a sequence of three coordinate transformations:

   • A deformation of the section $S_2$ (moving the null hypersurface).
   • A reparameterization of the angular coordinates (gauge transformation on the sphere).
   • A redefinition of the radial coordinate to maintain the Bondi determinant condition.
     These transformations are parameterized by "deformation-and-gauge fields" ($\psi_i, X_A$) which are controlled via an implicit function theorem.

4. Radial Charges and Obstructions: The authors identify specific "radial charges" ($Q$) that act as obstructions to the gluing problem. These charges, denoted as $^{[1]}Q$ and $^{[2]}Q$, are derived from the transport equations of the Einstein constraints. In the linearized regime, these charges are conserved along the radial direction. The nonlinear analysis shows that these charges are invariant under gauge transformations up to second-order terms ($O(\epsilon^2)$).

5. Implicit Function Theorem: The existence of the gluing solution is reduced to solving a system of equations using the implicit function theorem. The authors demonstrate that the linearized gluing problem is surjective onto the space of data modulo the finite-dimensional space of radial charges. By introducing a "compensating family" of metrics (e.g., Kerr-(A)dS or Birmingham-Kottler metrics with varying mass parameters), they show that the radial charges can be adjusted to satisfy the necessary matching conditions.

Key Contributions and Results

• High-Differentiability Gluing: The paper proves a $C^k_u C^\infty_{(r, x^A)}$ gluing theorem for vacuum Einstein equations. This allows the construction of spacetimes with arbitrarily high finite differentiability classes, resolving the differentiability issues present in the earlier $C^2$ constructions.
• Generalization to $\Lambda$ and Dimension: The results hold for any spacetime dimension $n \ge 3$ and any cosmological constant $\Lambda \in \mathbb{R}$.
• Compensating Families: The authors establish that for mass parameters $m \neq 0$, the family of Kerr-(A)dS metrics (or Birmingham-Kottler metrics for negative curvature sections) provides sufficient degrees of freedom to compensate for the radial obstructions.
• Main Theorem (Theorem 1.2 / 8.1): The conjecture that a smooth, spacelike, vacuum codimension-two data set sufficiently close to a member of a compensating family can be glued to a deformation of another such data set is proven true near Birmingham-Kottler metrics with non-zero mass.
• Handling of Obstructions: The paper explicitly characterizes the dimension of the obstruction space (Table 7.1) and demonstrates how the mass parameter and Killing vectors of the background geometry determine the number of required compensating charges.

Significance
The paper claims that its primary contribution is the proof that characteristic gluing in asymptotically Minkowskian (and more generally, Birmingham-Kottler) spacetimes can be performed with an arbitrary number of transverse derivatives. This resolves the issue of poor differentiability in spacetimes evolved from characteristic data constructed in previous works [1, 2], thereby enhancing the utility of such spacetimes for further theoretical constructions.

The authors note that while the generalization to higher dimensions and arbitrary cosmological constants is of independent interest, the restriction to non-zero mass parameters ($m \neq 0$) is currently necessary. This is due to the lack of known families of metrics with sufficient parameters to compensate for obstructing radial charges in the cases of Ricci-flat sections or Einstein sections with positive Ricci tensors distinct from the round sphere (Remark 1.3). The existence of such families would extend the validity of their results without further modification.

The work relies heavily on the linearized analysis of the gluing problem presented in [3, 4], extending those results to the fully nonlinear regime through a sophisticated application of the implicit function theorem and careful control of function space regularity.